A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion

A systematic metho d for cons tructing time discretizations of in tegrable lattice systems: lo c al equations of motion T ak a yuki Tsuchid a ∗ Okayama Institute f or Quantum Physics, Kyoyama 1-9-1, Okayama 700-0015, Jap an Octob er 26 , 20 18 Abstract W e prop ose a new metho d for discretizing the time v ariable in in tegrable lattice systems while main taining the lo c ality of the equa- tions of motion. The method is based on the zero-c ur v ature (Lax p air) represent ation and the lo we st-order “conserv ation la ws”. In con trast to the pioneering w ork of Ablo witz and Ladik, o ur metho d allo ws the auxiliary dep enden t v ariables app earing in the stage of time dis- cretizatio n to b e expressed lo c al ly in terms of th e original dep endent v ariables. The time-discretized lattice systems ha ve the same s et of conserv ed quantiti es and the same structures of the solutions as the con tin uous-time lattic e systems; only the time evo lution of the pa- rameters in the solutions that corresp ond to the angle v ariables is dis- cretized. The eﬀectiv eness of our method is illustrated u sing examples suc h as the T o da lattice , the V olterra lattic e, the mo diﬁed V olterra lattice , the Ab lo witz–Ladik lattice (an integ rable semi-discrete non- linear Schr¨ odinger system), and the lattice Heisen b erg ferromagnet mo del. F or the V olterra lattice and mod iﬁed V olterra lattice, we also present th eir ultr adiscr ete analogues. Keywor d s: in tegrable lattices, time discretization, Lax pa ir , T o da lattice, V olterra lattice, Ablowitz–Ladik latt ice, lo cal fully discrete NLS, ultradis- cretization, ultradiscrete (mo diﬁed) KdV hierarc h y P A CS numb ers: 02.30 .Ik, 02.70.Bf, 05.45.Yv, 45.05.+x ∗ E-mail: surna me at ms.u-tokyo.ac.jp 1 Con ten ts 1 In tro duction 3 2 Metho d for time discretization 5 2.1 Lax pair and a conserv ation la w in the con tin uous-time case . 6 2.2 Lax pair in the discrete-time case and auxiliary v ariables . . . 7 2.3 F undamen tal “conserv ation laws” in t he discrete-time case . . 9 2.4 Algebraic system for the auxiliary v ariables and lo cal equa- tions of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Remarks on nonautonomous extensions . . . . . . . . . . . . . 11 3 Examples 14 3.1 The T o da lat t ice in F lasc hk a–Manako v co ordinates . . . . . . 14 3.2 The Ablo witz–Ladik la ttice . . . . . . . . . . . . . . . . . . . 2 0 3.3 The V olterra lattice . . . . . . . . . . . . . . . . . . . . . . . . 36 3.4 The mo diﬁed V o lterra lattice . . . . . . . . . . . . . . . . . . . 45 3.5 The lattice Heisen b erg f erromagnet mo del . . . . . . . . . . . 51 4 Concluding remarks 55 References 58 2 1 In tro duction The quest for a ﬁnite-diﬀerence analog ue of a giv en diﬀeren tial equation can b e justiﬁed for sev eral sound reasons. A suitable discretization can repro duce most of t he imp ortan t prop erties of the diﬀeren tial equation in the small- v alue range of the diﬀerence in terv al a nd can b e considered a “ g eneralization” of t he original contin uous equation. Suc h a discretization facilitates a better and more in tuitiv e understanding of the diﬀerential equation without using a limiting procedure, whic h is needed to deﬁne diﬀeren tiation, and is ideal for p erforming nume rical experimen ts. The suitable discretization of a completely in tegrable system is usually required to retain the integrabilit y; if this is satisﬁed, it is called an in tegrable discretization. An integrable sys tem o ften admits more than one integrable discretization; in suc h a case , w e can consider the prop erties of eac h integrable discretization other t ha n inte gra bility and discuss whic h one is the most fa v orable for our purp ose. The problem o f integrable discretization has b een sp oradically studied since the mid-1970s, i.e., the da wn o f the mo dern theory of in tegrable systems. F or more than thirty ye ars, v ario us tec hniques ha v e b een deve lop ed t o obtain in tegrable discretizations of con tin uous systems. Readers in terested in the history of in tegrable discretizations are referred to the preface of Suris’s b o ok [1 ]. P artial diﬀeren tial equations (PDEs) inv olve more than one independen t v ariable. The discretization of an in tegrable no nlinear PDE is generally p er- formed in tw o steps; in t he ﬁrst step, we discretize the spatial v ariable(s) and in the second step, the time v ariable is discretized. Of course, for some PDEs suc h as the sine-Gordon equation u xt = sin u , the roles of t he space and time v ariables can b e sw app ed a nd it is not meaningful t o discuss the order of discretization. Ho w ev er, for a n in tegrable nonlinear PDE wherein the roles of individual v ariables are essen tially diﬀerent and not in terc hangeable, the order in whic h the independen t v ariables ma y b e discretized app ears to b e unique. Th us, we can consider that the time v ariable is alw a ys discretized last, after the spatial v ar iable(s) hav e been discretized. In this pap er w e fo cus on the problem of the in tegrable full discretiza- tion of diﬀeren tial- diﬀerence equations in 1 + 1 dimensions. The con tin uous indep enden t v aria ble to b e discretized is regar ded as time, as noted abov e. Most of the diﬀeren tial-diﬀerence equations conside red reduce to in tegrable PDEs in a prop er contin uous limit; ho w ev er, t his is not necessary and we can also start with inte gra ble lattice systems that hav e no con tinuous coun- terpart. Th e problem of time discretization is, b y it s nature, distinct from the problem of space discretization. In fact, the former problem has its o wn p eculiarities and diﬃculties that the latter do es not ha v e. This p oin t was 3 unco v ered by Ablo witz and Ladik [2 , 3] in their at t empt to fully discretize the space-discretize d nonlinear Schr¨ odinger (semi-discrete NLS) equation [4]. It turned out that unexp ected nonlocality emerges in the stage of time dis- cretization; the fully discrete NLS equation in v olv es inﬁnite sums and/or inﬁnite pro ducts (in the case of an inﬁnite c hain) with resp ect to t he discrete spatial v a r ia ble and is thus a global-in-space sc heme. The fully discrete NLS equation can sup erﬁcially be written in a lo cal form using additional dep en- den t v ariables called auxiliary varia bles , but it do es not provide an y essen tial resolution of the nonlo cality problem. The subseq uen t pap er b y T aha and Ablo witz [5] reinforces the impression that the app earance of glo bal terms is a general feature of the problem of time discretization f or in tegrable lattice systems . The pioneering work of Ablo witz and co w ork ers [2 , 3, 5] is based on the zero-curv ature (Lax pair) represen tation; the guiding principle is tha t the time discretization do es not c hang e the spatial par t of the Lax pair for an in tegrable lattice system. This automatically guarantees t he ma jor adv an- tages of the full discretization; t hat is, t he time-discretized lattice systems ha v e the same in tegrals of motion and the same structures of the solutions as the original contin uous-time latt ice systems. In more mo dern terms, eac h of their time discretizations b elongs to the same integrable hierarc hy as the underlying con tin uous-time system [6, 7]. Despite the elegance of this result, the a pp earance of inﬁnite sums/pro ducts is a shortcoming not acceptable to ev eryb o dy , a nd new ideas are needed to remo v e the nonlo calit y . In this regard, Suris recen t ly in tro duced the notion of lo c alizing changes o f vari- ables , a pplied it to a larg e num b er of in tegrable latt ice systems, and obtained their time discretiz atio ns written in lo cal equations of mo t io n [1]. Note that some of his discretizations coincide with earlier results obtained using Hi- rota’s bilinear metho d [8], as describ ed in the bibliogra phical remarks in his b o ok [1]. Suris’s idea w as to ﬁnd, by g uessing , a c hange of v a riables suc h that the equations of motion as w ell as the auxiliary v ariables can b e expres sed lo cally in terms of the new dep enden t v ariables; the c hange o f v ariables in- v olv es the step size of time a s a parameter and is considered as a discrete Miura transformation giving a one-parameter deformation of the original lat- tice hierarc h y . Th us, the in tegrals of motion and the solution for mulas for the time-discretized lattice system are deformed accordingly . Moreo v er, al- though Suris’s approach has successfully provid ed man y in teresting examples, its applicabilit y is rather limited. In pa r ticular, it is not applicable to the time discretization of NLS-ty p e lattices, suc h as the semi-discrete NLS equa- tion (also called the Ablo witz–Ladik latt ice) [4 ], wherein the t w o dep endent v ariables can b e related b y a complex conjugacy reduction. The main ob jective of this pap er is to prop ose a systematic metho d for 4 constructing time discretizations of in tegrable lattice systems written as lo c al equations of motion. In con trast to the other kno wn methods, our method generally requires no ad ho c treatmen t on a case-b y-case basis and app ears to ha v e no serious limitatio ns in its applicabilit y; it can b e applied to p o ssibly all lattice systems in 1 + 1 dimensions p ossessing a Lax pair represen t ation. In pa rticular, it can b e used to obtain lo cal full discretizations of NLS-t yp e lattices, including the Ablo witz–Ladik lattice [4]. Actually , our metho d can b e considered a s a c omplete d version of the Ablo witz–Ladik approac h [2, 3]; it b oth reﬁnes and extends their w ork in an essen tial w ay . A decisiv e breakthrough has been made b y considering the lo w est-order “conserv a t io n la ws”, deriv ed from t he zero- curv ature condition written in matrix form. In the pro cess, a critical role is play ed b y a n arbitrar y parameter in the Lax pair, called the sp ectral par a meter. The requiremen t that all the ﬂuxes corresp onding to the same conserv ed densit y ha v e to es sen tially coincide results in an “ultralo cal” algebraic system for the auxiliary v a riables; the simpler case where the conserv ed densit y is trivially a constan t can b e treated in a similar manner. Th us, b y solving this algebraic system, w e can restore the lo calit y of the equations; that is, the global terms app earing in the stage of time discretization can b e replaced b y lo c al expressions in terms of the original dep enden t v ariables. This pap er is organized as follo ws. In section 2, w e describe the general metho d f or discretizing the time v ariable. In section 3, w e construct time discretizations of the T o da lattice in Flasc hk a–Manak o v co ordinates [9 – 11], the Ablow itz–Ladik lattice, the V olterra lattice, the mo diﬁed V olterra lat- tice, and the lattice Heisen b erg ferromagnet mo del. As a spin-oﬀ, w e obtain ultradiscrete analogues [1 2, 13] o f the V olterra lattice and mo diﬁed V olterra lattice. In addition, w e unco v er unexp ected relationships with the w ork of Nijhoﬀ, Quisp el, Cap el et al. [14 – 17]. Section 4 is dev oted to concluding remarks. 2 Metho d for t ime discre tization In this section, w e discuss the problem of time discretization for a given in te- grable la ttice system. W e start with a Lax pair fo r m ulation in the contin uous- time case and then pro ceed to discretize the time v aria ble. A set of auxiliary v ariables is intro duced to express the time-discretized lattice as a closed sys- tem of equations. Using the fundamen t al “conserv a tion laws ” deriv ed from the Lax pair, w e can obtain lo cal expres sions for the auxiliary v ariables in terms of the original v a r iables. 5 2.1 Lax pair and a conserv ation la w in the con tin u ous- time case The Lax pair formulation in a semi-discrete space-time comprises a pair of linear equations, Ψ n +1 = L n ( λ )Ψ n , Ψ n,t = M n ( λ )Ψ n . (2.1) Here, Ψ n is a column-v ector function and n is the discrete spatial v ariable. The subs cript t denotes diﬀeren tiation with respect to the con tin uous time v ariable t . The square matrices L n and M n dep end on the sp ectral parameter λ , whic h is an arbitrary constan t indep endent of n and t . The compatibilit y condition of the ov erdetermined system (2.1) is giv en by [2, 4, 6, 7, 18, 19] L n,t + L n M n − M n +1 L n = O , (2.2) whic h is (a semi-discrete v ersion of ) the zero- curv ature condition. The sym- b ol O is used to stress that this is a matrix equation. If we sp ecify the λ -dep enden t matrices L n and M n appropriately , (2 .2 ) results in a close d diﬀeren tial-diﬀerence system for some λ - indep enden t quan tities in L n and M n . In suc h a case, the pair of matrices L n and M n is called a Lax pair. The matrix L n is usually ultralo cal in the dep enden t v ariables; that is, if L n in v olv es some v ariable, sa y u n , then it do es not in v olv e shifted v ar ia bles suc h a s u n ± 1 and u n ± 2 . In additio n, the determinan t of L n is required to b e nonzero f o r generic λ so that the sp ectral problem is w ell-p osed on the entire inﬁnite c hain. The zero-curv ature condition (2.2) generates a conserv ation la w of the follo wing for m: ∂ ∂ t log(det L n ) = ∆ + n (tr M n ) . (2.3) Here, ∆ + n is the forw ard diﬀerence op erator in the spatial direction, i.e., ∆ + n f n := f n +1 − f n . (2.4) F or a pr op e r integrable lattice system, det L n ( λ ) has to b e either a time- indep enden t function of λ or the expo nen tial of a λ - indep enden t conserv ed densit y multiplied by a n ov erall t -independent factor. Indeed, if this w as not satisﬁed, e.g. , det L n ( λ ) = 1 + λu n , then the expansion of log det L n ( λ ) with resp ect to λ would yield an inﬁnite n umber of (almost) ultralo cal conserv ed densities, say u n , u 2 n , . . . , and thus the lattice system w ould b e trivial in some sense. Note that the zero-curv ature condition (2.2) has the following in v ariance prop erties: 6 (a) L n → f ( λ ) L n , where f ( λ ) is a t -indep enden t scalar function, (b) M n → M n + g ( λ ) I , where g ( λ ) is an n -indep enden t scalar function and I is the identit y matrix, (c) L n → exp( α ρ n ) L n and M n → M n + αj n I , where α is a parameter, ρ n is a conserv ed densit y and j n is the corresp o nding ﬂux (up t o a sign), namely , ∂ t ρ n = ∆ + n j n . In particular, using (2.3) and the ab ov e prop erties, w e can con v ert the Lax pair to a normalized form, i.e., det L n = 1 and tr M n = 0. 2.2 Lax pair in the discrete-time case and auxiliary v ariables No w, w e discuss ho w to construct the time discretization of a give n lattice system ha ving the Lax pair L n and M n . The natural discrete-time analo gue of the linear system (2.1) is giv en b y Ψ n +1 = L n ( λ )Ψ n , e Ψ n = V n ( λ )Ψ n , (2.5) where the tilde denotes the forward shift ( m → m + 1) in the discrete time co ordinate m ∈ Z . Here and hereafter, the dep endenc e on m is usually sup- pressed unless it is shifted. The compatibility condition o f this ov erdeter- mined linear system is given by [2, 3, 5 – 7, 19, 20] e L n V n = V n +1 L n , (2.6) whic h is (a f ully discretized v ersion of ) the zero-curv ature condition. Note that (2.6) can b e rewritten as V n = e L − 1 n V n +1 L n or e L n V n L − 1 n = V n +1 . (2.7) F ollowing the w ork of Ablowitz and co w ork ers [2 , 3, 5], the matrix L n ( λ ) is assumed to b e the same as that in the semi-discrete case. Then, w e lo ok for a V n ( λ ) suc h that the zero-curv ature condition (2.6) results in a closed system of pa rtial diﬀerence equations pro viding a discrete -time analogue of the semi-discrete system. F or t his purp ose, w e assume tha t the matrix V n has asymptotic b eha vior, V n = I + h [ M n + O ( h )] , (2.8) where h is a suﬃcien tly small (but no nzero) parameter indep enden t of λ a nd is considered the diﬀerenc e in terv al of time (cf. (2.2) and (2.6)). Mor e pre- cisely , h approximates the “true” step size o f discrete time up to an o ( h ) error. 7 Moreo v er, we assu me that V n ( λ ) has esse ntially the same λ -dep endence as I + hM n ( λ ). Note, ho w ev er, that M n ( λ ) can only b e determined up to the addition of an n -indep enden t scalar matrix (cf. (b)). Th us, this arbitrariness has to b e taken in to accoun t in determining V n ( λ ); this corresp onds to the freedom (no nunique ness) of c ho osing the linear part of the time-discretized lattice system that determines the disp ersion relation [3]. Mor eov er, some n -indep enden t quan tities (usually set as constants ) in M n translate in to n - dep enden t quan tities in V n , whic h t ypically constitute, up to the reformula- tion of t he dep enden t v ariables, new auxiliary variables . Then, w e sp ecify appropriate b o undar y conditions fo r these new v ariables in V n , whic h should retriev e the corresponding n -independent v alues in M n . In f act, w e usually assume “constan t” b o undary conditions for V n , lim n →−∞ V n = lim n → + ∞ V n = ﬁnite. (2.9) Ho w ev er, the r ig h t-hand side is allo w ed to dep end on the time v ariable m ∈ Z . In the application of the inv erse scattering metho d based on the Lax pair, w e need to sp ecify the b oundary conditions for L n as n → ±∞ . In suc h a case, it is r e dundant to imp ose the b oundary conditions on V n at b oth spatial ends as giv en in (2.9), and it is non trivial that the redundant b oundary conditions are compatible. In fact, it is suﬃcien t to kno w only one of the t w o b oundary v alues, lim n →−∞ V n or lim n → + ∞ V n . In the existing literature [1 – 3, 6, 7, 19], it is hy p o thesized that these tw o limits indeed coincide; a preliminary consideration without using this h yp othesis is give n in [2 1 ]. In section 3, w e demonstrate for sp eciﬁc examples that this is not a hypothesis but a v eriﬁable fact. Let us decompose L n ( λ ) in to a sum of terms, eac h of whic h is the pro duct of an ( n, m )-indep enden t scalar function of λ and a λ -indep enden t matrix, i.e., L n ( λ ) = i max X i = i min f i ( λ ) L ( i ) n . The scalar functions f i ( λ ) ( i min ≤ i ≤ i max ) are linearly indep endent; typi- cally , they a r e p o w ers o f λ , e.g. , f i ( λ ) = λ i . The nonzero elemen ts of t he matrices L ( i ) n are classiﬁed into tw o t yp es, that is, constan ts (or, at most, arbitrary functions of only one indep enden t v ariable) and dynamical v ari- ables dep ending on b oth indep enden t v ariables. W e express the entire set of functionally indep enden t dynamical v a riables in L ( i ) n ( i min ≤ i ≤ i max ) as { l n } . The set of dynamical v ariables { v n } is deﬁned from V n ( λ ) in exactly the same w ay . Then, t he zero-curv ature condition (2.6) pro vides a (typically bilinear a lgebraic) system for { e l n , l n } and { v n , v n +1 } . In particular, this sys- tem con tains a useful subsystem, that is, an ultralo cal and linear system in 8 { v n } , wherein the co eﬃcien ts in v olv e { l n } and its shifts; this subsystem can also b e deriv ed b y noting that (2.7) holds true as an iden tity in λ . W e solv e this subsystem to express a subset of { v n } in terms of the remaining { v n } as w ell as { l n } and its shifts suc h as { e l n } and { l n − 1 } . Th us, w e can reduce the n um b er of independen t dynamical v aria bles in V n while main taining the ultralo cality of V n with resp ect to { v n } . It remains to b e v eriﬁed that the zero-curv ature condition (2.6) indeed provides a meaningful fully discrete system for the reduced set of dep enden t v ar ia bles; it should deﬁne a con- sisten t and unique time ev olution for generic initial data under appro pria te b oundary conditions ( cf. (2.9)). 2.3 F undamen tal “conserv ation la ws” in the discrete- time case W e consider the determinan t of b oth sides of (2.6) to obtain the equality (det e L n )(det V n ) = (det L n )(det V n +1 ) . (2.10) This relation can b e written more explicitly in the form of a discrete conser- v ation law, ∆ + m log(det L n ) = ∆ + n log(det V n ) . Here, ∆ + m is the forw ard diﬀerence op erator in t he time direction, i.e., ∆ + m f n := e f n − f n . This is the discrete-time v ersion of (2.3). Note that after cancelling the m - indep enden t factor of det L n , the relation (2.10) should reduce to either det V n = det V n +1 (2.11) or exp ( e ρ n ) det V n = exp ( ρ n ) det V n +1 , (2.12) where ρ n is a non trivial conserv ed densit y . In the ﬁrst case (2.11), one may sp eculate that the ab o v e conse rv ation la w ma y b ecome the meaningless re- lation ∆ + m (const.) = ∆ + n (const . ). Ho w ev er, in all the examples that require the intro duction of auxiliary v ariables, this app ears not to b e the case; it is not immediately eviden t that the determinan t of V n is n -indep enden t, and th us the relation (2.10) still con tains meaningful info r ma t io n. W e employ a simpliﬁed but still ultralo cal (with respect to { v n } ) form of V n , compute its determinan t, and expand it with respect to λ in the summe d form det V n ( λ ) = j max X j = j min g j ( λ ) a ( j ) n . (2.13) 9 Here, g j ( λ ) ( j min ≤ j ≤ j max ) are linearly indep endent functions of λ , e.g. , g j ( λ ) = λ 2 j , and their co eﬃcien ts a ( j ) n are λ -indep endent functions of { v n } , { l n } , and the space/time shifts of { l n } . Substituting (2.13) in to (2.1 1), we obtain the n -indep endence o f the co eﬃcien ts of g j ( λ ) for all j , i.e., a ( j ) n = lim n →±∞ a ( j ) n , j min ≤ j ≤ j max . (2.14) In the second case (2.12), whic h is mor e common than (2.11), the substitution of (2.13) giv es the set o f relations exp ( e ρ n ) a ( j ) n = exp ( ρ n ) a ( j ) n +1 , j min ≤ j ≤ j max . Th us, there exist seemingly more than one ﬂux log a ( j ) n asso ciated with the same conserv ed densit y ρ n , but the a ( j ) n should coincide up to trivial pro - p ortionality facto r s. Indeed, calculating the rat io of the abov e eq uality for diﬀeren t v alues of j o n b oth sides, w e obta in a ( j 1 ) n a ( j 2 ) n = lim n →±∞ a ( j 1 ) n a ( j 2 ) n , j min ≤ j 1 6 = j 2 ≤ j max . (2.15) That is, the ratio a ( j min ) n : · · · : a ( j max ) n is indep enden t of n . In b o th the ab o v e cases, the right-hand side of (2.14) or (2.15) is determined b y t he b ound- ary conditions fo r V n , in pa rticular, the b oundary v alues of the dep enden t v ariables con tained in V n ; eac h rig h t-hand side is set as a deﬁnite v alue in- dep enden t of n (cf. (2.9)). This results in an “ultralo cal” algebraic system for a subset o f { v n } that essen tially constitutes the auxiliary v ariables; the n um b er of indep endent unkno wns is usually equal to that of the indep en- den t equalities so that this algebraic system is neither o v erdetermined nor underdetermined. 2.4 Algebrai c system for the auxiliary v ariables and lo cal equations of motion W e solv e the obtained algebraic sys tem for the auxiliary v a riables app earing in V n (and not in M n ). By eliminating all but one of the auxiliary v ariables, this sys tem b ecomes a scalar algebraic equation in the remaining auxiliary v ariable; the other auxiliary v ariables are express ible in terms of the solution of this equation. The degree and complexit y of this alg ebraic equation dep end on the b oundary conditions for V n , whic h determine the disp ersion relation of the time-discretized lattice system. In this pap er, we mainly consider the case for a degree of tw o so that the equation is solv able b y the quadratic 10 form ula. F or suﬃcien tly small h , referring to the prescrib ed b eha vior (2.8) of V n , w e can discard one of the tw o solutions as improp er, and obtain a unique prop er solution of the quadratic equation. In general, t he larger t he n umber of grid p oints deﬁning the lattice sy stem as well as the disp ersion relation, the higher the degree of the algebraic equation determining the a uxiliary v ariables. The hig her- degree case can b e in terpreted as a comp o sition of lo w er-degree cases; that is, the time ev olution in the higher-degree case can b e factor ized into sequen tial applications of more elemen tary time ev olutions. This p oin t will b e illustrated in section 3. Once all the auxiliary v ariables ha v e been expressed lo cally in terms of the original dep enden t v ariables that ha v e alr eady app eared in the semi-disc rete case, w e only hav e to substitute them in to a n a ppropriate subset of equations arising f r o m t he zero-curv ature condition (2.6). Because of the use of the “conserv ation la ws” (cf. (2.11) or (2.12) ) , not all of the equations arising from (2.6) are indep enden t and necessary an y longer. W e choo se a minimal subset of these equations so that the substitution of the lo cal expressions in the auxiliary v ariables pro duces the unique discrete-time ev olution of the lattice syste m. Last but not least, the ab ov e set of conserv ation law s used to determine the auxiliary v ariables can, in principle, b e deriv ed from the o r iginal system of partial diﬀerence equations resulting fro m t he zero-curv ature condition (2.6). Ho w ev er, in practice, this is an extremely diﬃcult task; any computation conducted at the comp onen t leve l without f o llo wing a set pro cedure is highly unlik ely to arriv e at the nontrivial conserv ation laws to b e deriv ed. Thus , our deriv ation p erfor med at the matrix lev el using a determinan ta l formula and the sp ectral parameter is po ssibly t he o nly w a y of obta ining them. 2.5 Re marks on n on autonomous extensions Actually , the “step size” parameter h introduced in (2.8) need not be a con- stan t and can dep end arbitrarily on the discrete time co ordinate m ∈ Z ; in other w ords, the discrete-time ﬂow in v olving an a rbitrary parameter h b e- longs to the same hierarc hy , and w e can specify a n y v alue of h at eve ry step of t he time ev olution. Indeed, this do es not result in an y essen t ial diﬀerence in the subseque nt computations, b ecause the zero-curv a t ur e condition ( 2.6) in v olv es equal-time V n only . In this w a y , w e can obtain nona uto nomous ex- tensions of time-discretized lattice sys tems in v olving one arbit r a ry function of t he discrete t ime (see, for example, [22] and references therein). More- o v er, if the time discretization considered can b e factorized in to a comp osi- tion of M elemen tary time ev olutions (cf. [6, 7, 19]), then it generally allo ws an extension inv olving M arbitrary functions of t ime; ho w ev er, w e do not proactiv ely discus s this p ossibilit y to av oid unnecessary confusion. F rom the 11 mo dern p oint of view (cf. [20, 23 – 25]), a discrete-time ﬂo w can b e identi- ﬁed with the spatial part of an a uto-B¨ ac klund transformation of the or ig inal con tin uous-time ﬂo w or an auto-B¨ ac klund transformation of the whole hier- arc h y of con tin uous-time ﬂows . Th us, the consistency of this nonautonomous (“non uniform in time”) extension can b e understo o d as the comm utativit y of auto-B¨ ac klund transformatio ns for diﬀerent v alues of the B¨ ac klund param- eter [26, 27], kno wn as Bianc hi’s p erm utabilit y theorem. R eaders interes ted in B¨ acklund transformations are referred to the Pro ceedings [28]. Bianc hi’s permutabilit y of ten implies an ultralo cal relation among t he four solutions of the same system, called a nonlinear sup erp osition form ula; its deriv ation is based on the compatibility condition of tw o “diﬀerent” time ev olutions: e Ψ = V ( λ, h 1 )Ψ and b Ψ = V ( λ, h 2 )Ψ. If there exist tw o or more distinct one-parameter auto-B¨ ac klund tra nsfor ma t io ns, then t he ma- trix V ( λ, h 2 ) may b e generalized to W ( λ, h 2 ). The nonlinear sup erp osition form ula motiv ates us to place the four solutions appropria tely at the four ver- tices of a rectangle [15, 16] a nd to a ssign each v alue of the B¨ acklund para meter to eac h pair of parallel sides; this is often referred to as the Lam b diagram (see Fig. 3 in [26]). It is con v enien t to iden tify eac h v alue o f the B¨ ac klund parameter with the length of eac h side, although v alues are not restricted to p ositiv e n um b ers. W e can use this rectangle as an elemen tary cell deﬁning t w o directions of n ew indep enden t v ariables [15 – 17]. In fact, rep eated ap- plications of the a uto-B¨ ac klund transformation at (g enerally) distinct v alues of t he B¨ ac klund para meter g enerate a t w o-dimensional unequally spaced (in b oth directions) lattice in the quadran t (cf. Fig. 12 in [27] a nd the main ﬁg ure in [29]), or ev en in the plane, with elemen ta ry cells of v arious sizes. With this understanding, w e can rein terpret ev ery nonlinear sup erp osition form ula as a fully discretized lattice system in v olving one arbitrary function of one in- dep enden t v ariable and another a rbitrary function of the o ther indep enden t v ariable. Th us, if w e encoun ter a time discretization of an integrable lattice system that has the same form as a nonlinear sup erp osition form ula for some con- tin uous/discrete in tegrable sys tem, then it allows a natural nonautonomous extension inv o lving t w o arbitrary f unctions originating from tw o v alues of the B¨ ack lund parameter. F or example, the nonlinear superp osition form ula for the po ten tial KdV hierarc h y (cf. (14) or (16) in [30]) suggests a nonau- tonomous extension of a discrete p otential KdV equation (cf. (5) and (6) in [3 1]; (5.16) in [32]; (68) in [33]), while the nonlinear superp osition form ula for the sine-Gordon equation (or the p otential mKdV hierarc h y) with sim- ple sign handling implies a nonautonomous extension of the fully discretized 12 sine-Gordon equation [20, 34] giv en b y tan  u n +1 ,m +1 + u n,m 4  = f ( n ) + g ( m ) f ( n ) − g ( m ) tan  u n +1 ,m + u n,m +1 4  , (2.16) or equiv alen tly , f ( n ) g ( m ) sin  u n +1 ,m +1 − u n +1 ,m − u n,m +1 + u n,m 4  = sin  u n +1 ,m +1 + u n +1 ,m + u n,m +1 + u n,m 4  . (2.17) If w e rescale/redeﬁne the arbitrary f unctions and v ariables as f ( n ) = (4 / ∆ ) F ( n∆ ) and g ( m ) = h G ( mh ), and u n,m = u ( n∆, mh ), n∆ =: x , and mh =: t , resp ec- tiv ely , then the contin uous limit ∆, h → 0 reduces ( 2 .17) to the v ariable- co eﬃcien t sine-Gordon equation F ( x ) G ( t ) u xt = sin u, whic h is ob viously equiv alen t to t he constan t-co eﬃcien t sine-Gordon equation u X T = sin u . Th us, this t yp e of nona utonomous extension in the discrete case is though t to b e a ve stige of co ordina t e transformations in the con tin uous case that do not mix the tw o independen t v aria bles. It is still unclear whether a n y of the time-discretized lat tice systems ob- tained in the next sec tion in their presen t form can b e iden tiﬁed with a nonlinear sup erp o sition form ula. Alternative ly , w e pro p ose an intriguing pro cedure for constructing fully discrete nonautonomous systems in volv ing t w o arbitrary functions; this pro cedure is closely related to the pro p ert y of three-dimensional consistency [35 – 37]. It is assumed that L n ( λ ) is ultr alo cal in t he set of dynamical v ariables { l n } , while V n ( λ, h ) can b e written in terms of { l n − 1 , e l n } or { e l n − 1 , l n } . W e only consider the former case, b ecause the latter case can b e dealt with in a similar manner. Then, the linear problem (2.5) with the “step size” h set as h = h 1 and h = h 2 resp ectiv ely implies the relations e Ψ n = N (1) n Ψ n − 1 , N (1) n := V n ( λ, h 1 ) L n − 1 ( λ ) , (2.18a) and b Ψ n = N (2) n Ψ n − 1 , N (2) n := V n ( λ, h 2 ) L n − 1 ( λ ) . (2.18b) Here, the newly introduced matrix N (1) n connects the v alues of Ψ at tw o lattice p o in ts ( n − 1 , m ) and ( n, m + 1), a nd dep ends o nly on the dynamical v ariables at these p oin ts, namely , { l n − 1 , e l n } . Similarly , N (2) n connects the 13 t w o p oints ( n − 1 , m ) and ( n, m + 1 ′ ), and depends on the dynamical v ari- ables at these po in ts: { l n − 1 , b l n } . Note that ( n, m + 1 ′ ) can b e iden tiﬁed with ( n, m + 1) only if h 1 = h 2 . The compatibilit y condition of (2.18) is giv en by a “new” zero-curv at ure equation, e N (2) n +1 N (1) n = b N (1) n +1 N (2) n . Th us, the substitution of the explicit forms of t he matrices N (1) n ( { l n − 1 , e l n } , h 1 ) and N (2) n ( { l n − 1 , b l n } , h 2 ) in to this equation should provide a closed system for the dynamical v a riables at the four lattice p oints l n − 1 , e l n , b l n , b e l n +1 (= e b l n +1 ) . W e can regard the corresponding parallelogram as deﬁning t wo directions of new indep enden t v aria bles, so tha t the system now in v olv es tw o arbitrary functions originating fro m h 1 and h 2 . 3 Examples In this section, w e apply the g eneral metho d for time discretization in section 2 to ﬁv e imp ortant examples: the T o da lat t ice, the Ablo witz–Ladik latt ice (semi-discrete NLS), the V olterra lattice, the mo diﬁed V olterra lattice, and the lattice Heisen b erg ferromagnet mo del. W e are only in terested in time discretizations that can accurately appro x- imate the contin uous-time dynamics in the small-v alue range of the “step size” parameter h . This implies that the discrete-time Lax matrix V n al- lo ws the asymptotic expansion with resp ect to h given in ( 2 .8); in pa rticular, ( V n − I ) /h do es not inv olve O ( 1 /h ) terms. This requiremen t plays a cru- cial role in o btaining suitable lo cal expressions fo r the auxiliary v ariables app earing in V n . The parameter h is g enerally assume d to b e nonzero; alternative ly , one can allo w the case h = 0 as the trivial iden tity mapping e l n = l n . 3.1 The T o da lattice in Flasc hk a–Manak o v co ordinates W e consider the T o da lattice written in F lasc hk a–Manako v co ordinates [9– 11]: u n,t = u n ( v n − v n − 1 ) , v n,t = u n +1 − u n . (3.1) The parametrization u n = e x n − x n − 1 , v n = x n,t , (3.2) 14 enables the system (3.1) to b e rewritten as the Newtonian equations of motion for the T o da lattice, x n,tt = e x n +1 − x n − e x n − x n − 1 . The Lax pair for the T o da lattice (3.1) in Flasc hk a–Manako v co o r dina t es is giv en by [1] L n =  λ + v n u n − 1 0  , (3.3a) M n =  0 − u n 1 λ + v n − 1  . (3.3b) Indeed, the substitution of (3.3) in to the zero-curv ature conditio n (2.2) re- sults in (3.1). Let us mo v e to the discrete-time case. A comprehensiv e o v erview of t he existing results on time discretizations of the T o da lattice is giv en in § 3.22 and § 5.11 of [1], th us w e do not rep eat it here. In view of the zero-curv ature condition (2.6), we assume the Lax mat r ix V n of the following form: V n = I + h  − λa + α n − au n − e u n b n a + b n λb n + α n − 1 + av n − 1 + b n v n − 1  . (3.4) Here, α n and b n are auxiliary v ariables. The zero-curv ature condition (2.6) for the L a x pair (3.3a) and (3.4 ) amoun ts to the following system of partial diﬀerence equations:                1 h ( e u n − u n ) = α n +1 u n − e u n α n − 1 + e v n ( au n + e u n b n ) − ( a e u n + e u n b n ) v n − 1 , 1 h ( e v n − v n ) = α n +1 v n − e v n α n + a ( u n +1 − e u n ) + e u n +1 b n +1 − e u n b n , α n − α n +1 = a ( e v n − v n ) , e u n b n = b n +1 u n . (3.5) The general form ( 3 .5) of the time-discretized T o da lattice is in tegrable for matrix-v alued dep enden t v ariables, but in the following, w e consider only the case of scalar dep enden t v ariables. In view of (2.9) and (3 .2), we imp ose the follo wing b oundary conditions for u n , v n , α n , and b n : lim n →±∞ u n = 1 , lim n →±∞ v n = 0 , lim n →±∞ α n = 0 , lim n →±∞ b n = b. (3.6) The b oundary v alue of α n is set as zero b y redeﬁning h . T o b e precise, the b o undar y conditions (3.6) contain r e dundant info r mation. Indeed, it can 15 b e sho wn that the auxiliary v ar ia bles ha v e the same limit v alues for n → −∞ and n → + ∞ . Therefore, it is suﬃcien t to assume either lim n →−∞ ( α n , b n ) = (0 , b ) or lim n → + ∞ ( α n , b n ) = (0 , b ). In the con tin uum limit o f time h → 0, the aux- iliary v ariables reduce to constan ts, i.e., α n → 0 and b n → b . Th us, in this limit, the time discretization ( 3 .5) reduces to u n,t = ( a + b ) u n ( v n − v n − 1 ) , v n,t = ( a + b )( u n +1 − u n ) , whic h is, if a + b 6 = 0, equiv alen t to the T o da lattice (3.1) up to a redeﬁnition of t . Note that in the case a + b = 0 , (3.5) has the trivial solution e u n = u n , e v n = v n , α n = 0, b n = b . The determinan t of the 2 × 2 Lax matrix L n (3.3a) can b e immediately computed as det L n = u n . Using t he recurrence form ula for α n in (3.5), w e can rewrite the Lax matrix V n (3.4) in an ultralo cal form with resp ect to the auxiliary v ariables α n and b n . Th us, it s 2 × 2 determinant is computed as det V n ( λ ) = [1 + h ( − λa + α n )] [1 + h ( λb n + α n + a e v n − 1 + b n v n − 1 )] + h 2 ( au n + e u n b n ) ( a + b n ) = − λ 2 h 2 ab n − λh  ( a − b n ) (1 + hα n ) + ha 2 e v n − 1 + hab n v n − 1  + (1 + hα n ) 2 + (1 + hα n ) ( ha e v n − 1 + hb n v n − 1 ) + h 2 ( au n + e u n b n ) ( a + b n ) . Therefore, the equality (2.10) com bined with the b oundary conditions (3.6) (or the streamlined ve rsion as stated ab o v e) implies the set of relations            b n = b Λ n , ( a − b n ) (1 + hα n ) + ha 2 e v n − 1 + hab n v n − 1 = ( a − b ) Λ n , (1 + hα n ) 2 + (1 + hα n ) ( ha e v n − 1 + hb n v n − 1 ) + h 2 ( au n + e u n b n ) ( a + b n ) =  1 + h 2 ( a + b ) 2  Λ n . (3.7) Here, Λ n is the quan tit y t ha t satisﬁes the recurrence form ula e u n Λ n = u n Λ n +1 , (3.8) and has the normalize d b o undar y v alue, lim n →±∞ Λ n = 1. More precisely , only one of the tw o conditions lim n →−∞ Λ n = 1 and lim n → + ∞ Λ n = 1 is re- quired, in accordance with the b o undar y conditions for α n and b n . Then, the other condition can b e conﬁrmed. Thus , Λ n can b e written explicitly (but globally) as Λ n = n − 1 Y j = −∞ e u j u j = + ∞ Y j = n u j e u j . 16 Note that the ﬁrst equalit y in (3.7) is v alid ev en if a = 0 (cf. (3.5 ) ). The al- gebraic system (3.7) essen tially comprises t w o non trivial relatio ns containing the t w o unkno wns, α n and Λ n . It is rather cum b ersome t o iden tify the disp ersion relation of ( 3 .5) from its linear leading part (cf. subsection 3.2). Instead, w e can compute the asymptotic form of the Lax matrix V n (3.4) as n → −∞ or n → + ∞ and consider its factorization (cf. [1, 6, 7 , 19]). The results imply that the cases a = 0 and b = 0 form the basis for the general case; that is, t he time ev olution for general a and b is equiv alen t to the order-indep enden t comp osition of tw o time evolutions corresp onding to a = 0 and b = 0, resp ectiv ely . In addition, these t w o cases are related to eac h other throug h the time reﬂection, as w e will see b elo w. No w, w e consider the tw o fundamen tal cases: a = 0 or b = 0 . • The c ase a = 0, b 6 = 0. In this case, the auxiliary v ariable α n v anishes. Th us, the discrete-time system (3.5) reduces to (cf. (3.6) in [38])              1 h ( e u n − u n ) = b Λ n +1 u n ( e v n − v n − 1 ) , 1 h ( e v n − v n ) = b Λ n +1 ( e u n +1 − u n ) , e u n Λ n = u n Λ n +1 , lim n →−∞ Λ n = 1 or lim n → + ∞ Λ n = 1 . (3.9) The algebraic system (3.7) simpliﬁes to a quadra t ic equation in Λ n , ( hb ) 2 e u n Λ 2 n −  1 + ( hb ) 2 − hbv n − 1  Λ n + 1 = 0 . (3.10) The asymptotic behavior (2.8 ) of the Lax ma t rix V n implies that the prop er solution of (3.10) is giv en by  1 + ( hb ) 2  Λ n = 2 1 − ǫ v n − 1 + q (1 − ǫ v n − 1 ) 2 − 4 ǫ 2 e u n , (3.11) where ǫ := hb/ [1 + ( hb ) 2 ]. Th us, if hb ∈ R , t hen − 1 / 2 ≤ ǫ ≤ 1 / 2. In this case, the lo cal expres sion ( 3 .11) is v alid only if − 1 < hb < 1, whic h cov ers the range − 1 / 2 < ǫ < 1 / 2. The bo rderline cases hb = ± 1 ar e excluded from our consideration. If ( hb ) 2 > 1, (3.11) is inconsisten t with the b oundary con- ditions, and the other solution of (3 .1 0) should b e adopted. In any case, the b oundary conditions for u n and v n imply that lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. 17 Substituting (3 .11) into the ﬁrst and second equations in (3.9), w e obtain a time discretization of the T o da lattice (3.1) in the lo cal form,              e u n − u n ǫ = 2 u n ( e v n − v n − 1 ) 1 − ǫ v n + q (1 − ǫ v n ) 2 − 4 ǫ 2 e u n +1 , e v n − v n ǫ = 2 ( e u n +1 − u n ) 1 − ǫ v n + q (1 − ǫ v n ) 2 − 4 ǫ 2 e u n +1 . (3.12) Under the b oundary conditions lim n →±∞ u n = 1 and lim n →±∞ v n = 0, e v n is determined from u n , v n , and e u n +1 , and subseq uen tly , e u n is determined f rom v n − 1 , u n , v n , e v n , and e u n +1 . If hb ∈ R , the u n are nonzero and real-v alued, a nd the v n are real-v alued at the initial time, then (3.9) implies that the auxil- iary v ariable Λ n is alwa ys real-v a lued. Consequen tly , the discriminan t of the quadratic equation (3.10 ) m ust b e nonnegative. Th us, as long as the ampli- tudes of u n − 1 and v n are suﬃcien tly small and their eﬀects can b e regarded as p erturbations, the real-v aluedness o f u n and v n is preserv ed under the time ev olution of the discrete-time T o da lattice (3.12) for − 1 / 2 < ǫ < 1 / 2. T o express the bac kw ard time ev olution explicitly , w e only ha v e t o re- place Λ n +1 in the ﬁrst and second equations of (3.9) with Λ n e u n /u n and then substitute the lo cal expression (3.11). The resulting system is              e u n − u n ǫ = 2 e u n ( e v n − v n − 1 ) 1 − ǫ v n − 1 + q (1 − ǫ v n − 1 ) 2 − 4 ǫ 2 e u n , e v n − v n ǫ = e u n u n × 2 ( e u n +1 − u n ) 1 − ǫ v n − 1 + q (1 − ǫ v n − 1 ) 2 − 4 ǫ 2 e u n . (3.13) Using the ﬁrst equation, the second equation in (3.1 3) can also b e written as e v n − v n ǫ = 2 ( e u n +1 − u n ) 1 − ǫ (2 e v n − v n − 1 ) + q (1 − ǫ v n − 1 ) 2 − 4 ǫ 2 e u n . • The c ase b = 0, a 6 = 0. In this case, the auxiliary v ariable b n v anishes. Th us, t he discrete -time system (3.5) reduces to            1 h (1 + hα n + ha e v n − 1 ) ( e u n − u n ) = au n ( v n − e v n − 1 ) , 1 h (1 + hα n + hav n ) ( e v n − v n ) = a ( u n +1 − e u n ) , α n − α n +1 = a ( e v n − v n ) , lim n →−∞ α n = 0 or lim n → + ∞ α n = 0 . (3.14) 18 The algebraic system (3.7) simpliﬁes to ( 1 + hα n + ha e v n − 1 = Λ n , (1 + hα n ) 2 + (1 + hα n ) ha e v n − 1 + ( ha ) 2 u n =  1 + ( ha ) 2  Λ n . (3.15) Note that, using Λ n instead of α n , system ( 3 .14) with (3.8) and (3.15) can b e iden tiﬁed with the discrete-time T o da lattice give n in § 3.8 of [1] (also see [39]). By eliminating Λ n , (3.15) reduces to a quadratic equation in 1 + hα n , (1 + hα n ) 2 −  1 + ( ha ) 2 − ha e v n − 1  (1 + hα n )+( ha ) 2 u n −  1 + ( ha ) 2  ha e v n − 1 = 0 . The prop er solution o f this quadratic equation is giv en by 1 + hα n 1 + ( ha ) 2 = 1 − δ e v n − 1 + q (1 + δ e v n − 1 ) 2 − 4 δ 2 u n 2 , (3.16) where δ := ha/ [1 + ( ha ) 2 ]. Thus, if ha ∈ R , then − 1 / 2 ≤ δ ≤ 1 / 2. In t his case, the lo cal expression (3.16) is v alid o nly if − 1 < ha < 1, whic h co v- ers the ra nge − 1 / 2 < δ < 1 / 2. The b orderline cases ha = ± 1 are exc luded from our consideration. If ( ha ) 2 > 1, (3 .1 6) is inconsisten t with the b ound- ary conditions, and the other solution of the quadra t ic equation should b e emplo y ed. In any case , the b oundary conditions for u n and v n imply that lim n →−∞ α n = lim n → + ∞ α n = 0, and consequen tly , lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. Substituting the lo cal expression (3 .16) in to the ﬁrst and second equations of (3.14), w e obtain a t ime discretization of the T o da lattice (3 .1),              e u n − u n δ = 2 u n ( v n − e v n − 1 ) 1 + δ e v n − 1 + q (1 + δ e v n − 1 ) 2 − 4 δ 2 u n , e v n − v n δ = 2 ( u n +1 − e u n ) 1 + δ (2 v n − e v n − 1 ) + q (1 + δ e v n − 1 ) 2 − 4 δ 2 u n . (3.17) Under the b oundary conditions lim n →±∞ u n = 1 and lim n →±∞ v n = 0, e u n is determined from e v n − 1 , u n , and v n , and subsequen tly , e v n is determined fr o m e v n − 1 , e u n , u n , v n , and u n +1 . As lo ng as u n and v n can b e regarded a s p er- turbations around the b oundary v alues 1 and 0, their real- v aluedness is pr e- serv ed under the time evolution of the disc rete-time T o da lattice (3.17) for − 1 / 2 < δ < 1 / 2. T o obta in the backw ard time ev olution explicitly , w e rewrite α n in the ﬁrst and second equations of (3.14) as α n +1 + a ( e v n − v n ) and then substitute 19 the lo cal expression ( 3 .16). The resulting system is              e u n − u n δ = 2 e u n ( v n − e v n − 1 ) 1 + δ e v n + q (1 + δ e v n ) 2 − 4 δ 2 u n +1 , e v n − v n δ = 2 ( u n +1 − e u n ) 1 + δ e v n + q (1 + δ e v n ) 2 − 4 δ 2 u n +1 . (3.18) It is easy to see that (3.17) is equiv alen t t o (3.13) through the time reﬂection and the iden tiﬁcation δ ↔ − ǫ . In the same manner, (3.18) can b e identi- ﬁed with (3.1 2) through the time reﬂection. Th us, t he forw ard/backw ard time ev olution in the case b = 0 corresponds to the bac kw ard/forward time ev olution in the case a = 0, up to a redeﬁnition of t he pa r a meters. 3.2 The Ablo witz–Ladik lattice In t his subsection, w e discuss the time discretization of the Ablowitz–Ladik lattice, whic h w e consider to b e the most instructiv e example to illustrate our general metho d. The (nonreduced form of the) Ablow itz–Ladik lattice is ( q n,t − aq n +1 + bq n − 1 + ( a − b ) q n + aq n +1 r n q n − bq n r n q n − 1 = 0 , r n,t − br n +1 + ar n − 1 + ( b − a ) r n + br n +1 q n r n − ar n q n r n − 1 = 0 , (3.19) and its Lax pair (cf. (2.1) and (2.2)) is given b y L n = λ  1 0  +  0 q n r n 0  + 1 λ  0 1  =  λ q n r n 1 λ  , (3.20a) M n =  λ 2 a − a (1 + q n r n − 1 ) λaq n + b λ q n − 1 λar n − 1 + b λ r n b λ 2 − b (1 + r n q n − 1 )  . (3.20b) Here, the free parameters a a nd b are usually set as constants, but they can dep end on t he time v ariable t in an arbitrary manner, namely , a := a ( t ), b := b ( t ) . The fa miliar form o f the Ablow itz–Ladik lattice is obtained f r om (3.19) through the reduction of the complex conjugate: b = a ∗ , r n = σ q ∗ n , where σ is a r eal constan t. The case where a and b are purely imaginary leads to an in tegrable semi-discretization of the NLS equation. T aking in to accoun t the λ - dep endence of the Lax matr ix M n (3.20b), w e lo ok for a La x matrix V n with the follo wing dep endence on λ : V n ( λ ) = I + h  λ 2 V (2) n + λV (1) n + V (0) n + 1 λ V ( − 1) n + 1 λ 2 V ( − 2) n  . (3.21 ) 20 Here, V ( j ) n ( j = 2 , 0 , − 2) are diago na l matrices, while V ( j ) n ( j = 1 , − 1) are oﬀ-diagonal matrices. The nonzero parameter h ma y dep end on the discrete time co o rdinate m ∈ Z . Substituting (3.20a) and (3 .2 1) into ( 2.6), w e ﬁnd that the matrix V n should assume the form V n = I + h      λ 2 a + e q n c n r n − 1 + α n + 1 λ 2 d n λ ( aq n − e q n c n ) + 1 λ ( b e q n − 1 − d n q n − 1 ) λ ( a e r n − 1 − c n r n − 1 ) + 1 λ ( br n − e r n d n ) λ 2 c n + e r n d n q n − 1 + β n + b λ 2      . (3.22) Here, α n , β n , c n , and d n are auxiliary v ariables. The zero-curv a ture condition (2.6) is equiv alen t to the follo wing system o f partial diﬀerence equations:                                      1 h ( e q n − q n ) − aq n +1 + b e q n − 1 − α n +1 q n + e q n β n + e q n +1 c n +1 (1 − r n q n ) − ( 1 − e q n e r n ) d n q n − 1 = 0 , 1 h ( e r n − r n ) − br n +1 + a e r n − 1 − β n +1 r n + e r n α n + e r n +1 d n +1 (1 − q n r n ) − ( 1 − e r n e q n ) c n r n − 1 = 0 , α n +1 − α n = a ( e q n e r n − 1 − q n +1 r n ) , β n +1 − β n = b ( e r n e q n − 1 − r n +1 q n ) , (1 − e r n e q n ) c n = c n +1 (1 − r n q n ) , (1 − e q n e r n ) d n = d n +1 (1 − q n r n ) . (3.23) Actually , the general nonreduced fo rm (3.23) of the time-discretized Ablo witz– Ladik lattice is in tegrable f o r matrix-v a lued dep enden t v ariables. Ho we v er, in this pap er, we consider only the case of scalar dep enden t v aria bles. In ad- dition to (2.9), we impo se rapidly deca ying b o undary conditions for q n and r n , that is, lim n →±∞ q n = lim n →±∞ r n = 0 , lim n →±∞ ( α n , β n , c n , d n ) = ( α , β , c, d ) . (3.24 ) Similarly to the se mi-discrete case, the parameters a , b , α , β , c , and d are allo w ed to dep end on t he discrete time co ordinate m ∈ Z . Th us, the time dep endence of h can b e a bsorb ed b y a , b , α n , β n , c n , and d n . F or the same reason, it is p ossible to set h as unit y , but we prefer to leav e it as a small pa- rameter, usually 0 < | h | ≪ 1. T o b e precise, the b o undary conditions (3.24) con tain r e dundant information. Indeed, it can be sho wn that the auxiliary v ariables hav e the same limit v alues for n → −∞ and n → + ∞ . There- fore, it is suﬃcien t to assume either lim n →−∞ ( α n , β n , c n , d n ) = ( α , β , c, d ) or 21 lim n → + ∞ ( α n , β n , c n , d n ) = ( α , β , c, d ). The last f o ur relatio ns in (3.23) imply the following g lo bal express ions for the a uxiliary v ariables α n , β n , c n , and d n in terms of q n and r n [1, 3, 5 , 7, 19]: α n = α − aq n r n − 1 + a n − 1 X j = −∞ ( e q j e r j − 1 − q j r j − 1 ) = α − a e q n e r n − 1 − a + ∞ X j = n +1 ( e q j e r j − 1 − q j r j − 1 ) , β n = β − br n q n − 1 + b n − 1 X j = −∞ ( e r j e q j − 1 − r j q j − 1 ) = β − b e r n e q n − 1 − b + ∞ X j = n +1 ( e r j e q j − 1 − r j q j − 1 ) , c n = c n − 1 Y j = −∞ 1 − e q j e r j 1 − q j r j = c + ∞ Y j = n 1 − q j r j 1 − e q j e r j , d n = d n − 1 Y j = −∞ 1 − e q j e r j 1 − q j r j = d + ∞ Y j = n 1 − q j r j 1 − e q j e r j . The equiv alence of the tw o expre ssions fo r each auxiliary v aria ble is re- lated to the fact that the three quan tities P + ∞ j = −∞ q j r j − 1 , P + ∞ j = −∞ r j q j − 1 , and Q + ∞ j = −∞ (1 − q j r j ) are conserv ed as in t he con tinuous -time case. These con- serv ed quantitie s are a ssumed to be ﬁnite, and the last one should b e nonzero. The substitution of these expressions into t he ﬁrst t w o equations in (3.23) pro vides a global-in-space time discretization of the nonreduced Ablo witz– Ladik lattice (3.19). In the con tin uum limit of time h → 0, w e obtain α n → α − aq n r n − 1 , β n → β − br n q n − 1 , c n → c, d n → d. Th us, with an additional but unesse ntial constraint on the parameters, the time discretization (3.23) reduces to the nonreduced Ablo witz–Ladik lattice (3.19) in the limit h → 0. The determinant of the 2 × 2 Lax matrix L n (3.20a) can be immediately computed as det L n = 1 − q n r n . W e compute the determinan t of the 2 × 2 22 Lax matrix V n (3.22) as det V n ( λ ) = λ 4 h 2 ac n + 1 λ 4 h 2 bd n + λ 2  ha − h 2 a 2 q n e r n − 1 + h 2 aβ n + h 2 aq n − 1 e r n d n + hc n [1 + hα n + ha ( e q n e r n − 1 + q n r n − 1 )]  + 1 λ 2  hb − h 2 b 2 e q n − 1 r n + h 2 bα n + h 2 b e q n r n − 1 c n + hd n [1 + hβ n + hb ( e q n − 1 e r n + q n − 1 r n )]  + h 2 ab + h 2 c n d n + (1 + h e q n r n − 1 c n + hα n ) (1 + hq n − 1 e r n d n + hβ n ) − h 2 ( aq n − e q n c n ) ( br n − e r n d n ) − h 2 ( b e q n − 1 − q n − 1 d n ) ( a e r n − 1 − r n − 1 c n ) . Th us, the equality (2.10) com bined with the b oundary conditions (3.24) (or the streamlined ve rsion as stated ab o v e) implies the set of relations                                    c n = c Λ n , d n = d Λ n , ha − h 2 a 2 q n e r n − 1 + h 2 aβ n + h 2 aq n − 1 e r n d n + hc n [1 + hα n + ha ( e q n e r n − 1 + q n r n − 1 )] =  ha + h 2 aβ + hc (1 + hα )  Λ n , hb − h 2 b 2 e q n − 1 r n + h 2 bα n + h 2 b e q n r n − 1 c n + hd n [1 + hβ n + hb ( e q n − 1 e r n + q n − 1 r n )] =  hb + h 2 bα + hd (1 + hβ )  Λ n , h 2 ab + h 2 c n d n + (1 + h e q n r n − 1 c n + hα n ) (1 + hq n − 1 e r n d n + hβ n ) − h 2 ( aq n − e q n c n ) ( br n − e r n d n ) − h 2 ( b e q n − 1 − q n − 1 d n ) ( a e r n − 1 − r n − 1 c n ) =  h 2 ab + h 2 cd + (1 + hα )(1 + hβ )  Λ n . (3.25) Here, Λ n is the quan tit y t ha t satisﬁes the recurrence form ula (1 − e q n e r n ) Λ n = (1 − q n r n )Λ n +1 , (3.26) and has t he normalize d b o undar y v alue lim n →±∞ Λ n = 1. More precisely , only one of the tw o conditions lim n →−∞ Λ n = 1 and lim n → + ∞ Λ n = 1 is re- quired, in a ccordance with the b oundary conditions for ( α n , β n , c n , d n ). Then, the other condition can b e conﬁrmed. Thus , Λ n can b e written explicitly as Λ n = n − 1 Y j = −∞ 1 − e q j e r j 1 − q j r j = + ∞ Y j = n 1 − q j r j 1 − e q j e r j . (3.27) Note that the ﬁrst tw o equalities in (3.25) are v alid ev en if a = 0 or b = 0 (cf. (3.23)). The algebraic sys tem (3.25) essen tially comprises three non triv- ial relations for t he three unkno wns: α n , β n , and Λ n . Before discussing the general case, w e consider tw o sp ecial cases: a = b = 0 or c = d = 0. 23 • The c ase a = b = 0. In this case, the tw o auxiliary v ariables, α n and β n , b ecome n -indep enden t, i.e., α n = α , β n = β . Thus , the discrete-time system (3.2 3) reduces to              1 h ( e q n − q n ) + (1 − q n r n )Λ n +1 ( c e q n +1 − dq n − 1 ) − αq n + β e q n = 0 , 1 h ( e r n − r n ) + (1 − q n r n )Λ n +1 ( d e r n +1 − cr n − 1 ) − β r n + α e r n = 0 , (1 − e q n e r n ) Λ n = (1 − q n r n )Λ n +1 , lim n →−∞ Λ n = 1 or lim n → + ∞ Λ n = 1 , (3.28) and the algebraic system (3.25) simpliﬁes to h 2 cd (1 − e q n e r n ) (1 − q n − 1 r n − 1 ) Λ 2 n −  (1 + hα ) (1 + hβ ) + h 2 cd − (1 + hα ) hdq n − 1 e r n − (1 + hβ ) hc e q n r n − 1 ] Λ n + (1 + hα ) ( 1 + hβ ) = 0 . (3.29) System (3.28) is in v arian t under the fo llo wing transformation: q n 7→ µ n q n , r n 7→ µ − n r n , hc 7→ µ − 1 hc , hd 7→ µ hd , hα 7→ hα , hβ 7→ hβ , where µ is a nonzero constan t. Using a similar transformation with respect t o the time direction, it is p ossible to remo v e the parameters α and β , but w e prefer to retain them. In terms of the normalized parameters ˆ c := hc 1 + hα , ˆ d := hd 1 + hβ , (3.30) (3.29) can b e rewritten a s ˆ c ˆ d (1 − e q n e r n ) (1 − q n − 1 r n − 1 ) Λ 2 n − h 1 + ˆ c ˆ d − ˆ dq n − 1 e r n − ˆ c e q n r n − 1 i Λ n + 1 = 0 . (3.31) When cd 6 = 0 ( a nd th us ˆ c ˆ d 6 = 0) , this is a quadratic equation in Λ n , whic h has t w o solutions. The simplest w ay to reject the improp er solutio n is to recall the asymptotic b eha vior of the Lax matrix V n for small h (2.8), but here we tak e a diﬀeren t route. Let us ﬁrst consider the “trivial” case where q n and r n are zero for all n ; if t his is satisﬁed at some instan t m = m 0 , then it holds true iden tically fo r any time m . Th us, the tw o solutions of (3.31) are giv en by Λ n = 1 , 1 / ( ˆ c ˆ d ). The recurrence form ula for Λ n in ( 3 .28) implies that these t w o solutions a re unconnected, i.e., Λ n tak es the same v alue for all n . The solution Λ n = 1 / ( ˆ c ˆ d ) can b e discarded if ˆ c ˆ d 6 = 1, b ecause it is inconsisten t with the b oundary condition for Λ n . Nex t, in the g eneral case 24 where q n and r n are not iden tically zero, w e a ssume t hat their amplitudes are alw a ys so small that their eﬀects can b e regarded a s w eak p erturbations of the identic ally zero case. In pa r t icular, the v alue of Λ n is restricted to a neigh b orho o d of unity . Thu s, w e obtain the pro p er solution of the quadra t ic equation (3.31) as Λ n = 2 1 + C n + q (1 + C n ) 2 − 4 D n , (3.32) with C n := ˆ c ˆ d − ˆ dq n − 1 e r n − ˆ c e q n r n − 1 , D n := ˆ c ˆ d (1 − e q n e r n ) (1 − q n − 1 r n − 1 ) . (3.33) Note that this solution is also v alid for the linear case ˆ c ˆ d = 0. The deca ying b oundary conditions for q n and r n imply that lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. When ˆ c ˆ d ∈ R , the local expres sion (3.32) is v alid only if ˆ c ˆ d ≤ 1. If ˆ c ˆ d > 1, the o ther solutio n of (3.31) should b e adopted; alternatively , one can un- derstand the r ig h t-hand side of (3.32) as b eing deﬁned b y a T a ylor series for small ˆ c and ˆ d , a nd its ana lytic contin uation. A t the “threshold” v alue ˆ c ˆ d = 1, the discrete-time system (3.28) has the trivial solution e q n = ˆ dq n − 1 , e r n = ˆ cr n − 1 , Λ n = 1 / (1 − q n − 1 r n − 1 ). Th us, the discriminan t of the quadratic equation (3.31 ) v anishes at ˆ c ˆ d = 1, and the tw o solutions indeed in tersect. Unless ˆ c ˆ d = 1, a uniﬁed expression for Λ n , Λ n = 2 1 + C n +  1 − ˆ c ˆ d  r  1+ C n 1 − ˆ c ˆ d  2 − 4 D n (1 − ˆ c ˆ d ) 2 , can resolv e the sign problem of the square ro ot. In a ddition, the ab o v e square ro ot allows t he T a ylor expansion with resp ect to { e q n , e r n , q n − 1 , r n − 1 } . Ho w ev er, w e do not use this unw ieldy form ula. The ﬁrst and second equations in (3.28) are linear in e q n and e r n , resp ec- tiv ely . Th us, the forw ard time evolution can b e expressed as f ollo ws (cf. (3.30)):          e q n = 1 + hα 1 + hβ q n + (1 − q n r n ) Λ n +1  − 1 + hα 1 + hβ ˆ c e q n +1 + ˆ dq n − 1  , e r n = 1 + hβ 1 + hα r n + (1 − q n r n ) Λ n +1  − 1 + hβ 1 + hα ˆ d e r n +1 + ˆ cr n − 1  , (3.34) where Λ n is giv en by (3.32) with (3.33). In the simplest case of c = ˆ c = 0 or d = ˆ d = 0, the forw ard time ev olution (3.34) is giv en by a simple rational 25 mapping. This fact was disclosed b y Suris [7, 19]. In the general case, using the new parameters ˇ c := ˆ c  1 + ˆ c ˆ d  − 1 and ˇ d := ˆ d  1 + ˆ c ˆ d  − 1 , the mapping (3.34) can b e rewritten in a slightly simpler form:          e q n = 1 + hα 1 + hβ q n + 2 (1 − q n r n ) F n  − 1 + hα 1 + hβ ˇ c e q n +1 + ˇ dq n − 1  , e r n = 1 + hβ 1 + hα r n + 2 ( 1 − q n r n ) F n  − 1 + hβ 1 + hα ˇ d e r n +1 + ˇ cr n − 1  , with F n := 1 − ˇ dq n e r n +1 − ˇ c e q n +1 r n + q  1 − ˇ dq n e r n +1 − ˇ c e q n +1 r n  2 − 4 ˇ c ˇ d (1 − e q n +1 e r n +1 ) (1 − q n r n ) . That is, the ﬁrst term ˆ c ˆ d of C n +1 is remo ve d. When ˆ c ˆ d ∈ R , this expression for F n is v alid only if − 1 < ˆ c ˆ d ≤ 1; this corresp o nds to the rang e ˇ c ˇ d ≤ 1 / 4. If ˆ c ˆ d < − 1 o r ˆ c ˆ d > 1, the sign in front of the square ro ot has to be c hanged. When h is real, β = α ∗ , and d = c ∗ , w e can impose the complex con- jugacy reduction r n = σ q ∗ n with a real constan t σ . In particular, setting α = − β = − i γ / ∆ 2 , c = − d = − i / ∆ 2 , and r n = − ∆ 2 q ∗ n , w e obtain the fully discretized NLS equation i h ( e q n − q n ) + 2  1 + ∆ 2 | q n | 2  1 + C n +1 + q (1 + C n +1 ) 2 − 4 D n +1 e q n +1 + q n − 1 ∆ 2 − γ ∆ 2 ( e q n + q n ) = 0 , (3.35) where C n +1 = h 2 ( ∆ 2 + i γ h ) ( ∆ 2 − i γ h ) + i ∆ 2 h ∆ 2 + i γ h q n e q ∗ n +1 − i ∆ 2 h ∆ 2 − i γ h e q n +1 q ∗ n , D n +1 = h 2 ( ∆ 2 + i γ h ) ( ∆ 2 − i γ h )  1 + ∆ 2 | e q n +1 | 2   1 + ∆ 2 | q n | 2  . The c hoice of γ = 1 f or the real pa r a meter γ is the most natural when taking the con tinuum limit, while the c hoice of γ = 0 simpliﬁes the equation consid- erably . The aforemen tioned condition ˆ c ˆ d ≤ 1 implies that (1 − γ 2 ) h 2 ≤ ∆ 4 ; this is automatically satisﬁed if γ 2 ≥ 1. At ﬁrst glance, it is far from evide nt that the quantit y in the square ro ot is nonnegativ e: (1 + C n +1 ) 2 − 4 D n +1 ≥ 0. Ho w ev er, this inequalit y should hold true because the left-hand side repre- sen ts the discriminan t of t he quadratic equation in Λ n +1 (cf. (3.31)). In fact, 26 the reduction r n = − ∆ 2 q ∗ n with ∆ 2 > 0 guarantees the auxiliary v ariable Λ n to b e p ositiv e (cf. (3.27)). Th us, w e also ha v e the inequalit y 1 + C n +1 > 0. T o summarize, e q n +1 and q n are not fully indep enden t and satisfy the inequal- it y 1 + C n +1 ≥ 2 √ D n +1 . Note that (3.35) with n → n + 1 determines e q n +1 from q n +1 , e q n +2 , and q n . Similarly to the con tin uous NLS equation, this full discretization is homo- geneous under t he follo wing weigh ting sc heme: w eigh t ( ∆ ) = − 1, w eigh t ( h ) = − 2, w eigh t ( q n ) = 1. A t presen t, it is unclear whether the correspo nding one- parameter gro up of scaling symmetries, ( ∆, h, q n ) 7→ ( ∆/k , h/k 2 , k q n ), can deﬁne meaningful “self-similar” solutions to ( 3 .35). As men tioned previously , it might b e aesthetically pleasing to eliminate the ﬁrst “constant” term of C n +1 b y a suitable redeﬁnition of the parameters. F or example, in the simple case of γ = 0 and ∆ 2 = 1, (3.35) can b e rewritten as i δ ( e q n − q n ) + 2  1 + | q n | 2  ( e q n +1 + q n − 1 ) 1 + i δ C n + q (1 + i δ C n ) 2 − 4 δ 2 D n = 0 , (3.36) with C n := q n e q ∗ n +1 − e q n +1 q ∗ n , D n :=  1 + | e q n +1 | 2   1 + | q n | 2  . Here, δ := h/ (1 + h 2 ) is a new parameter. Because − 1 ≤ h ≤ 1, t he range of δ is − 1 / 2 ≤ δ ≤ 1 / 2; at the end p oin ts δ = ± 1 / 2, the time ev olution is trivial b ecause ˆ c ˆ d = 1. Note that (3.36) is in v arian t under the transformation q n 7→ ( − 1) n q n , δ 7→ − δ . T o expres s the backw ard time ev olution explicitly , w e only hav e to replace (1 − q n r n )Λ n +1 in the ﬁrst and second equalities of (3.28) with (1 − e q n e r n ) Λ n , and then substitute the lo cal expression (3.32) with (3.33). In the complex conjugacy reduction wherein h is real, β = α ∗ , and d = c ∗ , w e can “normalize” t he scaling of the dep enden t v ariable by setting r n = − q ∗ n . In this case, w e rewrite the ﬁrst equation for q n in (3.28) as (1 + hα ∗ ) e q n − (1 + hα ) q n + (1 + | e q n | 2 )Λ n ( hc e q n +1 − hc ∗ q n − 1 ) = 0 . (3.37) Moreo v er, (3.29) can b e rewritten as | 1 + hα | 2 + h 2 | c | 2 + (1 + hα ) hc ∗ q n − 1 e q ∗ n + (1 + hα ∗ ) hc e q n q ∗ n − 1 = | 1 + hα | 2 1 Λ n + h 2 | c | 2  1 + | e q n | 2   1 + | q n − 1 | 2  Λ n . (3.38) Using (3.37), we can replace 1 / Λ n and Λ n in (3.38) with rational expressions in q n as we ll as its shifts and complex conjugate. Th us, we obtain a rational 27 form of the fully discrete NLS equation, | 1 + hα | 2 + h 2 | c | 2 + 2 Re [(1 + hα ) hc ∗ q n − 1 e q ∗ n ] = | 1 + hα | 2  1 + | e q n | 2  hc e q n +1 − hc ∗ q n − 1 (1 + hα ) q n − (1 + hα ∗ ) e q n + h 2 | c | 2  1 + | q n − 1 | 2  (1 + hα ) q n − (1 + hα ∗ ) e q n hc e q n +1 − hc ∗ q n − 1 . (3.39) Surprisingly , this coincides with the double-discrete NLS equation prop osed b y Quisp el, Nijhoﬀ, Cap el, and v an der Linden [15 , 17], up to a minor c hange o f co ordinates and parameters; despite its “ elegance”, this rational v ersion ha s the drawbac k that t he forw ard/back w ard time evolution cannot b e uniquely determined. In addition, one cannot immediately recognize that (3.39) reduces to the NLS equation in a contin uous limit. Thus , w e prefer our v ersion, whic h is seem ingly less elegan t b ecause of the existence of t he square ro ot but can deﬁne the unique time evolution prop erly and a llow an easy-to-follow contin uous limit. Actually , using a co ordinate transformation, it is p o ssible to “iden tify” our time discretiz atio n [(3.3 4 ) with ˆ d = ˆ c ∗ , α = β = 0 , and r n = − q ∗ n ] with the a ut o -B¨ a c klund transformation of the Ablow itz–Ladik hierarc hy deriv ed b y Nijhoﬀ, Quisp el, and Cap el [14]. How ev er, their expression (see (19) in [14]) inv olves the indeﬁnite sign ± in fron t of the square ro ot and it is not clear ho w to understand and determine it. Once the lo cal expressions for the auxiliary v ar ia bles hav e b een deriv ed, w e can normalize the 2 × 2 matrix La x pair , L n and V n , so that det L n and det V n b ecome equal to 1. Indeed, this is easily ac hiev ed by dividing L n and V n b y √ det L n and √ det V n , resp ectiv ely ( cf. (2.6) and (2.10)). How ev er, the normalized Lax pa ir not in volv ing the auxiliary v ariables app ears t o b e rather cum b ersome and we do not presen t it here. • The c ase c = d = 0. In this case, the t w o auxiliary v ariables c n and d n v anish. Th us, the discrete-time system (3.23) reduces to                  1 h ( e q n − q n ) − aq n +1 + b e q n − 1 − a ( e q n e r n − 1 − q n +1 r n ) q n − α n q n + e q n β n = 0 , 1 h ( e r n − r n ) − br n +1 + a e r n − 1 − b ( e r n e q n − 1 − r n +1 q n ) r n − β n r n + e r n α n = 0 , α n +1 − α n = a ( e q n e r n − 1 − q n +1 r n ) , β n +1 − β n = b ( e r n e q n − 1 − r n +1 q n ) , (3.40) 28 where lim n →−∞ ( α n , β n ) = ( α, β ) or lim n → + ∞ ( α n , β n ) = ( α , β ). The algebraic system (3.25) simpliﬁes to            1 − haq n e r n − 1 + hβ n = (1 + hβ ) Λ n if a 6 = 0 , 1 − hb e q n − 1 r n + hα n = (1 + hα ) Λ n if b 6 = 0 , h 2 ab ( 1 − q n r n − e q n − 1 e r n − 1 ) + (1 + hα n ) (1 + hβ n ) =  h 2 ab + (1 + hα )(1 + hβ )  Λ n . (3.41) System (3.40) is in v arian t under the fo llo wing transformation: q n 7→ µ n q n , r n 7→ µ − n r n , ha 7→ µ − 1 ha , hb 7→ µ hb , hα 7→ hα , hβ 7→ hβ , where µ is a nonzero constan t. It can b e easily v eriﬁed that the ﬁrst equalit y in (3.41) is also v alid for a = 0, and the second equalit y is v alid for b = 0. Using the ﬁrst and second equalities, the third equality in (3.41) results in a quadratic equation in Λ n , (1 + hα ) (1 + hβ ) Λ 2 n −  (1 + hα ) (1 + hβ ) + h 2 ab − (1 + hα ) haq n e r n − 1 − (1 + hβ ) hb e q n − 1 r n ] Λ n + h 2 ab (1 − q n r n ) (1 − e q n − 1 e r n − 1 ) = 0 . (3.42) In terms of the normalized parameters ˆ a := ha 1 + hβ , ˆ b := hb 1 + hα , (3.43) (3.42) can b e rewritten a s Λ 2 n −  1 + ˆ a ˆ b − ˆ aq n e r n − 1 − ˆ b e q n − 1 r n  Λ n + ˆ a ˆ b (1 − q n r n ) (1 − e q n − 1 e r n − 1 ) = 0 . (3.44) W e recall that the Lax matrix V n giv en b y (3.22) with c n = d n = 0 is required to allo w the asym ptotic expansion (2.8) for small h . This implies that the auxiliary v aria bles α n and β n determined by (3.4 1) are at most of order O ( 1 ) and do not in v olv e 1 /h . Th us, one of the t w o solutio ns of the quadratic equation (3.44) is rejected, a nd w e obtain its prop er solution as Λ n = 1 + A n + q (1 + A n ) 2 − 4 B n 2 , (3.45) with A n := ˆ a ˆ b − ˆ aq n e r n − 1 − ˆ b e q n − 1 r n , B n := ˆ a ˆ b (1 − q n r n ) (1 − e q n − 1 e r n − 1 ) . (3.46) 29 The deca ying b oundary conditions for q n and r n imply that lim n →−∞ Λ n = lim n → + ∞ Λ n = 1, and consequen tly , lim n →−∞ ( α n , β n ) = lim n → + ∞ ( α n , β n ) = ( α , β ). When ˆ a ˆ b ∈ R , the lo cal expression (3.45) is v alid only if ˆ a ˆ b ≤ 1. If ˆ a ˆ b > 1 , the other solu- tion of (3.44) should b e employ ed; alternativ ely , one can understand the righ t-hand side of (3.45) as b eing deﬁned b y a T a ylor series for small ˆ a and ˆ b , and its analytic contin uation. A t the “threshold” v alue ˆ a ˆ b = 1 , the discrete- time system (3.40) has the trivial solution e q n = ˆ aq n +1 , e r n = ˆ br n +1 , α n = α , β n = β . Th us, the discriminan t of the quadratic equation (3.44) v anishes at ˆ a ˆ b = 1 , and the t w o solutions indeed in tersect. Unless ˆ a ˆ b = 1, a uniﬁed expression for Λ n , Λ n = 1 + A n +  1 − ˆ a ˆ b  r  1+ A n 1 − ˆ a ˆ b  2 − 4 B n (1 − ˆ a ˆ b ) 2 2 , can resolv e the sign problem of the square ro ot. In a ddition, this expression allo ws the T ay lor expans ion with resp ect to { q n , r n , e q n − 1 , e r n − 1 } . Ho w ev er, w e a v oid the use o f this un wieldy f orm. Using the ﬁrst t w o relations in (3.4 1), w e can rewrite α n and β n in t he ﬁrst t w o relations in (3.40) in terms of Λ n ,        1 h ( e q n − q n ) + β e q n − α q n + 1 − q n r n Λ n ( − aq n +1 + b e q n − 1 ) = 0 , 1 h ( e r n − r n ) + α e r n − β r n + 1 − q n r n Λ n ( − br n +1 + a e r n − 1 ) = 0 . (3.47) This system is linear in e q n and e r n . Th us, the forw ard time ev olution can be expresse d as follow s (cf. (3 .4 3)):          e q n = 1 + hα 1 + hβ q n + 1 − q n r n Λ n  ˆ aq n +1 − 1 + hα 1 + hβ ˆ b e q n − 1  , e r n = 1 + hβ 1 + hα r n + 1 − q n r n Λ n  ˆ br n +1 − 1 + hβ 1 + hα ˆ a e r n − 1  , (3.48) where Λ n is giv en by (3.45) with (3.46). In t he simplest case of a = ˆ a = 0 or b = ˆ b = 0, the quadratic equation (3.44) in Λ n reduces to a linear equation, and the fo rw ard time ev olution (3.48) is g iven b y a simple ratio nal mapping. This fact w as disco v ered by Suris [7 , 19]; actually , sligh tly prior to his w ork, relev ant results in a prelimi- nary form w ere r ep orted by P empinelli, Boiti, a nd Leon [40 ]. When h is real, β = α ∗ , and b = a ∗ , w e can impose the complex con- jugacy reduction r n = σ q ∗ n with a real constan t σ . In particular, setting 30 α = − β = − i γ / ∆ 2 , a = − b = i / ∆ 2 , and r n = − ∆ 2 q ∗ n , w e obtain the fully dis- cretized NLS equation i h ( e q n − q n ) + 2  1 + ∆ 2 | q n | 2  1 + A n + q (1 + A n ) 2 − 4 B n q n +1 + e q n − 1 ∆ 2 − γ ∆ 2 ( e q n + q n ) = 0 , (3.49) where A n = h 2 ( ∆ 2 + i γ h ) ( ∆ 2 − i γ h ) + i ∆ 2 h ∆ 2 + i γ h q n e q ∗ n − 1 − i ∆ 2 h ∆ 2 − i γ h e q n − 1 q ∗ n , B n = h 2 ( ∆ 2 + i γ h ) ( ∆ 2 − i γ h )  1 + ∆ 2 | q n | 2   1 + ∆ 2 | e q n − 1 | 2  . Th us, this is equiv alen t to the fully dis crete NLS equation (3.35) in the case a = b = 0, up to the space reﬂection n → − n ; this corresp ondence can b e readily noticed b y considering the disp ersion relation [3]. The c hoice of γ = 1 for the real parameter γ is the most natural for taking the con tin uum limit, while the c hoice of γ = 0 simpliﬁes the equation considerably . The aforemen tioned condition ˆ a ˆ b ≤ 1 implies that (1 − γ 2 ) h 2 ≤ ∆ 4 ; this is auto- matically satisﬁed if γ 2 ≥ 1. Note that e q n − 1 and q n are correlated and alw a ys satisfy the inequalit y 1 + A n ≥ 2 √ B n if ∆ 2 > 0. T o express the back w ard time ev olutio n explicitly , w e hav e to rewrite Λ n in (3.47) in terms of Λ n +1 using (3.26), and then substitute the lo cal expression (3.45) with (3.46). By this pro cedure, the fully discrete NLS equation ( 3 .49) in the case c = d = 0 can also b e iden tiﬁed with the f ully discrete NLS equation (3 .35) in the case a = b = 0 through t he time reﬂection m → − m and the sign in v ersion of h , h → − h [3, 7, 19]. Note that the auxiliary v ariables app earing in the stag e of time discretiz a- tion can b e in terpreted, after a minor redeﬁnition, as giving the ﬂuxes of the ﬁrst f ew conserv ation laws. Indeed, ( 3 .26) a nd (3.40) prov ide the following conserv ation la ws: ∆ + m log (1 − q n r n ) = ∆ + n log Λ n , (3.50a) ∆ + m ( q n r n − 1 ) = ∆ + n [( α n − α ) /a + q n r n − 1 ] , (3.50b) ∆ + m ( q n − 1 r n ) = ∆ + n [( β n − β ) /b + q n − 1 r n ] . (3.50c) The cases a = 0 and b = 0 can b e understo o d a s limiting cases; the ﬁnal lo cal expressions for the ﬂuxes do not con tain 1 /a and 1 /b , and a lso hold true for these cases. Using (3 .2 6), (3.41), (3.4 3 ), (3.45), a nd (3.48) , w e can express 31 the ﬂuxes in (3.50b) and (3.50c) explicitly as 1 a ( α n − α ) + q n r n − 1 = 1 + hα ha h Λ n − 1 + ˆ b e q n − 1 r n i + q n r n − 1 = 1 + hα ha   − 1 + A n − q (1 + A n ) 2 − 4 B n 2 + ˆ a ˆ b − ˆ aq n e r n − 1   + q n r n − 1 = 1 + hα 1 + hβ   − 2 ˆ b (1 − q n r n ) (1 − e q n − 1 e r n − 1 ) 1 + A n + q (1 + A n ) 2 − 4 B n + ˆ b − q n e r n − 1   + q n r n − 1 = h 1 + hβ   2 (1 − e q n − 1 e r n − 1 ) ( − b + aq n e r n − 2 ) 1 + A n + q (1 + A n ) 2 − 4 B n + b   , 1 b ( β n − β ) + q n − 1 r n = 1 + hβ hb [Λ n − 1 + ˆ aq n e r n − 1 ] + q n − 1 r n = 1 + hβ hb   − 1 + A n − q (1 + A n ) 2 − 4 B n 2 + ˆ a ˆ b − ˆ b e q n − 1 r n   + q n − 1 r n = 1 + hβ 1 + hα   − 2ˆ a (1 − q n r n ) (1 − e q n − 1 e r n − 1 ) 1 + A n + q (1 + A n ) 2 − 4 B n + ˆ a − e q n − 1 r n   + q n − 1 r n = h 1 + hα   2 (1 − e q n − 1 e r n − 1 ) ( − a + b e q n − 2 r n ) 1 + A n + q (1 + A n ) 2 − 4 B n + a   , where A n and B n are giv en b y (3.46). Note that using (3.26) and (3.48), we can further rewrite these ﬂuxes in many diﬀerent forms. • The gener al c ase ( a, b ) 6 = 0 and ( c, d ) 6 = 0 . Let us consider the (nonreduced) discrete -time Ablowitz–Ladik lattice (3.23) with (3.24) in the general case of ( a, b ) 6 = 0 and ( c, d ) 6 = 0 . Before dealing with the general case of the para meters a , b , c , d , α , and β , w e ﬁrst consider the sp ecial case where the conditions c = − a ( 6 = 0), d = − b ( 6 = 0) , and β = − α are satisﬁed. This is a par t icularly in teresting case from the p ersp ectiv e o f applications suc h as n umerical exp eriments , b ecause the linear part of the equations for q n and r n has symmetry with resp ect to the t ime reﬂection m → − m com bined with the sign in v ersion of h . Moreo v er, the symmetry with resp ect to the space reﬂection n → − n can also b e achie v ed b y imp o sing the further condition b = − a [1, 3, 5 , 19, 4 1]. 32 In the case of c = − a ( 6 = 0), d = − b ( 6 = 0), and β = − α , the algebraic sys- tem (3.25) can b e reformulated a s                        (1 − hbq n − 1 e r n Λ n + hβ n ) − Λ n (1 − ha e q n r n − 1 Λ n + hα n ) = haq n e r n − 1 + [ − 2 hα + ha ( e q n e r n − 1 + q n r n − 1 )] Λ n + ha e q n r n − 1 Λ 2 n , (1 − ha e q n r n − 1 Λ n + hα n ) − Λ n (1 − hbq n − 1 e r n Λ n + hβ n ) = hb e q n − 1 r n + [2 hα + hb ( e q n − 1 e r n + q n − 1 r n )] Λ n + hbq n − 1 e r n Λ 2 n , (1 − ha e q n r n − 1 Λ n + hα n ) (1 − hbq n − 1 e r n Λ n + hβ n ) = − h 2 ab (1 − q n r n − e q n − 1 e r n − 1 ) − h 2 ab (1 − e q n e r n − q n − 1 r n − 1 ) Λ 2 n +  h 2 ab (2 + q n e r n + e q n r n + e q n − 1 r n − 1 + q n − 1 e r n − 1 ) + (1 + hα ) ( 1 − hα )  Λ n , (3.51) with c n = − a Λ n and d n = − b Λ n . Using the ﬁrst t w o relatio ns, w e can express 1 − ha e q n r n − 1 Λ n + hα n and 1 − hbq n − 1 e r n Λ n + hβ n in terms of Λ n . Substitut- ing these expressions in to the third relation, w e obtain a sixth-degree equation in only Λ n . How ev er, this sextic equation turns o ut to b e t o o complicated for exact treatmen t. Th us, w e need to tak e a diﬀeren t route. One se nsible approac h is to solv e the algebraic system (3.51) appro ximately; for exam- ple, using p erturbativ e expansions with resp ect to h , we o btain approxim ate expressions for the auxiliary v ariables as follo ws:              α n − a e q n r n − 1 Λ n = α − a 2 ( e q n + q n ) ( e r n − 1 + r n − 1 ) + O ( h ) , β n − bq n − 1 e r n Λ n = − α − b 2 ( e q n − 1 + q n − 1 ) ( e r n + r n ) + O ( h ) , Λ n = 1 − ha 2 ( e q n + q n ) ( e r n − 1 + r n − 1 ) − hb 2 ( e q n − 1 + q n − 1 ) ( e r n + r n ) + O ( h 2 ) . (3.52) Com bining (3.5 2) with (3 .23) for c = − a , d = − b , and β = − α , we obtain a high- precision nume rical sc heme that is “almost in tegrable” (cf. [3 , 5, 41]). W e can also consider the next order in h and include higher-order corrections. W e now discuss the general case where no particular conditions o n the parameters a , b , c , d , α , and β are assumed. Instead of exploiting the compli- cated alg ebraic system (3.25), w e explain how to recreate the discrete -t ime Ablo witz–Ladik lattice (3.2 3) by the comp osition o f a f ew more elemen tary time ev olutions suc h as those studied earlier for the sp ecial cases a = b = 0 or c = d = 0. T o this end, w e recall some basic prop erties of in tegrable hierar- c hies. That is, under suitable b oundary conditions, an integrable hierarc h y generally comprises inﬁnitely man y comm uting ﬂo ws. Any comp osition of these ﬂows also b elong s to the same hierarc h y; eac h of the ﬂows is uniquely determined b y sp ecifying its linear part, or equiv alently , its disp ersion rela- tion. Under the b oundary conditions (3.24) or, equiv alently , the streamlined 33 v ersion, the discrete-time Ablowitz–Ladik lattice (3 .2 3) has the following lin- ear leading part:      1 h ( e q n − q n ) + c e q n +1 + β e q n + b e q n − 1 − aq n +1 − αq n − dq n − 1 ∼ 0 , 1 h ( e r n − r n ) + d e r n +1 + α e r n + a e r n − 1 − br n +1 − β r n − cr n − 1 ∼ 0 . (3.53) Substituting the ansatz q n ∼ λ 2 n , e q n ∼ λ 2 n ω , r n ∼ λ − 2 n , e r n ∼ λ − 2 n ω − 1 in to (3.53), w e obta in the corresponding disp ersion relation [3] ω  λ 2  = 1 + h  aλ 2 + α + d λ 2  1 + h  cλ 2 + β + b λ 2  . (3.54) Note that the Lax matrix V n (3.22) approac hes a diagonal matrix as n → ±∞ (cf. (3.24)), and ω ( λ 2 ) is equal to the limit v alue of the ratio b et w een the diagonal eleme nts [3]; this is natural b ecause the n -indep enden t ov erall factor of V n pla ys no role in the zero-curv ature condition (2.6). F or notational con v enience, w e express the forward time ev olution deﬁned b y (3.2 3) with (3 .2 4) as the mapping f ( a, b, c, d, α + 1 h , β + 1 h ) : ( q n , r n ) 7→ ( e q n , e r n ) . Note that only the ratio of the six argumen ts in f matters in iden tifying the corresp onding ﬂow as w ell as its linear part. Let us discuss how to factorize this mapping in to a comp o sition of simpler mappings b elonging to the same class. A rigo rous and p ersuasiv e discussion can b e giv en on the basis of the scattering dat a used in the inv erse scattering formalism. The scattering data are space-indep endent quan tities that ha ve very simple time dep endences de- termined b y the disp ersion relation, typically , e R ( λ ) = ω ( λ 2 ) R ( λ ) [3]. Th us, their time ev olution is essen tially linear and allo ws a comm ut a tiv e superp o - sition of diﬀeren t ﬂow s. Con v ersely , the decomp o sition of a complex ﬂow in to elemen tary ﬂows can b e uncov ered by examining the factor izat io n of the disp ersion relation, that is, whether ω ( λ 2 ) can b e factorized in to a pro duct of simpler quan tities suc h as ω 1 ( λ 2 ) ω 2 ( λ 2 ) or ω 1 ( λ 2 ) ω 2 ( λ 2 ) ω 3 ( λ 2 ). If the reader is unfamiliar with the in v erse scattering metho d, the same guideline can b e deriv ed by noting the one-to-one corresp ondence b et w een (the linear part of ) a ﬂo w and its dispersion relation. One can also understand this guideline “honestly” at the leve l of the asso ciated linear problem (cf. t he second equation in (2.5) and the sen tence b eneath (3.54)); that is, b y consid- ering whether the Lax matrix V n can b e factorized in to a pro duct of simpler matrices suc h as V (2) n ( λ ) V (1) n ( λ ) or V (3) n ( λ ) V (2) n ( λ ) V (1) n ( λ ) [1, 6, 7, 19 ]. 34 By elemen tary computations, the righ t-hand side of (3.54) can b e fa cto r - ized in to the pro duct of tw o or three simpler quan tities (as functions of λ ) as follow s: 1 + h  aλ 2 + α + d λ 2  1 + h  cλ 2 + β + b λ 2  = 1 + hα + p (1 + hα ) 2 − 4 h 2 ad + 2 haλ 2 1 + hβ + p (1 + hβ ) 2 − 4 h 2 bc + 2 hb λ 2 × 1 + 2 hd/λ 2 1+ hα + √ (1+ hα ) 2 − 4 h 2 ad 1 + 2 hcλ 2 1+ hβ + √ (1+ hβ ) 2 − 4 h 2 bc (3.55a) = 1 + 2 haλ 2 1+ hα + √ (1+ hα ) 2 − 4 h 2 ad 1 + 2 hb/λ 2 1+ hβ + √ (1+ hβ ) 2 − 4 h 2 bc × 1 + hα + p (1 + hα ) 2 − 4 h 2 ad + 2 hd λ 2 1 + hβ + p (1 + hβ ) 2 − 4 h 2 bc + 2 hcλ 2 (3.55b) = 1 + hα + p (1 + hα ) 2 − 4 h 2 ad 1 + hβ + p (1 + hβ ) 2 − 4 h 2 bc × 1 + 2 haλ 2 1+ hα + √ (1+ hα ) 2 − 4 h 2 ad 1 + 2 hb/λ 2 1+ hβ + √ (1+ hβ ) 2 − 4 h 2 bc × 1 + 2 hd/λ 2 1+ hα + √ (1+ hα ) 2 − 4 h 2 ad 1 + 2 hcλ 2 1+ hβ + √ (1+ hβ ) 2 − 4 h 2 bc . (3.55c) Th us, the aforemen tioned g uideline implies the corresp onding decompo sition of the mapping f ( a, b, c, d, α + 1 h , β + 1 h ) in to simpler ones, i.e., f ( a, b, c, d, α + 1 h , β + 1 h ) = f ( a, b, 0 , 0 , α ′ + 1 h , β ′ + 1 h ) ◦ f (0 , 0 , c ′ , d ′ , 1 h , 1 h ) (3.56a) = f ( a ′ , b ′ , 0 , 0 , 1 h , 1 h ) ◦ f (0 , 0 , c, d , α ′ + 1 h , β ′ + 1 h ) (3.56b) = f (0 , 0 , 0 , 0 , α ′ + 1 h , β ′ + 1 h ) ◦ f ( a ′ , b ′ , 0 , 0 , 1 h , 1 h ) ◦ f (0 , 0 , c ′ , d ′ , 1 h , 1 h ) , (3.56c) where a ′ := 2 a 1 + hα + p (1 + hα ) 2 − 4 h 2 ad , b ′ := 2 b 1 + hβ + p (1 + hβ ) 2 − 4 h 2 bc , c ′ := 2 c 1 + hβ + p (1 + hβ ) 2 − 4 h 2 bc , d ′ := 2 d 1 + hα + p (1 + hα ) 2 − 4 h 2 ad , α ′ := p (1 + hα ) 2 − 4 h 2 ad − 1 + hα 2 h , β ′ := p (1 + hβ ) 2 − 4 h 2 bc − 1 + hβ 2 h . 35 Note that the o rder o f comp osition in (3.56) do es not matt er, as the or der of multiplication in (3.55) do es not matter. In terestingly , the time ev olu- tion deﬁned on the six p oin ts ( n + i, m + j ), i = − 1 , 0 , 1 and j = 0 , 1, is now written a s the comp osition of tw o time ev olutions, eac h of whic h is a previ- ously described f our-p oint sc heme, and, if a ny (cf. (3.56c)), one trivial time ev olution. Moreo v er, if h is real, b = a ∗ , d = c ∗ , and β = α ∗ , the original mapping f ( a, b, c, d, α + 1 h , β + 1 h ) allows the reduction of the complex conju- gate r n = σ q ∗ n ( σ : real), and this reduction is consisten t with eve ry component mapping in v olve d in (3.56). Using the disp ersion relation (3.54), w e can readily ﬁnd the in v erse map- ping of f ( a, b, c, d, α + 1 h , β + 1 h ). Indeed, the trivial iden tity 1 + h  aλ 2 + α + d λ 2  1 + h  cλ 2 + β + b λ 2  × 1 + h  cλ 2 + β + b λ 2  1 + h  aλ 2 + α + d λ 2  = 1 implies the non trivial form ula f − 1 ( a, b, c, d, α + 1 h , β + 1 h ) = f ( c, d, a, b, β + 1 h , α + 1 h ) . In particular, this conﬁrms the already men tioned corresp ondence b etw een the case a = b = 0 and the case c = d = 0 through the time reﬂection. Actually , the righ t-hand side of (3.54) is a lready the pro duct of the n u- merator and the inv erse of the denominator; this implies the decomp osition f ( a, b, c, d, α + 1 h , β + 1 h ) = f ( a, 0 , 0 , d , α + 1 h , 1 h ) ◦ f (0 , b, c, 0 , 1 h , β + 1 h ) = f (0 , b, c, 0 , 1 h , β + 1 h ) ◦ f ( a, 0 , 0 , d, α + 1 h , 1 h ) . Th us, using this form ula, one can f urther decomp ose (3.56) in to “ elemen tary” ﬂo ws (cf. [1, 7 , 19]); ho w ev er, suc h ﬂo ws do not main tain the complex conju- gacy relation b etw een q n and r n , whic h may b ecome a serious b ottlenec k in practical applications. 3.3 The V olterra lattice The V olterra lattice u n,t = u n ( u n +1 − u n − 1 ) (3.57) is obtained from the Ablo witz–Ladik lattice (3.19) through the reduction a = b = 1, q n = u n − 1, and r n = − 1. Th us, the time discretization of the V olterra lattice can b e obtained from the discrete -t ime Ablo witz–Ladik lat- tice in subsection 3.2 through the r eduction q n = u n − 1 and r n = − 1 to - gether with a = b , α n = β n +1 , and c n = d n . The discrepancy b et w een this 36 reduction and the deca ying b oundary conditions (3.2 4) is nonesse ntial in this regard. Ho we v er, in the f ollo wing, w e prefer to consider the V olterra lattice indep enden tly as an in teresting example. The La x pair for the con tin uous- time V o lterra lattice (3.57) is giv en by [1, 42] L n =  λ u n − 1 0  , (3.58a) M n =  λ 2 + u n λu n − λ u n − 1  . ( 3 .58b) Indeed, the substitution of (3.58) in to the zero-curv ature condition (2.2) results in (3.57). In t he discrete-time case, w e assume the Lax ma t r ix V n of the follo wing form: V n = I + h  λ 2 a + α n + e u n b n λ ( au n + e u n b n ) − λ ( a + b n ) − λ 2 b n + α n − 1 + e u n − 1 b n − 1  . (3.59) Here, α n and b n are auxiliary v ariables. The zero-curv ature condition (2.6) for the Lax pair (3.58a) and (3.59) is equiv alen t to the follo wing sy stem of partial diﬀerence equations (cf. [5]):        1 h ( e u n − u n ) = α n +1 u n − e u n α n − 1 + e u n +1 b n +1 u n − e u n e u n − 1 b n − 1 , α n +1 − α n = a ( u n +1 − e u n ) , e u n b n = b n +1 u n . (3.60) The general form (3.60) o f the time-discretized V olterra lattice is in tegrable for matr ix- v alued dep enden t v ariables, but in the fo llo wing, w e consider only the case of scalar dep enden t v a riables. In view of (2.9), w e imp o se the fol- lo wing b oundary conditions for u n , α n , and b n : lim n →±∞ u n = 1 , lim n →±∞ α n = 0 , lim n →±∞ b n = b. (3.61) The b o undary v alue of u n is normalized b y scaling a and b , and the bo undary v alue of α n is set as zero b y redeﬁning h . T o b e precise, the b oundary conditions (3 .61) con tain r e dundant inf o rmation. Indeed, it can b e sho wn that the a uxiliary v ariables hav e the same limit v alues for n → −∞ and n → + ∞ . Therefore, it is suﬃcien t to assume either lim n →−∞ ( α n , b n ) = (0 , b ) or lim n → + ∞ ( α n , b n ) = (0 , b ). In the contin uum limit of time h → 0, w e obtain α n → au n − a and b n → b . Th us, in this limit, the time discretization (3.60) reduces to u n,t = ( a + b ) u n ( u n +1 − u n − 1 ) , 37 whic h is, if a + b 6 = 0, equiv alent to the V olterra la t t ice (3.57), up to a redef- inition of t . Note that in the case a + b = 0, (3.60) has the trivial solution e u n = u n , α n = au n − a , b n = b . The determinant of the 2 × 2 Lax matrix L n (3.58a) can be immediately computed a s det L n = u n . Using the recurrence f o rm ulas fo r α n and b n in (3.60), w e can rewrite the La x matrix V n (3.59) as an ultralo cal f orm with resp ect to the auxiliary v ariables α n and b n . Th us, its 2 × 2 determinan t is computed as det V n ( λ ) =  1 + h  λ 2 a + α n + e u n b n   1 + h  − λ 2 b n + α n − au n + a e u n − 1 + b n u n − 1  + h 2 λ 2 ( au n + e u n b n ) ( a + b n ) = − λ 4 h 2 ab n + λ 2 h  ( a − b n ) (1 + hα n ) + ha 2 e u n − 1 + hab n ( u n − 1 + e u n + u n )  + [1 + h ( α n + e u n b n )] [1 + h ( α n − au n + a e u n − 1 + b n u n − 1 )] . Therefore, the equalit y (2.10) com bined with the b oundary conditions (3.61) (or the streamlined ve rsion as stated ab o v e) implies the set of relations            b n = b Λ n , ( a − b n ) (1 + hα n ) + ha 2 e u n − 1 + hab n ( u n − 1 + e u n + u n ) =  a − b + ha 2 + 3 hab  Λ n , [1 + h ( α n + e u n b n )] [1 + h ( α n − au n + a e u n − 1 + b n u n − 1 )] = (1 + hb ) 2 Λ n . (3.62) Here, Λ n is the quan tit y t ha t satisﬁes the recurrence form ula e u n Λ n = u n Λ n +1 , (3.63) and has t he normalize d b o undar y v alue lim n →±∞ Λ n = 1. More precisely , only one of the tw o conditions lim n →−∞ Λ n = 1 and lim n → + ∞ Λ n = 1 is re- quired, in accordance with the b o undar y conditions for α n and b n . Then, the other condition can b e conﬁrmed. Thus , Λ n can b e written explicitly (but globally) as Λ n = n − 1 Y j = −∞ e u j u j = + ∞ Y j = n u j e u j . Note that the ﬁrst equalit y in (3.6 2) is v alid ev en if a = 0 (cf. (3.60) ). The algebraic system (3.62) essen tially comprises tw o non trivial relatio ns f o r the t w o unkno wns α n and Λ n . Because of the no nzero b o undar y v alue of u n (3.61), it is rather cum- b ersome to iden tify t he linear leading part in (3.6 0) and the corresponding 38 disp ersion relation (cf. subsection 3.2). Instead, w e can compute the asymp- totic form of the Lax matrix V n (3.59) as n → −∞ or n → + ∞ and consider its factorizatio n (cf. [1, 6, 7, 19]). The r esults imply that the cases a = 0 and b = 0 form the basis for the general case; that is, the time ev olution for gen- eral a and b is equiv alen t to the order-indep enden t comp osition of t w o time ev olutions corresp onding to a = 0 and b = 0, resp ectiv ely . In addition, these t w o cases are related to eac h other through the space/time reﬂection, a s w e will see b elo w. No w, w e consider the tw o fundamen tal cases: a = 0 or b = 0 . • The c ase a = 0, b 6 = 0. In this case, the auxiliary v ariable α n v anishes. Th us, the discrete-time system (3.60) reduces to    1 h ( e u n − u n ) = b Λ n +1 u n ( e u n +1 − u n − 1 ) , e u n Λ n = u n Λ n +1 , lim n →−∞ Λ n = 1 or lim n → + ∞ Λ n = 1 . (3.64) Note that in the case hb = − 1, (3.64) has the trivial solution e u n = u n − 1 , Λ n = 1 /u n − 1 . The algebraic syste m (3.6 2) simpliﬁes to a quadratic equation in Λ n , (1 + hb e u n Λ n ) (1 + hbu n − 1 Λ n ) = (1 + hb ) 2 Λ n , or equiv alen tly , ( hb ) 2 e u n u n − 1 Λ 2 n −  (1 + hb ) 2 − hb ( e u n + u n − 1 )  Λ n + 1 = 0 . (3.65) The asymptotic behavior (2.8 ) of the Lax ma t rix V n implies that the prop er solution of (3.65) is giv en by (1 + hb ) 2 Λ n = 2 1 − ǫ ( e u n + u n − 1 ) + q [1 − ǫ ( e u n + u n − 1 )] 2 − 4 ǫ 2 e u n u n − 1 , (3.66) where ǫ := hb/ (1 + hb ) 2 . Thus , if hb ∈ R , then ǫ ≤ 1 / 4. In this case, the lo cal ex pression (3.66) is v alid only if − 1 < hb < 1, whic h cov ers the range ǫ < 1 / 4. The case hb = 1, corresp onding to ǫ = 1 / 4, inv olve s a subtle sign problem of the square ro ot, whic h w e do not discuss here. If ( hb ) 2 > 1, (3.66) is inconsisten t with the b o undar y conditions, and the o t her solution of (3.65) should b e adopt ed. In a n y case, the b oundary conditions for u n imply that lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. As hb approaches 1, more restrictiv e conditions ha v e to b e imp osed on the u n to ensure a consisten t choice of the solution of (3.65). 39 Substituting (3.66 ) in to the ﬁrst equation in (3.64 ) , w e obtain a time discretization of the V olterra lattice (3.57) in the lo cal form, e u n − u n ǫ = 2 u n ( e u n +1 − u n − 1 ) 1 − ǫ ( e u n +1 + u n ) + q [1 − ǫ ( e u n +1 + u n )] 2 − 4 ǫ 2 e u n +1 u n . (3.67) Th us, t he v alue of e u n is uniquely determined by u n , e u n +1 , and u n − 1 . Note that (3.67) can b e rewritten a s e u n − u n ǫ = 2 ( e u n +1 e u n − u n u n − 1 ) 1 + ǫ ( e u n +1 − u n ) + q [1 + ǫ ( e u n +1 − u n )] 2 − 4 ǫ e u n +1 . (3.68) If hb ∈ R a nd the u n are nonzero a nd real-v alued at the initial time, then (3.64) implies that the auxiliary v ariable Λ n is alw a ys real-v alued. Con- sequen tly , the discriminan t of the quadratic equation (3.65) mu st b e non- negativ e. Th us, if ǫ < 1 / 4, u n approac hes 1 suﬃcien tly smo othly and fast as n → ±∞ , and the u n are close t o 1 at the initial time, then the real- v aluedness of u n is preserv ed under the t ime ev olution of the discrete-time V olterra la ttice (3.67) or (3.68). T o expres s the backw ard time ev olution explicitly , w e only hav e to replace Λ n +1 u n in the ﬁrst equation of (3.64) with Λ n e u n and then substitute the lo cal expression (3.66). The resulting equation is e u n − u n ǫ = 2 e u n ( e u n +1 − u n − 1 ) 1 − ǫ ( e u n + u n − 1 ) + q [1 − ǫ ( e u n + u n − 1 )] 2 − 4 ǫ 2 e u n u n − 1 . (3.69) • The c ase b = 0, a 6 = 0. In this case, the auxiliary v ariable b n v anishes. Th us, t he discrete -time system (3.60) reduces to    1 h ( e u n − u n ) + α n ( e u n − u n ) = a ( u n +1 u n − e u n e u n − 1 ) , α n +1 − α n = a ( u n +1 − e u n ) , lim n →−∞ α n = 0 or lim n → + ∞ α n = 0 . (3.70) Note that in the case ha = − 1, (3.70) has the trivial solution 1 + hα n = u n +1 , e u n = u n +2 . Th us, w e can assume ha 6 = − 1. The algebraic system (3.62) simpliﬁes to ( 1 + hα n + ha e u n − 1 = (1 + ha ) Λ n , (1 + hα n ) (1 + hα n − hau n + ha e u n − 1 ) = Λ n . (3.71) 40 By eliminating Λ n , (3.71) reduces to a quadratic equation in 1 + hα n , (1 + ha ) (1 + hα n ) 2 − [1 + ha (1 + ha ) ( u n − e u n − 1 )] (1 + hα n ) − ha e u n − 1 = 0 . The prop er solution o f this quadratic equation is giv en by 1 + hα n = 1 + F n − G n + q (1 + F n − G n ) 2 + 4 G n 2 (1 + ha ) , (3.72) with F n := ha (1 + ha ) u n , G n := ha (1 + ha ) e u n − 1 . When ha ∈ R , the lo cal express ion (3 .72) is v alid only if ha > − 1 / 2; the b orderline case ha = − 1 / 2 is excluded from o ur consideration. If ha < − 1 / 2, the other solutio n of t he quadratic equation should b e emplo y ed. Unless ha = − 1 / 2, a uniﬁed express ion for 1 + hα n , 1 + hα n = 1 + F n − G n + (1 + 2 ha ) q  1+ F n − G n 1+2 ha  2 + 4 G n (1+2 ha ) 2 2 (1 + ha ) , can res olve the sign pro blem of the square ro ot, but it is un wieldy and lo oks unattractive . Note that the b o undar y conditions for u n imply that lim n →−∞ α n = lim n → + ∞ α n = 0, and consequen tly , lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. Substituting the lo cal express ion (3 .72) for 1 + hα n in to the ﬁrst equa- tion of (3.70), w e obtain a time discretization o f the V olterra lattice (3.57). In t erms of the new parameter δ := ha (1 + ha ), w e can write this time dis- cretization as e u n − u n δ = 2 ( u n +1 u n − e u n e u n − 1 ) 1 + δ ( u n − e u n − 1 ) + q [1 + δ ( u n − e u n − 1 )] 2 + 4 δ e u n − 1 . (3.73) Th us, e u n is uniquely determined b y u n , u n +1 , and e u n − 1 . Note that if ha > − 1 / 2, then δ > − 1 / 4. If this inequalit y is satisﬁed, u n approac hes 1 suﬃcien tly smo othly and fast as n → ±∞ , and the u n are close to 1 at the initial time, then the real-v aluedness of u n is preserv ed under the time ev olutio n (3.73). T o obta in the backw ard time ev olution explicitly , w e rewrite α n in the ﬁrst equation of (3.70) as α n +1 − a ( u n +1 − e u n ) and then substitute the lo cal expression (3.72). The resulting equation is e u n − u n δ = 2 ( u n +1 u n − e u n e u n − 1 ) 1 − δ ( u n +1 − e u n ) + q [1 + δ ( u n +1 − e u n )] 2 + 4 δ e u n . (3.74) 41 It is easy to see that (3.73) is equiv alen t to (3.68) (or (3.69)) through the space (or time) reﬂection and the identiﬁcation δ ↔ − ǫ . In the same man- ner, (3.74) can be iden tiﬁed with (3.68) through the time reﬂec tion. Th us, the forw ard/back w ard t ime evolution in the case b = 0 corresp onds to the bac kw ard/forward time ev o lutio n in the case a = 0, up to a redeﬁnition of the parameters. Ultr adiscr etization [12, 13] . W e prop ose an ultradiscrete analo gue of the time- discretized V olterra lattice in the case b = 0. F or the forw ard time ev olution, w e ﬁrst rewrite (3 .73) as e u n u n = 1 + δ ( u n − e u n − 1 ) + 2 δ u n +1 + q 1 + 2 δ ( u n + e u n − 1 ) + δ 2 ( u n − e u n − 1 ) 2 1 + δ ( u n + e u n − 1 ) + q 1 + 2 δ ( u n + e u n − 1 ) + δ 2 ( u n − e u n − 1 ) 2 . (3.75) Then, w e assume δ > 0 so that the p ositivit y of the dep enden t v ariable, u n > 0, ∀ n ∈ Z , can b e preserv ed in the t ime evolution. By reparametrizing the parameter a nd the dep enden t v ariable a s δ =: exp( − L/ε ) and u n =: exp( U n /ε ), resp ectiv ely , and taking the logarithm, the ab o v e equation b ecomes e U n − U n = ε log  1 + e U n − L ε − e e U n − 1 − L ε + 2 e U n +1 − L ε + p X n  − ε log  1 + e U n − L ε + e e U n − 1 − L ε + p X n  , (3.76a) X n := 1 + 2 e U n − L ε + 2 e e U n − 1 − L ε +  e U n − L ε − e e U n − 1 − L ε  2 . (3.76b) F or brevit y , the ε -dep endenc e of the U n is suppressed. When taking the limit ε → +0, the w ell-kno wn f orm ula [13, 4 3, 44], lim ε → +0 ε lo g M X j =1 e A j ( ε ) ε ! = max 1 ≤ j ≤ M ( A j ) , where A j ( ε ) are real functions a llo wing the one-sided limit lim ε → +0 A j ( ε ) =: A j , do es no t apply directly; how eve r, the core idea b ehind this for mula is still v alid. After simple consideration of the cases U n R e U n − 1 when U n > L , (3.76) is sho wn to reduce, in the limit ε → +0, to e U n − U n = max  f ( U n , e U n − 1 ) , U n +1 − L  − max  0 , U n − L, e U n − 1 − L  . (3.77) 42 Here, U n := lim ε → +0 U n ( ε ) is a real v ariable, and the b oundary conditio ns lim n →±∞ u n = 1 translate in to lim n →±∞ U n = 0. The function f ( U n , e U n − 1 ) is deﬁned b y f ( U n , e U n − 1 ) := lim ε → +0 ε lo g  1 + e U n − L ε − e e U n − 1 − L ε + p X n  =      U n − L, U n > e U n − 1 and U n > L e U n − 1 − U n − 1 + g ( e U n − 2 , U n − 1 ) , U n = e U n − 1 > L 0 , U n < e U n − 1 or U n ≤ L ≥ 0 , where g ( e U n − 2 , U n − 1 ) is deﬁned in a similar manner as g ( e U n − 2 , U n − 1 ) := lim ε → +0 ε lo g  1 + e e U n − 2 − L ε − e U n − 1 − L ε + p X n − 1  =      e U n − 2 − L, e U n − 2 > U n − 1 and e U n − 2 > L f ( U n − 2 , e U n − 3 ) , e U n − 2 = U n − 1 > L 0 , e U n − 2 < U n − 1 or e U n − 2 ≤ L ≥ 0 . One migh t naiv ely think that f ( U n , e U n − 1 ) at U n = e U n − 1 > L is give n b y ( U n − L ) / 2; ho w ev er, this is not true in general, b ecause the equalit y U n (+0) = e U n − 1 (+0) do es not imply U n ( ε ) ≡ e U n − 1 ( ε ), 0 < ε ≪ 1. As indicated ab o v e, the correct v alue of f ( U n , e U n − 1 ) can b e computed recursiv ely using the form ulas f ( U n +1 , e U n ) = e U n − U n + g ( e U n − 1 , U n ) , g ( e U n , U n +1 ) = f ( U n , e U n − 1 ) . (3.78) W e no w brieﬂy explain how these formulas can b e deriv ed in the most general case, i.e., without any assumptions on the ar g umen ts suc h as U n +1 = e U n . Note that (3.78) can b e rega r ded a s a conserv ation law . Th us, in view o f the zero b o undary conditions for U n , the follo wing global expres sions are v alid if L > 0 : f ( U n +1 , e U n ) = ∞ X j =0  e U n − 2 j − U n − 2 j  = ∞ X j =1  U n +2 j − e U n +2 j  , g ( e U n , U n +1 ) = ∞ X j =0  e U n − 2 j − 1 − U n − 2 j − 1  = ∞ X j =1  U n +2 j − 1 − e U n +2 j − 1  . 43 T o v erify (3.78), w e compare (3.75) with (3 .63), wherein the expression for Λ n is giv en by the ﬁrst equation in (3.71) with (3.72). This results in the non trivial relation 1 + δ ( e u n − u n +1 ) + q 1 + 2 δ ( u n +1 + e u n ) + δ 2 ( u n +1 − e u n ) 2 = 1 + δ ( u n − e u n − 1 ) + q 1 + 2 δ ( u n + e u n − 1 ) + δ 2 ( u n − e u n − 1 ) 2 . It is a n easy exercise to rewrite this relation by rationalizing its n umerator as 1 + δ ( u n +1 − e u n ) + q 1 + 2 δ ( u n +1 + e u n ) + δ 2 ( u n +1 − e u n ) 2 = e u n u n  1 + δ ( e u n − 1 − u n ) + q 1 + 2 δ ( u n + e u n − 1 ) + δ 2 ( u n − e u n − 1 ) 2  . By substituting δ = exp( − L/ε ) and u n = exp( U n /ε ), and taking the loga- rithm, the ab ov e tw o equalities reduc e, in the limit ε → +0, to the sec ond and ﬁrst equalities in (3.78), resp ectiv ely . The par a meter L ma y dep end on t he discrete time co ordinate m ∈ Z . The c om mutativity of the ultradiscrete ﬂo ws deﬁned b y (3.77) fo r distinct v alues of L remains to b e established. In this connection, we conjecture that (3.77) and its time rev ersal (see b elo w) comprise an ultra discrete analogue o f the KdV hierar c h y , wherein the parameter L lab els each ﬂo w in the hierarch y . It is also b eyond the scope of this pap er to in v estigate the relationship b et w een the ultradiscretized V olterra lattice (3.77) and the “b ox and ball system” of T ak ahashi and Satsuma [45]. F or the bac kw ard time ev olution, we rewrite ( 3 .74) as u n e u n = 1 + δ ( e u n − u n +1 ) + 2 δ e u n − 1 + q 1 + 2 δ ( e u n + u n +1 ) + δ 2 ( e u n − u n +1 ) 2 1 + δ ( e u n + u n +1 ) + q 1 + 2 δ ( e u n + u n +1 ) + δ 2 ( e u n − u n +1 ) 2 . Th us, in the same wa y as that for the forw ard time ev olution, w e obtain an ultradiscrete analo gue of (3.74), U n − e U n = max  g ( e U n , U n +1 ) , e U n − 1 − L  − max  0 , e U n − L, U n +1 − L  , whic h is related to (3.77) through the com bined space a nd t ime reﬂection ( cf. (3.74) and (3.73)). 44 3.4 The mo diﬁed V olterra latti ce The mo diﬁed V o lterra lattice q n,t = (1 + q 2 n )( q n +1 − q n − 1 ) , (3.79) where q n is a scalar dep endent v ariable, w as in tro duced b y Hirota [4 6 ] (also see [4, 47, 4 8]); it is obtained f rom the Ablow itz–Ladik system ( 3 .19) through the reduction a = b = 1 and r n = − q n . The o v erall co eﬃcien t of the nonlinear terms, including its sign, is nonessen tial at the lev el o f the equation a nd the asso ciated Lax pair, b ecause it can b e c hanged by r escaling q n . Note that (3.79) is in v a rian t under the tr ansformation q n 7→ ( − 1) n q n , t 7→ − t . The La x pair for the con tinuous -time mo diﬁed V olterra lattice (3.79) is giv en by L n =  λ q n − q n 1 λ  , (3.80a) M n =  λ 2 + q n q n − 1 λq n + 1 λ q n − 1 − λq n − 1 − 1 λ q n q n q n − 1 + 1 λ 2  . (3.80b) Indeed, the substitution of (3.80) in to the zero-curv ature condition (2.2) results in (3.79). In the discrete-time case, we consider the reduction a = b , α n = β n , c n = d n , and r n = − q n of the Lax matrix V n (3.22). Th us, w e obtain V n = I + h      λ 2 a − e q n q n − 1 c n + α n + 1 λ 2 c n λ ( aq n − e q n c n ) + 1 λ ( a e q n − 1 − c n q n − 1 ) λ ( − a e q n − 1 + c n q n − 1 ) + 1 λ ( − aq n + e q n c n ) λ 2 c n − e q n q n − 1 c n + α n + a λ 2      , (3.81) where α n and c n are the auxiliary v ariables. The zero-curv ature condition (2.6) for the Lax pair (3.80a) and (3.81) amoun ts to the follow ing system of partial diﬀerence equations:              1 h ( e q n − q n ) − aq n +1 + a e q n − 1 − α n +1 q n + e q n α n + e q n +1 c n +1 (1 + q 2 n ) − ( 1 + e q 2 n ) c n q n − 1 = 0 , α n +1 − α n = a ( − e q n e q n − 1 + q n +1 q n ) , (1 + e q 2 n ) c n = (1 + q 2 n ) c n +1 . (3.82) The discussion in subsection 3 .2 implies that the general case of a 6 = 0 and c n 6 = 0 is equiv alen t to the comp osition of t wo simpler cases: the case a = 0 45 and α n = 0 and the case c n = 0. Moreov er, t he latter case can b e related with the former case through the time reﬂection. Th us, in this subsection, w e only consider t he elemen tary case a = 0, α n = 0, and c n = − Λ n under the b oundary conditions lim n →±∞ q n = 0 , lim n →±∞ Λ n = 1 . (3.83) Note that the b oundary v alue of c n is set as − 1 without an y loss of generalit y; it can b e c hanged t o an y nonzero v alue b y rescaling the “step size” parameter h . T o b e precise, the b oundary conditions (3 .83) con tain r e dunda nt informa- tion. Indeed, it can b e sho wn that Λ n has the same limit v alue for n → −∞ and n → + ∞ . Therefore, it is suﬃcien t to assume either lim n →−∞ Λ n = 1 or lim n → + ∞ Λ n = 1. No w, the g eneral system (3.82) reduces to the simpler system    1 h ( e q n − q n ) = (1 + q 2 n )Λ n +1 ( e q n +1 − q n − 1 ) , (1 + e q 2 n )Λ n = (1 + q 2 n )Λ n +1 . (3.84) The second relation in (3.8 4) implies that the global exp ressions for the auxiliary v ariable Λ n in terms of the q n are Λ n = n − 1 Y j = −∞ 1 + e q 2 j 1 + q 2 j = + ∞ Y j = n 1 + q 2 j 1 + e q 2 j . The substitution of eac h expression in to the ﬁrst relatio n in (3.84) pro vides a global-in-space time discretization of the mo diﬁed V olterra lattice (3.79). If h ∈ R and the q n are real-v alued at the initia l time, then the real-v aluedness of q n is preserv ed under the discrete-time evolution. Let us resolv e this nonlo cality . The determinan t of the 2 × 2 Lax matrix L n (3.80a) is giv en by det L n = 1 + q 2 n , while t he determinan t of the Lax matrix V n (3.81) with a = 0, α n = 0, and c n = − Λ n is computed as det V n ( λ ) = −  λ 2 + 1 /λ 2  h Λ n + h 2  1 + e q 2 n   1 + q 2 n − 1  Λ 2 n + 2 h e q n q n − 1 Λ n + 1 . Th us, the equality (2.10) com bined with the b oundary conditions (3.83) (or the streamlined v ersion as stated ab ov e) leads to the quadratic equation in Λ n , h 2  1 + e q 2 n   1 + q 2 n − 1  Λ 2 n −  1 + h 2 − 2 h e q n q n − 1  Λ n + 1 = 0 . (3.85) 46 By recalling the prescribed asymp totic b ehavior (2.8) of the Lax matrix V n for small h , w e obtain the prop er solution of this quadratic equation in Λ n , Λ n = 2 1 + h 2 − 2 h e q n q n − 1 + q (1 + h 2 − 2 h e q n q n − 1 ) 2 − 4 h 2 (1 + e q 2 n )  1 + q 2 n − 1  = 2 1 + h 2 − 2 h e q n q n − 1 + q (1 − h 2 ) 2 − 4 h ( e q n + hq n − 1 ) ( h e q n + q n − 1 ) . (3.86) When h ∈ R , the lo cal expression (3.86) is v alid only if − 1 ≤ h ≤ 1. If h 2 > 1, the other solution of (3.85) should b e used. Unless h = ± 1, a uniﬁed expres - sion for Λ n , Λ n = 2 1 + h 2 − 2 h e q n q n − 1 + (1 − h 2 ) q 1 − 4 h (1 − h 2 ) 2 ( e q n + hq n − 1 ) ( h e q n + q n − 1 ) , can resolv e the sign problem of the square ro ot, but w e do not use this form. In an y cas e, the decay ing b oundary conditions for q n imply that lim n →−∞ Λ n = lim n → + ∞ Λ n = 1. Th us, the b oundary conditions for Λ n giv en in (3.83) are compatible. Substituting (3.86 ) in to the ﬁrst equation in (3.84 ) , w e obtain a time discretization of the mo diﬁed V olterra lattice (3.79) in the lo cal fo rm, 1 h ( e q n − q n ) = 2 (1 + q 2 n ) ( e q n +1 − q n − 1 ) 1 + h 2 − 2 h e q n +1 q n + q (1 − h 2 ) 2 − 4 h ( e q n +1 + hq n ) ( h e q n +1 + q n ) . (3 .87) Note that the right-hand side of (3.87) do es not inv olve e q n . Similarly to the con tin uous-time case, (3.87) (or (3.84)) is in v arian t under the transformation q n 7→ ( − 1) n q n , h 7→ − h . If − 1 ≤ h ≤ 1, the real-v aluedness of q n is preserv ed under t he discrete-time ev olution (3 .87); the discriminan t in the square ro ot is nonnegativ e as long as we start with suﬃcien tly small real-v alued q n at the initial time. T o express the backw ard time ev olution ex plicitly , w e only ha v e to replace (1 + q 2 n )Λ n +1 in the ﬁrst equation of (3.84) with (1 + e q 2 n ) Λ n (cf. the second equation) and then substitute the lo cal express ion (3.86). In terms of the new parameter δ give n b y δ := h/ (1 + h 2 ), (3.87) can b e rewritten in a sligh tly mor e compact form, 1 δ ( e q n − q n ) = 2 (1 + q 2 n ) ( e q n +1 − q n − 1 ) 1 − 2 δ e q n +1 q n + q 1 − 4 δ e q n +1 q n − 4 δ 2  1 + e q 2 n +1 + q 2 n  . (3.8 8) 47 Note that if h ∈ R (or − 1 ≤ h ≤ 1), then − 1 / 2 ≤ δ ≤ 1 / 2. A t the “threshold” v alues of h , h = ± 1, corresp o nding to δ = ± 1 / 2, w e can extract the square ro ot in (3.87) to obtain a r a tional mapping. This is connected with the fact that in the cases h = ± 1, the discrete-time sys- tem (3 .84) under the boundar y conditions (3.83) has the trivial solution e q n = ∓ q n − 1 , Λ n = 1 / (1 + q 2 n − 1 ) (cf. § 4.6 in [1]). Th us, the discriminan t of the quadra t ic equation (3.85) v anishes at h = ± 1, and the tw o solutio ns in- tersect. T o obtain a non trivial mapping from this observ ation, w e set h = +1 in (3.84) and replace q n with i w n , namely , ( e w n − w n = (1 − w 2 n )Λ n +1 ( e w n +1 − w n − 1 ) , (1 − e w 2 n )Λ n = (1 − w 2 n )Λ n +1 . (3.89) Moreo v er, w e generalize the b oundary conditions (3.83) a s lim n →±∞ w n = γ , lim n →±∞ Λ n = 1 (1 + γ ) 2 , γ 6 = − 1 , and assume that | w n − γ | is suﬃcien tly small. Thus , the additional condition γ 6 = 0 excludes the trivial time ev olution e w n = − w n − 1 . F ollow ing the same pro cedure as ab ov e, w e obtain the quadratic equation in Λ n that can b e factorized as [(1 + e w n ) (1 + w n − 1 ) Λ n − 1] [(1 − e w n ) (1 − w n − 1 ) Λ n − 1] = 0 . Substituting its prop er solution Λ n = 1 / [(1 + e w n ) (1 + w n − 1 )] in to (3.89), w e obtain a single equation ( e w n − 1)( e w n +1 + 1) = ( w n − 1)( w n − 1 + 1). By a trivial change of the dep enden t v aria ble w n =: 1 + 2 ν y n ( ν 6 = 0), w e obt a in the w ell-kno wn “discrete V olterra equation” [17, 49], e y n (1 + ν e y n +1 ) = y n (1 + ν y n − 1 ) . (3.90) Therefore, the discrete-time equation (3 .90) b elongs to the mo diﬁed V olterra hierarc h y and not the original V olterra hierarc h y [1]; this corresp onds to the sp ecial case where the quadratic equation for the auxiliary v ar iable Λ n is factorized to provid e a rational solution. Ultr adiscr etization [12, 13] . W e presen t an intuitiv ely plausible deriv ation of an ultradiscrete a nalogue of the time-discretized mo diﬁed V olterra lattice, although this ma y not b e a unique ultradiscretization. F or the forward time 48 ev olution, we rewrite (3.88) as e q n = q n n 1 + ǫq n q n − 1 + p Y n o − ǫ ( e q n +1 − q n − 1 ) 1 + ǫ e q n +1 q n + p Y n = q n n 1 + ǫ ( q n + 1 /q n ) q n − 1 + p Y n − ǫ e q n +1 /q n o 1 + ǫ e q n +1 q n + p Y n , (3.91) where ǫ := − 2 δ and Y n := 1 + 2 ǫ e q n +1 q n − ǫ 2  1 + e q 2 n +1 + q 2 n  . Note that the equation e q n = q n in the trivial case ǫ = 0 preserv es the sign of the dep en- den t v ariable on eac h lattice site n ∈ Z . Thus , w e consider a class of solu- tions in the limit ǫ → +0 suc h that the p ositivit y of the dep enden t v ariable, q n > 0, ∀ n ∈ Z , is preserv ed in the non trivial time ev o lution. W e intro- duce the pa r a metrization ǫ =: exp( − L/ε ), L > 0 and q n,m =: ρ exp( Q n,m /ε ), ρ > 0 , where m ∈ Z is the discrete time co ordinat e, whic h is usually sup- pressed. The scaling parameter ρ can dep end w eakly on n , m , and ε , but for brevit y , w e treat it as a constan t. Once this parametrizatio n is substi- tuted in to (3.91), the solution { Q n,m } of the initial-v alue problem dep ends on ε via the time ev olution. W e h yp othesize that b y c ho osing ρ appropri- ately , the ε - dep endence of Q n in the considered class of solutions b ecomes negligible as ε → +0, and Q n is nonnegativ e, Q n ≥ 0. Probably , the simplest w a y to justify the latter condition is to mo dify the zero b oundary condi- tions (cf. (3.83)) to nonzero b oundary conditions, lim n →±∞ q n = ρ , and to assume q n ≥ ρ , ∀ n ∈ Z . W e are not intere sted in tracing inﬁnitely long tails of solitons tha t deca y exp onen tially as n → ±∞ ; rather, w e prefer to extract solitons with compact supp or t in the limit ε → +0. This is wh y w e impo se the zero b oundary conditions lim n →±∞ Q n = 0 on the new v ariable Q n . The condition Y n ≥ 0, guar a n teed by the nonnegativity of the discrimi- nan t of (3.85) , requires that ǫq n ≤ e q n +1 + q (1 − ǫ 2 )  1 + e q 2 n +1  and ǫ e q n +1 ≤ q n + p (1 − ǫ 2 ) (1 + q 2 n ) . Using the para metrization ǫ = exp( − L/ε ) and q n = ρ exp( Q n /ε ) with Q n ≥ 0, w e can interpret these conditions in the limit ε → +0 as − L ≤ e Q n +1 − Q n ≤ L . 49 By taking the logarithm, (3 .9 1) b ecomes e Q n = Q n + ε log  1 + e Q n + Q n − 1 − L ε  ρ 2 + e − 2 Q n ε  + p Y n − e e Q n +1 − Q n − L ε  − ε log  1 + ρ 2 e e Q n +1 + Q n − L ε + p Y n  , (3.92a) Y n = 1 − e − 2 L ε + ρ 2 e e Q n +1 + Q n − L ε  2 − e e Q n +1 − Q n − L ε − e Q n − e Q n +1 − L ε  . (3.92b) T aking the aforemen tioned assumptions into accoun t, (3.92) reduces in the limit ε → +0 to the follo wing equation: e Q n = Q n + max 0 , Q n + Q n − 1 − L, e Q n +1 + Q n − L 2 ! − max  0 , e Q n +1 + Q n − L  . (3.93) As men tioned ab ov e, t he b oundary conditions lim n →±∞ Q n = 0 are a ssumed. It is desirable to conﬁrm for eac h solution that the conditions Q n ≥ 0 and   e Q n +1 − Q n   ≤ L hold true in the time ev olution. No te that the parameter L ma y dep end on the discrete time. It is hop ed that the issue of the c ommuta- tivity of the ultradiscrete ﬂo ws deﬁned b y (3.93) for distinct v alues of L will b e in v estigated. In this regard, w e exp ect that (3.93 ) and its time rev ersal (see below) will comprise an ultradiscrete analogue of the mKdV hierarc h y , wherein the parameter L lab els eac h ﬂow in the hierarch y . The ultradiscrete equation (3.93) is “linear” in t he sense that it is in v ar ia n t under the rescal- ing Q n 7→ k Q n , L 7→ k L , k > 0. Th us, if L is a time-indep enden t constan t, it is p ossible to ﬁx L at unit y . How ev er, this normalization often c hanges an in teger-v a lued Q n to a fractiona l v alue; th us, w e r a ther prefer to lea v e L as a free parameter. Our deriv a tion of the ultradiscrete modiﬁed V olterra lattice (3.93) is more or less intuitiv e and is not mathematically rigoro us. A more detailed treatmen t of all the terms in the n umerator of (3.91) may lead to a more complicated ultradiscrete equation, but w e prefer the relativ ely simple equation (3.93). F ortunately , for the speciﬁc examples that w e considered, (3.93) allo ws the “stable” propagation of solitons a nd their elastic (but non- trivial) collisions; th us, (at least some o f ) the in tegrabilit y prop erties app ear to b e retained in this ultra discretization. The bac kw ard time ev olution of the time-discretized mo diﬁed V olterra lattice is obtained from (3.88) thr o ugh the com bined space and time reﬂection n → − n , m → − m . Th us, its ultradiscrete analogue is obtained from (3.93) in the same manner. 50 3.5 The lattice Heise n b erg ferromagnet mo del The lattice Heis enberg ferromagnet mo del w as prop osed in 1982 by sev eral diﬀeren t authors [50 – 53]. The equation o f mot io n fo r this semi-discrete mo del can b e deriv ed almost systematically using either Ishimori’s approac h [51] based on a gauge transformation from t he Ablo witz–Ladik lattice or the r - matrix f ormalism ba sed on the fundamen tal P oisson brac k et relation [50, 53]. Ho w ev er, for a concise and easy-to-understand deriv ation, we use a more heuristic approa ch based on the zero-curv ature represen tation. W e start with the Lax pair of the following form in the semi-discrete case: L n = I + λS n , (3.94a) M n = λ 1 − λ 2 A n L n , (3.94b) where the conditions ( S n ) 2 = I and A n S n = S n − 1 A n are assumed. The latter condition guaran tees the useful relation A n L n = L n − 1 A n . Th us, substituting the Lax pair (3.94) in to the zero- curv ature condition (2 .2), w e obtain ( L − 1 n ) t + λ 1 − λ 2 ( A n +1 − A n ) = O . Noting the iden tity ( I + λS n )( I − λS n ) = (1 − λ 2 ) I , this results in S n,t = A n +1 − A n . (3.95) Because ( S n ) 2 = I a nd A n S n = S n − 1 A n , (3.95) implies the relation A n +1 ( S n +1 + S n ) = ( S n + S n − 1 ) A n . Th us, we choose A n as A n = 2i aS n − 1 ( S n + S n − 1 ) − 1 + 2 b ( S n + S n − 1 ) − 1 , so that the ab o v e relation is automatically satisﬁed. Here, a and b a re n - indep enden t scalars, but they ma y dep end on the time v ariable t . Substitut- ing this expression for A n in to (3.95), w e o bt a in S n,t = ∆ + n  2i aS n − 1 ( S n + S n − 1 ) − 1 + 2 b ( S n + S n − 1 ) − 1  , (3.96) where ∆ + n is the forw ard diﬀerence o p erator in t he discrete space (cf. (2.4)). Note that the evolution equation (3.96) is consisten t with the condition 51 ( S n ) 2 = I for a general l × l matrix S n . W e no w consider t he simplest non- trivial case of l = 2 and express S n in terms of the P auli matrices as S n = 3 X j =1 S ( j ) n σ j (=: σ · S n ) = " S (3) n S (1) n − i S (2) n S (1) n + i S (2) n − S (3) n # . (3.97) Here, S n =  S (1) n , S (2) n , S (3) n  is a unit v ector, i.e., h S n , S n i = 1. Because 2 ( S n + S n − 1 ) − 1 = ( S n + S n − 1 ) / (1 + h S n , S n − 1 i ) [51], (3.96) reduces to a sin- gle v ector equation in v olving b oth the scalar pro duct and the v ector pro duct, S n,t = ∆ + n  a S n × S n − 1 1 + h S n , S n − 1 i + b S n + S n − 1 1 + h S n , S n − 1 i  , h S n , S n i = 1 . (3.9 8) The case b = 0 giv es the la ttice Heisen b erg ferromagnet mo del [51, 52], while the case a = 0 corresponds to its simples t higher symmetry [54]. Let us examine the discrete -t ime case. W e mainly consider the time discretization o f the reduced system (3.98), and not the general matrix system (3.96), b ecause the latt er problem is expected to b e to o complicated. W e start with the Lax matrix V n of the follo wing form: V n = I + h λ 1 − λ 2 A n L n , (3.99) where t he conditions ( S n ) 2 = I and A n S n = e S n − 1 A n are assumed. The latter condition guaran tees the useful relation A n L n = e L n − 1 A n . Th us, substituting the Lax pair, (3.94a) and (3.99), in to the zero-curv ature condition (2.6), we obtain 1 h  e L − 1 n − L − 1 n  + λ 1 − λ 2 ( A n +1 − A n ) = O , or equiv alen tly , 1 h  e S n − S n  = A n +1 − A n . (3.100) Under the condition ( S n ) 2 = I , the relation A n S n = e S n − 1 A n is auto matically satisﬁed if A n tak es the general f o rm A n = B n S n + e S n − 1 B n . Ho w ev er, in analogy with the semi-discrete case, we employ a more speciﬁc form of A n , A n = 2i a n e S n − 1  S n + e S n − 1  − 1 + 2 b n  S n + e S n − 1  − 1 , 52 whic h a lso satisﬁes the relation A n S n = e S n − 1 A n . Here, the scalar unkno wns a n and b n are auxiliary v ariables. Substituting this expression f or A n in to (3.100), w e o btain 1 h  e S n − S n  = ∆ + n h 2i a n e S n − 1  S n + e S n − 1  − 1 + 2 b n  S n + e S n − 1  − 1 i . (3 .101) It only remains necessary to ﬁx the a uxiliary v ariables a n and b n . T o this end, w e recall that the time ev o lution determined b y (3.101) has to b e consisten t with the condition ( S n ) 2 = I . In the follo wing, we fo cus on the case of the 2 × 2 matrix S n giv en by (3.97). The requiremen t tr S n = 0 results in the n -indep endence of a n , th us w e set a n = a . Therefore, (3.101) reduces to 1 h  e S n − S n  = ∆ + n " a S n × e S n − 1 1 + h S n , e S n − 1 i + b n S n + e S n − 1 1 + h S n , e S n − 1 i # . (3.102) T o ﬁx b n , we in v ok e the pro cedure presen ted in section 2; we assume the b oundary conditions lim n →−∞ h S n , e S n − 1 i = 1 , lim n →−∞ b n = b. (3.103) Because h S n , S n i = 1, w e ha v e det L n = 1 − λ 2 (cf. (3.94a) and (3.9 7)). Th us, the equalit y (2.11) derive d from (2.10) implies that the determinan t of V n , det V n = det  I + h λ 1 − λ 2 A n L n  = det ( I − λS n + hλA n ) det  1 1 − λ 2 L n  , is an n - indep enden t quan tity . Therefore, b oth the tr ace and the determinan t of S n − hA n m ust b e n -indep enden t. The n -indep endence of tr ( S n − hA n ) is already satisﬁed b y setting a n = a . The determinan t of S n − hA n can b e computed as det ( S n − hA n ) = det h S n − 2i ha e S n − 1  S n + e S n − 1  − 1 − 2 hb n  S n + e S n − 1  − 1 i = det h (1 − 2 hb n ) I + S n e S n − 1 − 2i ha e S n − 1 i det  S n + e S n − 1  =  1 + h S n , e S n − 1 i − 2 hb n  2 + 1 + 4( ha ) 2 − h S n , e S n − 1 i 2 − 2  1 + h S n , e S n − 1 i  , 53 whic h coincides with its b o undary v alue determined b y (3.103). This results in a quadratic equation in hb n , i.e., 2( hb n ) 2 − 2  1 + h S n , e S n − 1 i  hb n + 2( ha ) 2 +  hb (2 − hb ) − ( ha ) 2   1 + h S n , e S n − 1 i  = 0 . F or suﬃcien tly small h (0 < | h | ≪ 1 ) , t he prop er solution of this quadratic equation is giv en b y hb n = 2( ha ) 2 + [ hb (2 − hb ) − ( ha ) 2 ] (1 + g n ) 1 + g n + p (1 + g n ) 2 − 4( ha ) 2 − 2 [ hb (2 − hb ) − ( ha ) 2 ] (1 + g n ) , where g n := h S n , e S n − 1 i . Substituting this lo cal expression fo r hb n in to (3.1 02), w e obtain an in tegrable time discretization o f (3.98); this time discretization is essen t ia lly equiv alen t to (B.18 ) in [15] (see also (26) in [16 ] and (3.10) in [1 7]). W e write the equation of mot ion for three imp ortant cases: the case b = 0 , e S n − S n = ∆ + n ( ha S n × e S n − 1 1 + h S n , e S n − 1 i +   1 − v u u t 1 − 2( ha ) 2 1 − h S n , e S n − 1 i  1 + h S n , e S n − 1 i  2   S n + e S n − 1 2    , h S n , S n i = 1; the case hb (2 − hb ) = ( ha ) 2 , e S n − S n = ∆ + n ( ha S n × e S n − 1 1 + h S n , e S n − 1 i + " 1 − s 1 − 4( ha ) 2  1 + h S n , e S n − 1 i  2 # S n + e S n − 1 2 ) , h S n , S n i = 1; and the case a = 0, 1 δ  e S n − S n  = ∆ + n   S n + e S n − 1 1 + h S n , e S n − 1 i + q  1 + h S n , e S n − 1 i  2 − 2 δ  1 + h S n , e S n − 1 i    , δ := hb (2 − hb ) , h S n , S n i = 1 . 54 The ﬁrst and second cases provide time discretizations of t he lattice Heisen- b erg f erromagnet mo del [(3.98) with b = 0], while the third case g iv es a discrete-time analogue of its simplest higher symmetry [(3.98) with a = 0]. Note that the ﬁrst case b = 0 can b e identiﬁed with (6.8) in [1 5]. All these diﬀerence sc hemes are seemingly hi ghly implicit , that is, the v alue e S n is not written explicitly in a closed form in terms of S n , S n +1 , and e S n − 1 , but this is not a serious dra wbac k. Indeed, w e can obtain the p ow er series expansion of e S n in h (or δ ) to an y order succe ssiv ely . Moreov er, we can compute the exact v alue of e S n b y the follo wing steps. Fir st, we tak e the scalar pro duct b e- t w een the equation of motion and S n +1 to obtain an equation for h S n +1 , e S n i , wherein e S n app ears only through the form h S n +1 , e S n i . Then, the prop er so- lution of this eq uation can b e found in an O ( h ) (or O ( δ )) neigh b orho o d of h S n +1 , S n i . Substituting it back in to the original equation of motion, w e arriv e at a linear equation for e S n that can b e solv ed straigh tforw ardly . In t his subsec tion, w e hav e obtained the t ime discretizations of the ﬁrst t w o ﬂows of the lattice Heisen b erg ferromagnet hierarc h y in a uniﬁed w a y . Our deriv ation is o riginal and easy to follo w, but not fully systematic, as some parts are based on heuristic treatmen ts. It w ould b e interes ting to in v estigate whether these time discretizations can b e derive d in a systematic manner from the time discretizations of the Ablowitz–Ladik lattice obtained in subsection 3.2, using Ishimori’s approa c h [51]. 4 Conclud ing remarks In this pap er, w e ha ve dev elop ed an eﬀectiv e metho d for obtaining prop er time discretiz atio ns of integrable lat t ice systems in 1 + 1 dimensions. This metho d, whic h is ba sed on the zero-curv ature condition (2.6), allows us to ob- tain lo cal equations of motion t hat can determine the time ev olution uniquely . Using this metho d, w e cons tructed new time discretizations of the T o da lattice, the Ablowitz –La dik lattice, the V olterra lattice, and t he mo diﬁed V olterra la t t ice, while w e obtained the same time discretizations of the lat- tice Heisen b erg ferromagnet mo del and its symmetry as those in [15 – 17]. It should b e stressed that t his is a systematic metho d and also a pplies to o ther in tegrable lat t ice systems. As a b on us, w e w ere able to derive ultradiscrete analogues of the V olterra lattice a nd the mo diﬁed V o lterra lattice in v olv- ing one arbitrary parameter L , namely , (3.77) and (3.93). Eac h o f these ultradiscrete equations, as w ell as its time rev ersal, app ears to form a hier- arc h y of m utually commuting ﬂo ws lab elled b y the par a meter L . Note that the ultradiscrete equations suc h as (3.77) and (3.93) allo w straigh tforward “m ulticomp onen t” g eneralizations. Indeed, for example, if we express the de- 55 p enden t v ariable and the parameter in the form U n = U (1) n + U (2) n i , U ( j ) n ∈ Z and L = L (1) + L (2) i > 0, L ( j ) ∈ Z , where i is an irrational num b er, and sub- stitute them in to (3 .7 7), w e can uniquely determine the time ev olution of the “t w o-comp onen t” system starting from a giv en initial condition. It w ould b e v ery intere sting to in v estigate the sp ecial case wherein the irrationa l i is arbitrarily close to 1. A notable featur e o f our metho d is that the time discretization do es not mo dify t he inte grable hierarc hy to whic h the orig inal lattice sys tem b elongs. Th us, the in tegrals of motion and the functional form of the solutions remain in v aria n t; o nly the time dependence of certain pa rameters corresponding to the angle v ariables in the solutions is changed (cf. [2, 3, 18]). Such a time discretization can b e iden tiﬁed with the spatial part of an auto-B¨ ac klund transformation of the con tin uous-time la ttice system o r, from a more uniﬁed p oin t of view, an auto- B¨ acklund transforma t ion of the whole hierarc h y of comm uting ﬂo ws. Therefore, any time discretization obtained b y our metho d preserv es the qualitativ e prop erties of the o r ig inal integrable hierarc h y and th us is exp ected to serv e as an excellen t sc heme fo r nume rical in tegratio n of the con tinuous -time latt ice system. This is in contrast with other kno wn metho ds that generally mo dify the in tegrable hierarc hy to ﬁnd a time dis- cretization in the lo cal fo rm. As is illustrated in subs ection 3.2, our metho d can also provide “almost in tegrable” n umerical sc hemes that far surpass the Ablo witz–Ladik–T aha lo cal sc hemes [2 , 3, 5, 41] in approximation accuracy . In our approac h, a lo cal-in-space time discretization is alw a ys derived from a global-in-space time discretization in v olving “nonlo cal” auxiliary v ar i- ables. The obtained lo cal equations of motion g enerally determine b oth the forw ard and back w ard time ev olution uniquely under the sp eciﬁed b oundary conditions; th us, they indeed giv e a discrete-time analogue of evolutionary diﬀeren tial-diﬀerence equations. One migh t consider that the original nonlo- cal time discretization no longer has an y use once the lo cal discretization has b een deriv ed f rom it; ho w ev er, this is often not the case. In actually solving an initial- v alue problem, the nonlo cal time discretizations o f ten pro vide cru- cial information on the attributes of the solution, suc h as the real- v aluedness, p ositivit y , and ratio nalit y with respect to the initial data and the parameters; it app ears that the lo cal discretizations a re not useful fo r establishing suc h prop erties directly . Let us illustrate t his p oint with one instructiv e exam- ple, namely , the discrete -time Ablowitz–Ladik lattice in the case a = b = 0 studied in subsection 3.2. The global sc heme (3.28) (cf. ( 3 .27)) is f r ee fro m irrational functions; consequen tly , under the decay ing b oundary conditions (cf. (3.24)), the discrete-time up dates of the dependen t v ariables ar e give n b y a rational mapping. On the ot her hand, the lo cal sc heme (3.34) with 56 (3.32) inv olve s a square ro ot; thus, it is extremely diﬃcult to foresee that the time ev olution can b e desc rib ed b y a rational mapping. Ho w ev er, these t w o sc hemes naturally deﬁne the same time evolution. Therefore, the quan tit y inside the square ro ot in (3.32) is alw ays equal to the square of a ratio nal function of the dep enden t v ariables at the previous momen t and the param- eters; t ypically , one encoun ters the for m q  1 − ˆ c ˆ d  2 (1 + f n ) 2 , where ˆ c and ˆ d dep end on the “ step size” parameter h through (3.30). The rational function f n in v olv es the dep enden t v a riables and the parameters app earing in (3.3 4), and is o f order O ( h ). In addition, it v anishes when the dep enden t v aria bles are iden tically zero. F or suﬃcien tly small | h | ( ≪ 1), this square ro ot is extracted as (1 − ˆ c ˆ d )(1 + f n ). Ho w ev er, for relative ly la r ge | h | , this is not ob vious; ev en when ˆ c ˆ d < 1, the v alues of t he dep enden t v a r ia bles ha v e to satisfy rather restrictiv e conditions. One simple w a y to b ypass this sign problem is t o consider the square ro ot as b eing deﬁned by its T aylor expansion for small | h | or small amplitudes of the dep enden t v ariables and then to extend the domain of the deﬁnition by analytic contin uation; this is brieﬂy explained in subsection 3.2. Th us, this example also illustrates the diﬃcult y of the sign pro blem in computing up dates of the dep enden t v ariables using the lo cal equations of motion. In this pap er, we mainly considered the “constan t” b oundary conditions at spatial inﬁnit y (cf. (2 .9 )) and assumed simple b oundary v alues of the dep enden t v ariables. Note, ho w ev er, that our metho d is not sensitiv e to the bo undar y v alues of the dep enden t v ariables and is also applicable to other b oundary conditions, including p erio dic, non v anishing, or nonconstan t b oundary conditions. Indeed, as is clear f rom the description in subsec- tion 2.3, one can fr eely mo dify the b oundary conditions for V n as long as they determine a deﬁnite v alue for t he right-hand side of (2.14) or (2.15) that is n -indep enden t. Alternative ly , one can ﬁrst set eac h v alue of (2.14) or (2.15) as a “constan t” free parameter and then elicit the corresp onding b oundary conditio ns for the dep enden t v ariables. In either case, one should tak e care to iden tify t he prop er solution of the resulting algebraic system (cf. subsection 2.4). W e can also understand in a more in tuitiv e w ay that the time discretizations deriv ed in this pap er are integrable under other suit- able b o undary conditions; note that the conserv ation laws should b e deriv ed from the lo cal equations of motion using only lo cal op erat io ns, i.e., without referring to the b o undary conditions. V ery r ecen tly , Adler, Bob enk o, and Suris [36, 37] successfully classiﬁed dis- crete in tegrable systems on quad-graphs using the notion of three-dimensional 57 consistency [35]. In part icular, under some assumptions, they presen ted a short but complete list of one-ﬁeld in tegrable equations deﬁned b y p olyno- mial relations o f degree one in eac h of the four ar gumen ts. Their results ha v e b een a ttracting a lot of interest from researc hers; for ex ample, click on t he NASA ADS link at http://arxi v.org/abs/nlin/0202024 . The time discretizations of p olynomial lattice systems obtained by our metho d generally con tain irr ational nonlinearity; the nonlinear terms are determined through the prop er solution of an “ultralo cal” algebraic eq uatio n of degree higher than o ne. Th us, suc h time discretizations essen tially lie outside the class considere d by Adler et al. 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A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion

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