Integrable symplectic maps associated with discrete Korteweg-de Vries-type equations

In tegrable symplectic maps asso ciated with discret e Kortew eg-de V ries-t yp e equations Xiao xue Xu 1 , M engmeng Jiang 1 , F rank W N ijh oﬀ 2 1 Sc ho ol of Mathematics and Statistics, Zh en gzhou Univ ersit y , Zh engzhou 45000 1, P R China 2 Departmen t of Applied Mathematics, Univ ersit y of Leeds, L eeds LS2 9JT, UK Abstract In this pap er we present nov el integrable symplectic maps, asso ciated with or- dinary diﬀerence equations, a nd show how they determine, in a remar k ably diverse manner, the in tegrability , including Lax pairs and the explicit solutions, for integrable partial diﬀerence equations which a re the discrete count er pa rts of integrable partial diﬀerential equations of Kor teweg-de V r ies-type (K dV-type ). As a consequence it is demonstrated that several distinct H amiltonia n systems lead to one and the s ame diﬀerence equation by means of the Liouville integrability framework. Thu s, these int egr able symplectic ma ps may provide a n eﬃcient too l for characterizing , and de- termining the in tegrability of, partial diﬀer e nce equations. Keyw ords : discrete K ortew eg-de V ries-t yp e equations, integrable Hamiltonian systems, in tegrable symplectic maps , Bak er-Akhiezer fun ctions, ﬁnite gen us s olutions 1 In tro duc tion In tegrable symplectic maps [1–4] co mp rise some of the remark able outcomes from the the- ory of discrete integ rable systems: suc h maps allo w the constru ction of sp ecial solutions for the c orresp onding partial diﬀerence equatio ns by mea ns of algebro-geo metric metho ds [5–9]. Many of th e inﬁn ite-dimensional discrete integrable mo d els that are sup p orted by (in the sense that they can b e reduced to) integrable symplectic maps hav e interesting prop erties: t he existence of Lax pairs, B¨ ac klund transformations, symmetries and conser- v ation la ws, Hamiltonian structures, the constru ction of integrable algorithms, (elliptic) soliton solutions, ﬁnite gen us solutions [see 10–23, and references therein]. By deﬁnition, in the symplectic sp ace N = ( R 2 N , d p ∧ d q ) with asso ciated coord inates ( p, q ) = ( p 1 , . . . , p N , q 1 , . . . , q N ) T , wher e N is a p ositiv e in teger, a mapping S that sends ( p, q ) to ( ˜ p, ˜ q ) is c alled an int egrable symplectic map, iﬀ the induced map S ∗ on the space 1 of diﬀerentia l forms on N satisﬁes S ∗ (d p ∧ d q ) = d p ∧ d q a nd S ∗ F j = F j , 1 ≤ j ≤ N , wher e F 1 , . . . , F N are sm o oth, f unctionally in dep end en t, and pairwise in inv olution (with resp ect to the P oisson brac k et asso ciated with symplectic form), functions in a d ense op en subset of R 2 N . W e will c onstru ct in this pap er some no vel in tegrable symplectic maps, whic h turn out to constitute a p o werful tool to study the natural d iscrete analogues of Kd V-t yp e equations. The latter are mem b ers of the celebrated Adler-Bob enk o-Sur is (ABS) list [24] of integrable partial diﬀerence equations on the quadrilateral lattice . In [25, 26], some o f the in tegrabilit y c haracteristics of m ultidimensional mappings aris- ing by p er io dic reductions from the partial diﬀerence analogues of the KdV equation were in ve stigated. These include L ax pairs, classical r -matrix structures and Liouville integra- bilit y , bu t th e exp licit solutions f or the m ap remai ned to b e established. In [27], b y mea ns of th e ﬁnite-gap tec h nique, t he rational maps generated from p erio dic initial problems of the lattice Kd V equation, were parameterized in terms of Kleinian fun ctions, and the closed-form m o diﬁed Hamiltonians for one- and t wo- degree symplectic mappings arisin g from th e lattice KdV and mo diﬁed K dV equations we re constructed, combined with the application of the metho d of separation of v ariables [28–30]. The latter ga ve rise to a dis- crete analogue of the Ko wal ewski-Dubr o vin equations, describing the dynamics in terms of the separatio n v ariables. The quantiza tion of the latt er systems h a v e b een in vestig ated as we ll in connection w ith integ rable quan tum ﬁeld systems [31–3 3 ]. Sym plectic mappin gs related to h igher-order count erparts of the KdV t yp e, e.g ., the Boussinesq type, ha v e been studied as w ell [34]. The motiv ation for this present pap er originates from [16], in wh ic h ﬁnite gen us solu- tions for the lattice p otenti al Kd V (lpKdV) equation ( ¯ ˜ u − u )( ˜ u − ¯ u ) = β 2 − β 1 , (1.1) are obtained th r ough int egrable symplectic maps and wh ere the lattice K dV (lKdV) equa- tion is solv ed as we ll. Here we use the usual notation, ˜ h ( m, n ) = h ( m + 1 , n ) , ¯ h ( m, n ) = h ( m, n + 1), for an y fu nction h ( m, n ). A d iscrete sp ectral pr oblem asso ciated w ith (1.1) can b e constructed fr om the p r op erty of multi-dimensional consistency of (1.1), cf. [19], namely ˜ χ = ( λ − β ) − 1 / 2 D ( β ) ( λ ; a, b ) χ, D ( β ) ( λ ; a, b ) =   a − λ + β + ab 1 b   χ, (1.2) where χ is a 2-comp onent ve ctor fun ction, λ is a sp ectral parameter, β is the parameter 2 of the lattice and a, b are p oten tials (i.e., functions of the indep enden t v ariables), whic h is diﬀeren t from the ones in [16, 35 ]. F urthermore, fr om the Darb oux/B¨ ac klun d appr oac h, it is found that (1.2) also allo ws a compatible sp ectral problem [36–38] ∂ x χ = U ( λ ; v, w ) χ =   v − λ + w 1 − v   χ. (1.3) The r esolution of the p oten tials v , w in (1.3) are not ind ep enden t fr om the a, b in (1.2), and the relations b et w een them are the key to the problem (see Section 2). Another ingredien t in o ur treatmen t is the met ho dology of “nonlinearisation” [39, 40], whic h is r elated to the expansions of p oten tials a, b, v , w in terms of squared eigenfunc- tions. In the p resen t p ap er, follo win g this metho d we prov e that the spectral p roblem (1.3) can b e nonlinearised resulting in a ﬁ n ite dim en sional in tegrable Hamiltonian system whic h p r o vides the essentia l conditions for constructing the relev ant in tegrable symplectic map stemmin g from (1.2). Using this map, w e dedu ce several well- deﬁn ed meromorph ic functions on th e sp ectral curv e. Finally the lpKd V equation (1.1) is solve d by solving the relev ant Jacobi inv ersion problem in terms of theta functions on the h yp erelliptic Riemann surface. In con trasts to the usu al cases treated in [16, 41, 42], wh ere p oten tials themselves satisfy th e corresp ond ing discrete mo dels, the solutions here are expressed in terms of the deriv ativ e of a sp ecial theta function with resp ect to the au x iliary Darb oux v ariable. As it turns out, w e conclude that one and the same discrete mo del can b e solv ed through d iﬀeren t Liouville inte grable m o dels. In s pired by this, we will also inv estiga te the lattice p oten tial mo diﬁed KdV (lpmKdV) equation β 1 ( ¯ u ˜ ¯ u − u ˜ u ) = β 2 ( ˜ u ˜ ¯ u − u ¯ u ) , (1.4) whic h is closely related to the Hirota equation, i.e., the lattice sine-Gordon (lsG) equa- tion w h ose algebro-geometric solutions hav e been discussed [43, 44]. In [41], in tegrable symplectic maps and no vel theta function solutions for equation (1.4) were constructed through in tegrable Hamiltonian systems asso ciated with the con tin uous sG equation. In con tract, in th e present pap er, we start f rom the K aup-New ell sp ectral p roblem [45] ∂ x χ = V ( λ ; v , w ) χ =   λ 2 / 2 λv λw − λ 2 / 2   χ, (1.5) and the asso ciated d iscrete sp ectral problem ˜ χ = ( λ 2 − β 2 ) − 1 / 2 D ( β ) ( λ ; a ) χ, D ( β ) ( λ ; a ) =   λa β β λa − 1   . (1.6) 3 In this example w e actually hav e a d iﬀeren t typ e of situation f rom the one of the previous examples [16, 41, 42] since the r elation b et w een the discrete p oten tial a and the con tin uous p oten tials v , w i s implicit. Ho w ev er, based on the Lax stru ctur e of the Kaup -New ell equation, (1.6), the system can still b e nonlinearised, in th e sense menti oned ab ov e, as an in tegrable symplectic map. In addition, w e will inv estigate th e lattice Sc h w arzian KdV (lSKdV) e quation, ﬁrst giv en in [35], β 2 1 ( ˜ ¯ u − ˜ u )( ¯ u − u ) = β 2 2 ( ˜ ¯ u − ¯ u )( ˜ u − u ) , (1.7) whic h exp resses the cross-ratio of four p oin ts in the complex plane b eing equ al to a con- stan t. Equ ation (1.7) wa s u sed in [44] to d eﬁne a d iscr ete conformal map, consequently in [46] solutions in terms of the Riemann theta f u nction were wr itten do wn in the con text of the geometry of those conform al maps. Moreo v er, th e ﬁnite genus s olutions to lSK dV equation (1.7) are discu ssed, with the h elp of some ﬁnite dimensional integ rable sys tems arising from one Lax matrix for the deriv ativ e S c hw arzian K dV equation [47]. In terest- ingly , in [48] by using the so called direct linearisation metho d, a more general form of (1.7) w as deduced (no wa days sometimes r eferred to as NQC equation), wh ic h in d iﬀeren t sp ecial p arameter cases r educes to b oth (1.1), (1.4) and (1.7 ). In th e present case the relev ant Hamiltonian system is asso ciated w ith the con tin uou s sp ectral problem ∂ x χ = W ( λ ; v, w ) χ =   − λ 2 / 2 + v + w λv − λ λ 2 / 2 − v − w   χ, (1.8) whic h is curiously compatible with the dyn amic problem of the Kac-v an Mo erb eke hier- arc h y [49], and wh ose corresp ond ing d iscrete sp ectral problem is giv en by ˜ χ = ( λ 2 − β 2 ) − 1 / 2 D ( β ) ( λ ; a, s ) χ, D ( β ) ( λ ; a, s ) =   λa β s β s − 1 λa − 1   . (1.9) Similarly , the parametrization of the p oten tials p la ys an essent ial role in this case. The pr esen t pap er is organised as follo ws. In Section 2, th e constru ction of integrable symplectic maps and the resulting ﬁ nite gen us solutions to the lpKd V equation (1.1) are presented. In Sect ion 3, we d eal with the lpmKdV equation ( 1.4 ), and exploit the p ermutabilit y of the in tegrable discrete phase ﬂ o ws arising from the iteration of a n o v el parameter-family of in tegrable s ymplectic maps, leading to the corresp onding ﬁ nite-gen us solutions to the partial diﬀerence equation. In Section 4, we study the lSK dV case and establish a useful relation b et w een the t wo discr ete p oten tials present in the same sp ectral 4 problem for the constru ction of the r elev ant integrable sy m plectic map. By this relation, a n o v el Lax pair for lSKd V equation (1.7) is obtained, diﬀeren t from the ones give n in [35, 47]. As a result, a recursion r elation for the ﬁn ite gen us solutions is presented. 2 The lattice p oten tial KdV equation In this section w e will in v estigate the lpKd V equatio n (1.1) via in tegrable symplectic maps. F or f urther calculations an int egrable Hamiltonian system is needed, wh ich pr o vides the in tegrals, the s p ectral curv e, etc. 2.1 An integrable Hamiltonian system F or th e sak e of self-con tainedness we ﬁrst review some of the metho dology of [16]. T his is based on the observ ation, going b ack t o [50], th at ﬁnite-dimensional in tegrable sys- tems can b e obtained by restricting the inﬁnite-dimensional int egrable systems to a ﬁ nite- dimensional inv arian t manifold. T o realise it, a go o d w a y is the nonlinearization of the sp ectral pr oblem [39, 40]. Here we u se th is tec hniqu e to construct a ﬁnite-dimensional in tegrable Hamiltonian system. Consider N copies of (1.3 ) with distinct eigen v alues α 1 , · · · , α N , and wr ite them in the v ector form: ∂ x p = v p − Aq + wq , ∂ x q = p − vq . (2.1) where A = d iag( α 1 , · · · , α N ). According to the principle of nonlinearisation [39, 40], th e reﬂectionless p oten tial can be expressed by the squ ared sum of eigenfunctions. In the present case, w e shall imp ose some constr aint on th e p oten tials v , w , so that the linear equation (2.1) can b e nonlinearised to pro du ce a completely integ rab e Hamiltonian system. By [6, 45, 49], w e tak e the f ollo wing Lax equation as a starting p oin t: ∂ x L ( λ ; p, q ) = [ U ( λ ; v , w ) , L ( λ ; p, q )] , (2.2) whic h is the compatibilit y cond ition of (1.3) and the eigen v alue problem L ( λ ; p, q ) χ = κχ . The Lax matrix is computed in a similar wa y as in [6, 45, 49] from (2.2), and adopts the follo wing traceless f orm: L ( λ ; p, q ) =   √ < q , q > + Q λ ( p, q ) − λ − Q λ ( p, p ) 1 + Q λ ( q , q ) − √ < q , q > − Q λ ( p, q )   , (2.3) 5 where Q λ ( ξ , η ) = < ( λI − A ) − 1 ξ , η > , and < ξ , η > = Σ N j =1 ξ j η j is the usual inner pro duct of tw o N -dimensional vec tors ξ , η . It is we ll-kno wn that the charact eristic p olynomial d et( κI − L ( λ ; p, q )) = κ 2 +det L ( λ ; p, q ) represent s the sp ectral curve of the s ystem, w h ic h in turn give s rise to the inte grals of the corresp ondin g Hamiltonian system. In our case, we h a v e the d eterminan t F λ △ = det L ( λ ; p, q ) = λ + Q λ ( Aq , q ) + Q λ ( p, p ) − 2 √ < q , q >Q λ ( p, q ) + Q λ ( p, p ) Q λ ( q , q ) − Q 2 λ ( p, q ) , (2.4) where Q λ ( Aq , q ) = − < q , q > + λQ λ ( q , q ). Actually , F λ is a rational fun ction of λ , ha ving simple p oles at { α j } N j =1 , sin ce the coeﬃcient of ( λ − α j ) − 2 is zero. Th us , F λ = Q N +1 j =1 ( λ − λ j ) Q N j =1 ( λ − α j ) = Λ( λ ) α ( λ ) . (2.5) where Λ( λ ) = Q N +1 j =1 ( λ − λ j ) , α ( λ ) = Π N j =1 ( λ − α j ). By virtue of general results of the theory of algebraic curves, cf. [51–53], the sp ectral curve associated with a 2-sheeted Riemann su rface of genus g = N is constructed according to th e metho d elab orated in [54, 55], R : ξ 2 = − R ( λ ) , (2.6) where R ( λ ) = Λ( λ ) α ( λ ). F or v alues of λ n ot corresp ond ing to a branch p oint, there are t w o p oin ts p ( λ ), ( τ p ) ( λ ) on R , p ( λ ) =  λ, ξ = p − R ( λ )  , ( τ p )( λ ) = ( λ, ξ = − p − R ( λ )  . with τ : R → R th e map of c hanging sheets. F urth er m ore, fr om equation (2.2 ) it follo ws F λ is indep en d en t of th e argumen t x , th er e- fore, F λ can act as the generating f unction of the in tegrals asso ciated with the Hamilto n ian system (see b elo w). In f act, setting F λ = λ + Σ ∞ j =1 F j λ − j , (2.7) the co eﬃcients in the expansion are giv en by F 1 = < Aq , q > + < p, p > − 2 √ < q , q > < p, q > , F l = < A l q , q > + < A l − 1 p, p > − 2 √ < q , q > < A l − 1 p, q > + + X j + k +2= l ; j,k ≥ 0  < A j p, p >< A k q , q > − < A j p, q > < A k p, q >  , ( l ≥ 2) . (2.8) 6 Inspired by the stru cture of the sp ectral curve (2.6), we ﬁnd that F λ is prop ortional to a p erfect squ are of a qu antit y H λ , n amely λ H 2 λ = F λ . (2.9) By u sing the exp r ession (2.7), we obtain H λ = 1 + Σ ∞ j =1 H j λ − j − 1 , (2.10) where H 1 ( p, q ) = 1 2 ( < Aq , q > + < p, p > ) − √ < q , q > < p, q > . (2.11) No w we consider H 1 ( p, q ) as a Hamiltonian function and calculate the corresp onding Hamiltonian s y s tem. F ortunately , when choosing ( v , w ) = ( √ < q , q >, < p , q > / √ < q , q > ) , (2.12) in the reﬂectionless case, what the solution ( p j , q j ) T of the linear equation (2.1) satisﬁes, is actually a system of n onlinear equations, wh ic h can b e written in Hamiltonian form as: p x = − ∂ H 1 /∂ q , q x = ∂ H 1 /∂ p. (2.13) So far, w e ha ve ﬁn ished the nonlinearisation of th e eigen v alue p roblem (1.3), resu lting in the Hamiltonian system (2.13). In the fu ture analysis w e need the follo wing in gredien ts: 1) the canonical b asis a 1 , · · · , a g , b 1 , · · · , b g of the homology group of conto ur s. 2) the basis of the h olomorphic d iﬀeren tials, written in the vect or f orm as ~ ω ′ = ( ω ′ 1 , · · · , ω ′ g ) T , ω ′ j = λ g − j d λ/ (2 ξ ) , (2.14) whic h can b e normalized in to ~ ω = C ~ ω ′ , wher e C = ( a j k ) − 1 g × g , with a j k the in tegral of ω ′ j along a k and ~ C l the l -th column ve ctor of C . Near the p oin t at inﬁnity , the follo wing lo cal expansion holds: ~ ω = [ ~ Ω 1 + O ( t 2 )]d t, (2.15) where ~ Ω 1 = − ~ C 1 and t ( t − 2 = − λ ) is the lo cal co ordinate for the branch p oint ∞ . 3) the p erio d icit y ve ctors ~ δ k , ~ B k deﬁned as integ rals o f ~ ω along a k , b k , r esp ectiv ely . They sp an a lattice T of p erio d s, wh ic h d eﬁnes the Jacobian v ariet y J ( R ) = C g / T . The matrix B = ( ~ B 1 , . . . , ~ B g ) is used to construct the Riemann theta fu nction θ ( z , B ) = X z ′ ∈ Z g exp π √ − 1( < B z ′ , z ′ > +2 < z , z ′ > ) , z ∈ C g . (2.16) 7 The Ab el map A : Div( R ) → J ( R ) is giv en as A ( p ) = Z p p 0 ~ ω , A (Σ n k p k ) = Σ n k A ( p k ) . (2.17) Th u s b y equation (2.15), we solv e − A ( p ) = Z p 0 p ~ ω = Z p 0 ∞ ~ ω + Z ∞ p ~ ω = η − ~ Ω 1 t + O ( t 3 ) , η = Z p 0 ∞ ~ ω . (2.18) No w we discuss the complete integ rability of (2.13) in the Liouville sense. Here w e emplo y the r -matrix and the ev olution of the Lax matrix along a certain ph ase ﬂo w, whic h can b e used to enco de the in vol ution and ind ep enden ce of the inte grals in our case. Referring to [56–58], w e v erify by direct computation that there are t w o m atrix-v alued functions, r 12 and r 21 , on the symp lectic space, r 12 =            2 λ − µ 0 − 1 √ < q , q > 0 0 0 2 λ − µ 1 √ < q , q > 0 2 λ − µ 0 0 0 0 0 2 λ − µ            , r 21 =            2 µ − λ − 1 √ < q , q > 0 0 0 0 2 µ − λ 0 0 2 µ − λ 0 1 √ < q , q > 0 0 0 2 µ − λ            , in terms of w hic h the fund amen tal Poisson brac ke t b et w een the Lax matrices tak es the form: { L ( λ ) ⊗ , L ( µ ) } = [ r 12 , L 1 ( λ )] − [ r 21 , L 2 ( µ )] , (2.19) where L ( λ ) is the abbreviation for L ( λ ; p, q ), L 1 ( λ ) = L ( λ ) ⊗ I , L 2 ( µ ) = I ⊗ L ( µ ) and I is the usual u nit matrix. Lemma 2.1. The Lax matrix L ( µ ) satisﬁes the ev olution equation along the t λ -ﬂo w deﬁned by th e Hamiltonian ve ctor ﬁeld of F λ , d L ( µ ) / d t λ = [ W ( λ, µ ) , L ( µ )] , (2.20) 8 where W ( λ, µ ) satisﬁes d d t λ   p j q j   =   − ∂ F λ /∂ q j ∂ F λ /∂ p j   = W ( λ, α j )   p j q j   , W ( λ, µ ) = 2 λ − µ L ( λ ) + 2 L 11 ( λ ) √ < q , q > σ + , σ + =   0 1 0 0   . Pr o of. Since L 2 ( λ ) = −F λ I , w e obtain { L 2 ( λ ) ⊗ , L ( µ ) } = {−F λ I ⊗ , L ( µ ) } =   −{F λ , L ( µ ) } 0 0 −{F λ , L ( µ ) }   =   d L ( µ ) / d t λ 0 0 d L ( µ ) / d t λ   . (2.21) By equation (2.19), w e calculat e the left hand side of (2.21) again and get { L 2 ( λ ) ⊗ , L ( µ ) } = L 1 ( λ ) { L ( λ ) ⊗ , L ( µ ) } + { L ( λ ) ⊗ , L ( µ ) } L 1 ( λ ) = − L 1 ( λ ) r 21 L 2 ( µ ) + L 1 ( λ ) L 2 ( µ ) r 21 − r 21 L 2 ( µ ) L 1 ( λ ) + L 2 ( µ ) r 21 L 1 ( λ ) = − [ L 1 ( λ ) r 21 + r 21 L 1 ( λ ) , L 2 ( µ )] =   [ W ( λ, µ ) , L ( µ )] 0 0 [ W ( λ, µ ) , L ( µ )]   , (2.22) where we use the form ulas L 2 1 ( λ ) = −F λ I a nd L 1 ( λ ) L 2 ( µ ) = L 2 ( µ ) L 1 ( λ ) = L ( λ ) ⊗ L ( µ ) whic h are easily got by some calculations. Then comparing (2.21) w ith (2.22), equation (2.20) is v eriﬁed.  As a corollary of Lemma 2.1, we hav e d L 2 ( µ ) / d t λ = [ W ( λ, µ ) , L 2 ( µ )], wh ic h imp lies d F µ / d t λ = {F µ , F λ } = 0 , ∀ µ, λ ∈ C , (2.23) b y u sing th e formula L 2 ( µ ) = −F µ I and the fact th at F λ is Hamiltonian for the t λ -ﬂo w. As a consequence we ha ve {F µ , F λ } = {F µ , H λ } = {H µ , H λ } = 0 , ∀ µ, λ ∈ C , { F j , F k } = { F j , H k } = { H j , H k } = 0 , ∀ j, k = 1 , 2 , 3 , . . . , 9 implying that F 1 , . . . , F N are in pairwise inv olution, moreov er, th ey are in tegrals for Hamil- tonian sys tem (2.13). In the th eory of Liouville inte grabilit y the fu nctional indep endence of F 1 , . . . , F N pla ys a fu ndamenta l role [59–61]. In order to prov e the latter, w e in tro d u ce the elliptic v ariables ν j , i.e., curvilinear orthogonal co ordin ates [62], L 21 ( λ ) = 1 + Q λ ( q , q ) = N Y j =1 λ − ν j λ − α j = n ( λ ) α ( λ ) . (2.24) A r esolution of (2.24) is given in term s of the quasi-Ab el-Jacobi v ariable ~ φ ′ and the Ab el- Jacobi v ariable ~ φ , wh ic h are deﬁn ed as ~ φ ′ = g X k =1 Z p ( ν k ) p 0 ~ ω ′ , ~ φ = C ~ φ ′ = A ( g X k =1 p ( ν k )) , (2.25) taking v alues in the Jacobian v ariet y J ( R ), by u s ing the Ab el map A . Consider on e of the en tries of m atrix equation (2.20): d L 21 ( µ ) / d t λ = 2( W 21 ( λ, µ ) L 11 ( µ ) − W 11 ( λ, µ ) L 21 ( µ )) , (2.26) Since F λ = − ( L 11 ( λ )) 2 − L 12 ( λ ) L 21 ( λ ), w e ﬁnd L 11 ( ν k ) = p − R ( ν k ) /α ( ν k ) , as a consequence of equation (2.5). Ev aluating equation (2.26) at the p oin t µ = ν k , we obtain the ev olution of th e elliptic v ariables ν k along th e t λ -ﬂo w, 1 2 p − R ( ν k ) d ν k d t λ = − 2 α ( λ ) n ( λ ) ( λ − ν k ) n ′ ( ν k ) , (1 ≤ k ≤ g ) , whic h are the Dubro vin equations for our case [8, 29, 63]. Then b y means of the Lagrange in terp olat ion formula for p olynomials, we h av e g X k =1 ν g − l k 2 p − R ( ν k ) d ν k d t λ = − 2 α ( λ ) g X k =1 ν g − l k n ( λ ) ( λ − ν k ) n ′ ( ν k ) = − 2 α ( λ ) λ g − l , (1 ≤ l ≤ g ) , whic h can b e rewritten in a s imple form { φ ′ l , F λ } = d φ ′ l d t λ = − 2 α ( λ ) λ g − l , (1 ≤ l ≤ g ) , (2.27) using the quasi-Ab el-Jacobi v ariable ~ φ ′ = ( φ ′ 1 , · · · , φ ′ g ) T giv en by (2.25). Expanding b oth sides of equation (2.27), we ob tain ∞ X j =1 { φ ′ l , F j } λ − j = − 2 λ − l Π g k =1 (1 − α k λ − 1 ) = − 2 ∞ X i =0 A i λ − ( i + l ) = − 2 ∞ X j = −∞ A j − l λ − j , 10 with A 0 = 1 , A − l = 0 , ∀ l ∈ N . Hence, we get ∂ ( φ ′ 1 , . . . , φ ′ g ) ∂ ( t 1 , . . . , t g ) =  { φ ′ l , F j }  g × g = − 2             1 A 1 A 2 . . . A g − 1 1 A 1 . . . A g − 2 . . . . . . . . . . . . A 1 1             , (2.28) where t j is the ﬂo w v ariable, i.e., d G/ d t j = { G, F j } for an y smo oth function G ( p, q ). W e note in passing that the matrix (2.28) is non-degenerate, h ence d F 1 , . . . , d F N are linearly ind ep endent throughout eac h cotangen t space T ∗ y R 2 N , ∀ y ∈ R 2 N . In deed su pp ose Σ N j =1 c j d F j = 0, th en Σ N j =1 c j { φ ′ l , F j } = 0 , ∀ l , which imp lies that c j = 0 , ∀ j . Thus w e arrive at the main resu lt: Prop osition 2.1. Th e Hamiltonian s ystem (2.13 ) is completely in tegrable in the sense of Liouville, p ossessing the inte grals F 1 , . . . , F N , w hic h are in inv olution w.r.t. the canonical P oisson brac k ets and functionally indep end en t on N = ( R 2 N , d p ∧ d q ). 2.2 An integrable symplectic map W e no w use the results in S ection 2.1 to construct an in tegrable symplectic map. Motiv ated b y [16], we deﬁne th e follo wing linear map on N , S β :   ˜ p j ˜ q j   = ( α j − β ) − 1 / 2 D ( β ) ( α j ; a, b )   p j q j   , (1 ≤ j ≤ N ) , (2.29) where D ( β ) is the relev ant Darb oux matrix giv en in (1.2). Similarly , w e will ﬁn d the constrain t on the d iscrete p otentia ls a, b , un der whic h S β can b e nonlin earised to d er ive an i nteg rable symplectic co rr esp ondence. This can be see n to arise from th e follo wing discrete Lax equation: D △ = L ( λ ; ˜ p, ˜ q ) D ( β ) ( λ ; a, b ) − D ( β ) ( λ ; a, b ) L ( λ ; p, q ) = 0 . (2.30) In fact, b y direct calculations, we h a v e D =   ˜ v − λ 1 − ˜ v   D ( β ) ( λ ) − D ( β ) ( λ )   v − λ 1 − v   + N X j =1 1 λ − α j ( ˜ ε j D ( β ) ( λ ) − D ( β ) ( λ ) ε j ) , 11 where v is giv en b y equation (2.12) and ε j =   p j q j − p 2 j q 2 j − p j q j   satisﬁes ˜ ε j D ( β ) ( α j ; a, b ) − D ( β ) ( α j ; a, b ) ε j = 0. Then the en tries of the matrix D are expressed as D 11 = a ( ˜ v − v − b ) + v 2 − β , D 12 = − λ ( b − a + v + ˜ v ) + ( β + ab )( v + ˜ v ) − < ˜ p, ˜ q > − < p, q >, D 21 = a − ˜ v − v − b, D 22 = b ( a − ˜ v + v ) − v 2 + β . Hence, from the formula for D 21 w e c ho ose the r estriction a = b + v + ˜ v , (2.31) Substituting (2.31) in to the other comp on ents, we obtain D 12 = b D 11 + a D 22 , D 11 = −D 22 = − P ( β ) ( b ; p, q ) , where P ( β ) ( b ; p, q ) = L 21 ( β ) b 2 + 2 L 11 ( β ) b − L 12 ( β ) , (2.32) b y using ˜ p j = ( α j − β ) − 1 / 2 [ ap j + ( − α j + β + ab ) q j ] , ˜ q j = ( α j − β ) − 1 / 2 ( p j + bq j ) , deriv ed from (2.29). Th erefore, w e assert that the r o ots of the qu adratic equ ation P ( β ) ( b ; p, q ) = 0 give rise to an explicit constrain t on b , b = f 2 β ( p, q ) = 1 1 + Q β ( q , q ) ( − √ < q , q > − Q β ( p, q ) ± p − R ( β ) α ( β ) ) , (2.33) Actually th ey are the v alues of a meromorp hic fun ction on the curv e R deﬁn ed by (2.6), B ( p ) = 1 1 + Q β ( q , q ) ( − √ < q , q > − Q β ( p, q ) + ξ α ( β ) ) , at the p oints p ( β ) and ( τ p )( β ), r esp ectiv ely . T hen by the relation (2.31), we get a = f 1 β ( p, q ) = f 2 β ( p, q ) + p < ˜ q , ˜ q > + √ < q , q >. (2.34) 12 Though doubled-v alued as fu nctions of β ∈ C , (2.33) and (2.34) are single-v alued as functions of p ( β ) ∈ R . Th us we get the f ollo wing expression for the discrete p oten tials in term of the s quare of eigenfun ctions: ( a, b ) = f β ( p, q ) = ( f 1 β ( p, q ) , f 2 β ( p, q )) , (2.35) b y whic h the linear map S β giv en b y (2.29) b ecomes a n onlinear map S β :   ˜ p ˜ q   = ( A − β ) − 1 / 2   ap + ( − A + β + ab ) q p + bq        ( a,b )= f β ( p,q ) . (2.36) Here we use the same symb ol S β for short. Prop osition 2.2. The ab o v e map (2.36) is an int egrable sy m plectic map, und er whic h the qu an tities F 1 , . . . , F N on ph ase space giv en by (2.8) are inv arian t. Pr o of. According to the ab o v e analysis, we ha v e L ( λ ; ˜ p , ˜ q ) D ( β )  λ ; f β ( p, q )  − D ( β )  λ ; f β ( p, q )  L ( λ ; p, q ) = 0 , (2.37 ) b y su bstituting (2.35) in to the left h and side of the discrete Lax equation (2.30). T aking the determinan t on (2.37), w e obtain ˜ F λ = F λ whic h implies F j ( ˜ p, ˜ q ) = F j ( p, q ), i.e., S ∗ β F j = F j , 1 ≤ j ≤ N . In ord er to get the symplectic prop ert y of (2.36), we calculate N X j =1 (d ˜ p j ∧ d ˜ q j − d p j ∧ d q j ) = − 1 2 d P ( β ) ( b ; p, q ) ∧ d p < ˜ q , ˜ q >, (2.38) where P ( β ) ( b ; p, q ) is giv en by (2.32). Thus S ∗ β (d p ∧ d q ) = d p ∧ d q un der the constraint (2.35).  Prop osition 2.2 implies the Liouville integrabilit y of the symplectic map [1–4]. Con- sidering th e map (2.36) as an iterativ e map, we can create a discrete orbit sta rting from an initia l data ( p 0 , q 0 ) ∈ R 2 N . Thus we are able to deﬁne a discrete phase ﬂo w  p ( m ) , q ( m )  = S m β ( p 0 , q 0 ) r ep eated application of the map , i.e., S m β = S β ◦ S m − 1 β . T he p oten tials along the S m β -ﬂo w are, ( a ( m ) , b ( m )) = ( a m , b m ) = f β  p ( m ) , q ( m )  = f β  S m β ( p 0 , q 0 )  , (2 .39) a m = b m + v m + v m +1 , o r a = b + v + ˜ v , (2.40) where v m = p < q ( m ) , q ( m ) > . Then equation (2.37) can b e written in the form L m +1 ( λ ) D ( β ) m ( λ ) = D ( β ) m ( λ ) L m ( λ ) , (2.41) 13 where w e h a v e used the abbreviations L m ( λ ) = L ( λ ; p ( m ) , q ( m )) , D ( β ) m ( λ ) = D ( β ) ( λ ; a m , b m ). W e remark that since F λ  S β ( p, q )  = F λ ( ˜ p, ˜ q ) = F λ ( p, q ), we ha v e F λ  p ( m ) , q ( m )  = F λ ( p 0 , q 0 ). Th us the sp ectral curve R is in v arian t under the S m β -ﬂo w. 2.3 Th e ﬁnite gen us solutions to the lpKdV equation Our aim is to calculate ﬁnite gap classes of exact solutions for lpKd V equation (1.1). W e no w consider D ( β ) m ( λ ) as a diﬀerence op erator and L m ( λ ) as an algebra op erator. Th e ab o v e comm utativit y relation (2.41) b et wee n them remind s u s of the Burchnall-Chaundy theory [64], for comm utativ e diﬀerenti al op erators, the d iscrete analogue of wh ic h w e formulate b elo w [16], and the Bak er-Akhiezer f unctions [65, 66] as wel l. Th er efore, we in tro du ce a new discr ete sp ectral problem with p oten tials a m , b m as h β ( m + 1 , λ ) = D ( β ) m ( λ ) h β ( m, λ ) , (2.42) and inv estigate its fu ndamenta l solution matrix M β ( m, λ ) with M β (0 , λ ) = I . F ortun ately , the solution space ε λ of equation (2.42) is inv arian t un d er the action of the linear op erator L m ( λ ). In f act, if h ∈ ε λ , then b y (2.41), ( Lh ) m +1 = L m +1 ( D ( β ) m h m ) = D ( β ) m ( Lh ) m , whic h implies L h ∈ ε λ . In the in v arian t space ε λ , the linear op erator L m ( λ ) has t w o eigen v alues ρ ± λ , indep en- den t of the discrete argu m en t m du e to Prop osition 2.2, ρ ± λ = ± ρ λ = ± p −F λ = ± p − R ( λ ) /α ( λ ) , (2.43) whic h are t he v alues of a well-deﬁned meromorp hic function ξ ( p ) /α  λ ( p )  on R at the p oints p ( λ ) , ( τ p ) ( λ ) resp ectiv ely . The corresp onding eigen v ectors h β , ± are give n by  L m ( λ ) − ρ ± λ  h β , ± ( m, λ ) = 0 . (2.44) Sim ultaneously h β , ± are solutions of the equation (2.42), thus w e ha ve h β , ± ( m + 1 , λ ) = D ( β ) m ( λ ) h β , ± ( m, λ ) , (2.45) and in this sense the h β , ± are common eigen vec tors for D ( β ) m ( λ ) and L m ( λ ). Since the rank of L m ( λ ) ∓ √ −F λ is equal to 1, in eac h case th e common eigen v ector is uniquely 14 determined u p to a constan t factor. W e choose h β , ± ( m, λ ) =   h (1) β , ± ( m, λ ) h (2) β , ± ( m, λ )   = M β ( m, λ )   c ± λ 1   . (2.46) By letting m = 0 in equations (2.44) and (2.46), w e solv e c ± λ = L 11 0 ( λ ) ± ρ λ L 21 0 ( λ ) = − L 12 0 ( λ ) L 11 0 ( λ ) ∓ ρ λ . (2.47) Hence c + λ c − λ = − L 12 0 ( λ ) /L 21 0 ( λ ), and c + λ , c − λ are tw o br anc hes of a m eromorphic fu nction on t wo sheets of R , since L j k 0 , j, k = 1 , 2 are r ational functions of λ apart fr om ρ λ . F ollo wing [16, 65–67], w e no w in ve stigate a w ell-deﬁned meromorphic fun ction h (2) β ( m, p ) on R , i.e., Bak er-Akhiezer function, with v alues h (2) β , + ( m, λ ) an d h (2) β , − ( m, λ ) at the p oin ts p ( λ ) an d ( τ p )( λ ) r esp ectiv ely . It turns out that the explicit expression of h (2) β ( m, p ) is the ke y to th e solutions f or lpKdV equation (1.1), in terms of theta fu nctions. According to the theory of Riemann sur faces [51–53], w e need to consider the zeros and p oles of h (2) β ( m, p ). Lemma 2.2. T he follo wing formula h olds [16]: h (2) β , + ( m, λ ) · h (2) β , − ( m, λ ) = ( λ − β ) m g Y j =1 λ − ν j ( m ) λ − ν j (0) . (2.48) Pr o of. S ince M β ( m, λ ) is the solution matrix of the equation (2.42), we ha ve M β ( m + 1 , λ ) = D ( β ) m ( λ ) M β ( m, λ ) , (2.49) and by induction, we obtain M β ( m, λ ) = D ( β ) m − 1 ( λ ) D ( β ) m − 2 ( λ ) . . . D ( β ) 0 ( λ ) . (2.50) Then from the comm utativit y relation (2.41), the action of the algebra op erator L m ( λ ) on M β ( m, λ ) giv es r ise to L m ( λ ) M β ( m, λ ) = M β ( m, λ ) L 0 ( λ ) . (2.51) Finally we calculate,   h (1) β , + h (1) β , − h (1) β , + h (2) β , − h (2) β , + h (1) β , − h (2) β , + h (2) β , −   = M β ( m, λ )   c + λ c − λ c + λ c − λ 1   M T β ( m, λ ) = 1 L 21 0 ( λ ) M β ( m, λ )[ L 0 ( λ ) + ρ λ ]i σ 2 M T β ( m, λ ) = 1 L 21 0 ( λ ) [ L m ( λ ) + ρ λ ] M β ( m, λ )i σ 2 M T β ( m, λ ) = 1 L 21 0 ( λ ) [ L m ( λ ) + ρ λ ]i σ 2 ( λ − β ) m , 15 where σ 2 is the P auli m atrix. Th us, h (2) β , + h (2) β , − = ( λ − β ) m L 21 m ( λ ) /L 21 0 ( λ ) , wh ic h implies (2.48) by using (2.24).  Lemma 2.2 giv es the total zeros and some p oles. W e no w exhibit the remaining p oles stemming f rom the asym p totic b eha viors [68]. Lemma 2.3. I n the n eigh b orh o od of ∞ , the follo wing formula reads: h (2) β , ± ( m, λ ) = ( ± t ) − m [1 + O ( t )] . (2.5 2) Pr o of. By using equation (2.46), w e ha v e h (2) β , ± ( m, λ ) = c ± λ M 21 β ( m, λ ) + M 22 β ( m, λ ) . (2.53) Th u s the asymptotic b eha viors for c ± λ and M β ( m, λ ) are needed. F rom equation (2.47), w e solv e c ± λ = ± √ − λ [1 + O ( λ − 1 / 2 )] , ( λ → ∞ ) . (2.54) Moreo v er, as λ → ∞ , we get M β (2 k, λ ) =   ( − λ ) k [1 + O ( λ − 1 )] O ( λ k ) O ( λ k − 1 ) ( − λ ) k [1 + O ( λ − 1 )]   , M β (2 k + 1 , λ ) =   O ( λ k ) ( − λ ) k +1 [1 + O ( λ − 1 )] ( − λ ) k [1 + O ( λ − 1 )] O ( λ k )   , (2.55) b y equation (2.50) and in duction. Substituting (2.54) and (2.55) in to (2.53 ) , we obtain h (2) β , ± (2 k, λ ) = O ( λ k − 1 / 2 ) + ( − λ ) k [1 + O ( λ − 1 )] = ( ± t ) − 2 k [1 + O ( t )] , h (2) β , ± (2 k + 1 , λ ) = ± ( − λ ) k +1 / 2 [1 + O ( λ − 1 / 2 )] = ( ± t ) − 2 k − 1 [1 + O ( t )] , whose un iﬁed form is exactly equation (2.52).  The sp ectral curv e R has a lo cal co ordin ate t = ( − λ ) − 1 / 2 at the br anc h p oints ∞ . Th u s , b y equation (2.52), h (2) β ( m, p ) has a p ole at ∞ of order m . Considering zeros and other p oles b y equation (2.48) , we arr iv e at Prop osition 2.3. The Bake r-Akhiezer function h (2) β ( m, p ) has divisors Div( h (2) β ( m, p )) = g X j =1  p ( ν j ( m )) − p ( ν j (0))  + m  p ( β ) − ∞  . (2.5 6) According to [51–53], for any tw o distinct p oin ts q , r ∈ R , there exists a dip ole ω [ q , r ], an Ab el diﬀerentia l of the third kind , with r esidues 1 , − 1 at the p oles q , r , resp ectiv ely , 16 satisfying Z a j ω [ q , r ] = 0 , Z b j ω [ q , r ] = Z q r ω j , ( j = 1 , · · · , g ) . Decomp osing the meromorph ic diﬀeren tial as the com bination dln h (2) β ( m, p ) = g X j =1 ω [ p ( ν j ( m )) , p ( ν j (0))] + mω [ p ( β ) , ∞ ] + g X j =1 γ j ω j + Ω , where γ j are constants, and Ω is the Ab elian diﬀerent ial of the second kind , with residues equal to zero at all p oles. Refer to [69], the diﬀeren tial leads to g X j =1 Z p ( ν j ( m )) p ( ν j (0)) ~ ω + m Z p ( β ) ∞ ~ ω ≡ 0 , (mo d T ) . (2.57) No w w e consider the f ormula (2.57) from the Ab el-Jacobi v ariables, and ﬁnd th at the S m β -ﬂo w viewed in th e Jacobian v ariet y J ( R ) is linear, i.e., ~ φ ( m ) ≡ ~ φ (0) + m ~ Ω β , (mo d T ) , (2.58) where ~ Ω β = R ∞ p ( β ) ~ ω , and ~ φ ( m ) = g X j =1 Z p ( ν j ( m )) p 0 ~ ω = A  Σ g j =1 p ( ν j ( m ))  , ( 2.59) whic h is th e Ab el-Jacobi v ariable (2.25) along the S m β -ﬂo w. Hence, the Bak er-Akhiezer function h (2) β ( m, p ) can b e constructed, in terms of th e Riemann theta f u nctions corresp onding to the Riemann surface determined by the sp ectral curv e (2.6) [16, 51–53, 65, 66], h (2) β ( m, p ) = C m · θ [ − A ( p ) + ~ φ ( m ) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] e m R p p 0 ω [ p ( β ) , ∞ ] , (2.60 ) where C m , ~ K are constants, indep end en t of p ∈ R . By letting p → ∞ in equ ation (2.60), with the help of Lemma 2.3, w e solv e the constant factor as C m = θ [ − A ( ∞ ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ ) + ~ φ ( m ) + ~ K ] · 1 ( r ∞ β ) m , r ∞ β = lim p →∞ t ( p ) e R p p 0 ω [ p ( β ) , ∞ ] . (2.61) Th u s , h (2) β ( m, p ) = θ [ − A ( p ) + ~ φ ( m ) + ~ K ] θ [ − A ( ∞ ) + ~ φ ( m ) + ~ K ] · θ [ − A ( ∞ ) + ~ φ (0) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] ·  1 r ∞ β e R p p 0 ω [ p ( β ) , ∞ ]  m . (2.62) In [5, 7, 8, 65–67], it in dicates that the r econstruction of some ﬁ n ite-gap p oten tials can b e redu ced to the classical J acobi inv ersion p r oblem on hyp erelliptic Riemann s urfaces. 17 In the present case, w e reconstruct th e discrete p oten tials a m , b m b y using the exp ression (2.62), ev en through the constraint (2.35 ) for th em h as b een giv en. T hen the ﬁ nite genus solutions to the lpKdV equation (1.1) can b e d educed [16]. Hence, we consider equation (2.45) imp lying h (1) β ( m + 1 , p ) = a m h (1) β ( m, p ) + ( − λ + β + a m b m ) h (2) β ( m, p ) , h (2) β ( m + 1 , p ) = h (1) β ( m, p ) + b m h (2) β ( m, p ) . After eliminating h (1) β ( m, p ), we hav e h (2) β ( m + 1 , p ) = ( b m + a m − 1 ) h (2) β ( m, p ) − ( λ − β ) h (2) β ( m − 1 , p ) . (2.63 ) Note that the r elation (2.40) b et ween a m and b m is not enough f or further calculatio ns. Actually , w e need to combine it w ith the compatibilit y of sp ectral pr oblems (1.2) and (1.3). Then the follo wing relations are obtained: a m = z m + v m +1 , (2.64) b m = z m − v m , (2.65) ( z m + z m − 1 ) x = z 2 m − z 2 m − 1 , (2.66) where z m = q v 2 m +1 + v 2 m − β . Remark: Equation (2.66) can also b e deriv ed from a B¨ ac klund transformation for the p oten tial KdV equation [19, 70], ( u m +1 + u m ) x = 2 λ − 1 2 ( u m +1 − u m ) 2 , when selecting − 2 z m = u m +1 − u m . (2.67) According to the p erm utabilit y p rop erty of the B¨ ac klund transformations, u satisﬁes the lpKdV equation. Hence, we are supp osed to in tegrate z m . By equations (2.64) and (2.65), the co eﬃcien t b m + a m − 1 in equation (2.63) can b e written as z m + z m − 1 . Now w e calculate z m + z m − 1 in tw o wa ys. First, we ha ve z m + z m − 1 = lim p →∞  h (2) β ( m + 1 , p ) h (2) β ( m, p ) + λ ( p ) h (2) β ( m − 1 , p ) h (2) β ( m, p )  , (2.68) 18 and with the help of equ ation (2.15), w e get θ [ − A ( p ) + ~ φ ( m ) + ~ K ] θ [ − A ( ∞ ) + ~ φ ( m ) + ~ K ] = 1 − t Θ m + O ( t 2 ) , (2.69) where Θ m = ∂ x | x =0 log θ [ x ~ Ω 1 + ~ K ( m )], ~ K ( m ) = η + ~ φ ( m ) + ~ K with η giv en by (2.18). Th u s , h (2) β ( m + 1 , p ) h (2) β ( m, p ) = 1 t { 1 + [Θ m − Θ m +1 + ǫ β ] t + O ( t 2 ) } , λ ( p ) h (2) β ( m − 1 , p ) h (2) β ( m, p ) = 1 t {− 1 + [Θ m − 1 − Θ m + ǫ β ] t + O ( t 2 ) } , where ǫ β is given by t r ∞ β e R p p 0 ω [ p ( β ) , ∞ ] = 1 + ǫ β t + O ( t 2 ) . Therefore, we ha ve z m + z m − 1 = Θ m − 1 − Θ m +1 + 2 ǫ β . (2.70) Second, we ha ve z m + z m − 1 = lim p → p ( β ) ( λ − β ) h (2) β ( m − 1 , p ) h (2) β ( m, p ) = r ∞ β r β · θ 2 [ ~ K ( m )] θ [ ~ K ( m + 1)] θ [ ~ K ( m − 1)] , (2.71) with r β = lim p → p ( β ) 1 λ − β e R p p 0 ω [ p ( β ) , ∞ ] . then by equ ation (2.66), w e obtain z m − z m − 1 = 2Θ m − Θ m +1 − Θ m − 1 . (2.72) As a result, by adding (2.70) and (2.72), we arrive at the explicit f ormula z m = Θ m − Θ m +1 + ǫ β = ∂ x | x =0 log θ [ x ~ Ω 1 + ~ K ( m )] θ [ x ~ Ω 1 + ~ K ( m + 1)] + ǫ β . (2.73) Here the exact expressions for a m and b m are not obtained. Ho wev er, the formula (2.73) is enough when w e compute the solutions f or th e lp KdV equation (1.1) in terms of Riemann theta fun ctions. Let th e p arameters β = β 1 , β 2 b e distinct and non-zero, and app lying the theory in Section 2.2 to th e t wo parameter cases resp ectiv ely , the resulting inte grable maps S β 1 , S β 2 19 p ossess the same Liouville set of inte grals F 1 , · · · , F N whic h su bsequently determine the action-angle v ariables ( I , ϕ ) with I = I ( F 1 , · · · , F N ). T h us, in the n eigh b orh o o d of eac h lev el set M c = { ( p, q ) ∈ R 2 N : F 1 ( p, q ) = c 1 , . . . , F N ( p, q ) = c N } , the phase ﬂo ws S m β 1 and S n β 2 are linearised by the same action-angle v ariables [42 ]. As a corollary of the d iscrete v ersion of the Liouville-Arnold theorem [2–4], S m β 1 and S n β 2 comm ute. Th en w e get a well d eﬁned fun ction, and it can b e expressed in t wo w a ys, resp ectiv ely , as  p ( m, n ) , q ( m, n )  = S m β 1 S n β 2 ( p 0 , q 0 ) = S m β 1  p (0 , n ) , q (0 , n )  = S n β 2 S m β 1 ( p 0 , q 0 ) = S n β 2  p ( m, 0) , q ( m, 0)  . (2.74) Th u s by equations (2.29) in the t wo s p ecial cases, (2.64) and (2.65), th e j -th comp onent satisﬁes tw o equ ations simultaneo u s ly w ith λ = α j ,   ˜ p j ˜ q j   = ( α j − β 1 ) − 1 / 2 D ( β 1 ) ( α j ; z ′ + ˜ v , z ′ − v )   p j q j   , z ′ = p ˜ v 2 + v 2 − β 1 ,   ¯ p j ¯ q j   = ( α j − β 2 ) − 1 / 2 D ( β 2 ) ( α j ; z ′′ + ¯ v , z ′′ − v )   p j q j   , z ′′ = p ¯ v 2 + v 2 − β 2 . (2.75) Besides the ev olution of equ ation (2.58) along the ab o ve tw o d iscrete ﬂows gives ~ φ ( m, n ) = ~ φ (0 , 0) + m ~ Ω β 1 + n ~ Ω β 2 . Comparing equation (2.67) and the theta fun ction expression (2.73) of z m , we n o w d eﬁne Z mn = ∂ x | x =0 log θ ( x ~ Ω 1 + m ~ Ω β 1 + n ~ Ω β 2 + ~ K 00 ) , (2 .76) with ~ K 00 = η + ~ φ (0 , 0) + ~ K . Th en we ha ve z ′ = Z mn − ˜ Z mn + ǫ β 1 , z ′′ = Z mn − ¯ Z mn + ǫ β 2 , and str aightforw ard calculations tell us that ( ¯ z ′ ) 2 − ( ˜ z ′′ ) 2 = ( z ′′ ) 2 − ( z ′ ) 2 − 2( β 1 − β 2 ). Th e latter r elations can b e u sed to calculate the comm utator ¯ D ( β 1 ) D ( β 2 ) − ˜ D ( β 2 ) D ( β 1 ) =   1 − ¯ ˜ Z mn + ˜ Z mn + ¯ Z mn − Z mn + ¯ ˜ v + v 0 1   Ξ , (2.77) 20 where Ξ = ( ˜ Z mn − ¯ Z mn + ǫ β 2 − ǫ β 1 )( Z mn − ¯ ˜ Z mn + ǫ β 2 + ǫ β 1 ) + β 2 − β 1 . Prop osition 2.4. The lpK dV equation (1.1) h as the ﬁ nite gen us solutions u ( m, n ) = ∂ x | x =0 log θ ( x ~ Ω 1 + m ~ Ω β 1 + n ~ Ω β 2 + ~ K 00 ) − mǫ β 1 − n ǫ β 2 . (2.78) Pr o of. The commutati vity of the ﬂo w S m β 1 and S n β 2 guaran tees the compatibilit y of equ ation (2.75). T hus ¯ D ( β 1 ) D ( β 2 ) = ˜ D ( β 2 ) D ( β 1 ) whic h imp lies Ξ = 0. This leads to lpKdV equation (1.1) wh en c ho osing u ( m, n ) = Z mn − m ǫ β 1 − n ǫ β 2 .  So far, from a nov el Hamiltonian s ystem diﬀeren t from the one in [16], w e ha ve suc- ceeded in dedu cing th e explicit an alytic solutions, i.e. ﬁnite gen us solutions in our case, for the lpKd V equation via inte grable symp lectic maps. Next we will inv estiga te the lpm KdV equation and the lSKdV equation in a similar wa y . 3 The lattice p oten tial mo diﬁed KdV equation Let us now consider the lattice version of the p oten tial mKdV equation (1.4). Note that Lax p airs for (1.4) hav e b een written do wn in [30, 35], but we hav e n ot b een able to b lend those linear problems with the algebro-geometric tec hnique of nonlinearisation emp lo y ed in the p resen t pap er. Inspired by Section 2.3, her e we select a d iﬀeren t parametrization for the discrete p oten tial a giv en in the Lax matrix (1.6), whereby (1.4) then arises as th e compatibilit y condition of a pair of such linear problems associated with the s h ifts of the v ector-function χ in the m and n directions, namely ˜ χ = ( λ 2 − β 2 1 ) 1 / 2 D ( β 1 ) ( λ ) χ, ¯ χ = ( λ 2 − β 2 2 ) 1 / 2 D ( β 2 ) ( λ ) χ, (3.1) where D ( β 1 ) ( λ ) is giv en by D ( β 1 ) ( λ ) =    λ ˜ u u β 1 β 1 λ u ˜ u    , (3.2) and where D ( β 2 ) ( λ ) is give n b y a similar m atrix obtained from (3.2 ) by making the r e- placemen ts β 1 → β 2 and ˜ → ¯ . In fact, w e ha v e ¯ D ( β 1 ) D ( β 2 ) − ˜ D ( β 2 ) D ( β 1 ) =    0 − λ ˜ u ¯ u λ u ˜ ¯ u 0    Ξ , (3.3) 21 where Ξ = β 1 ( ¯ u ˜ ¯ u − u ˜ u ) − β 2 ( ˜ u ˜ ¯ u − u ¯ u ). It turn s out th at almost everything th at holds true f or the lpKd V equation also holds true for the lpm K dV equation. W e shall now discuss the integ rable symplctic maps and sho w ho w to solve lpmKdV equation (1.4) via the nonlinearisation approac h, whic h diﬀers from pr evious approac hes. 3.1 An integrable Hamiltonian system As the starting p oin t for the s ubsequent calculations, we no w review some results from [45]. Introducing a Lax matrix L ( λ ; p, q ) =   1 / 2 + Q λ ( A 2 p, q ) − λQ λ ( Ap, p ) λQ λ ( Aq , q ) − 1 / 2 − Q λ ( A 2 p, q )   , (3.4) where Q λ ( ξ , η ) = < ( λ 2 − A 2 ) − 1 ξ , η > , A = d iag( α 1 , . . . , α N ) with α 2 1 , . . . , α 2 N distinct in pairs and non-zero. It satisﬁes the r -matrix ansatz { L ( λ ) ⊗ , L ( µ ) } = [ r 12 ( λ, µ ) , L 1 ( λ )] − [ r 12 ( µ, λ ) , L 2 ( µ )] , r 12 ( λ, µ ) = λ λ 2 − µ 2  λ ( I + σ 3 ⊗ σ 3 ) + µ ( σ 1 ⊗ σ 1 + σ 2 ⊗ σ 2 )  = 2 λ λ 2 − µ 2         λ 0 0 0 0 0 µ 0 0 µ 0 0 0 0 0 λ         , where σ 1 , σ 2 , σ 3 are the u s ual Pauli matrices. Considering the determinant of the Lax matrix (3.4), F λ △ = det L ( λ ; p, q ) = − (1 / 2 + Q λ ( A 2 p, q )) 2 + λ 2 Q λ ( Ap, p ) Q λ ( Aq , q ) , (3.5) w e also hav e th e ev olution equation in th e present case, d L ( µ ) / d t λ = [ W ( λ, µ ) , L ( µ )] , W ( λ, µ ) = 2 µ λ 2 − µ 2   µL 11 ( λ ) λL 12 ( λ ) λL 21 ( λ ) − µ L 11 ( λ )   , (3.6) where t λ is the ﬂow v ariable corresp ond ing to the Hamiltonian fu nction F λ . I n a sim- ilar wa y as in Section 2, we obtain pairwise inv olutive quantiti es F 0 , F 1 . . . , F N − 1 , i.e., 22 { F j , F k } = 0, from the p o wer series expansions F λ = − 1 4 + ∞ X j =1 F j λ − 2 j , | λ | > max {| α 1 | , . . . , | α N |} , F λ = ∞ X j =0 F − j λ 2 j , | λ | < min {| α 1 | , . . . , | α N |} , (3.7) where F 0 = − (2 < p, q > − 1) 2 , F 1 = < Ap, p >< Aq , q > − < A 2 p, q > , F k = − < A 2 k p, q > − X i + j = k ; i,j ≥ 1 < A 2 i p, q >< A 2 j p, q > − X i + j = k +1; i,j ≥ 1 < A 2 i − 1 p, p >< A 2 j − 1 q , q >, ( k ≥ 2) . Besides, F λ is a rational fu nction of ζ = λ 2 and can b e factorized as F λ = − 1 4 R ( ζ ) α 2 ( ζ ) , (3.8) where α ( ζ ) = N Y j =1 ( ζ − α 2 j ) , Z ( ζ ) = N Y k =1 ( ζ − ζ k ) , R ( ζ ) = α ( ζ ) Z ( ζ ) , The relev an t sp ectral cur v e is deﬁn ed as R : ξ 2 − R ( ζ ) = 0 , (3.9) with genus g = N − 1 an d t wo inﬁnities ∞ + , ∞ − . F or an y ζ ∈ C , in the non-br anc h case (not equal to ζ j , α 2 j ) there are tw o corresp on d ing p oin ts on R : p ( ζ ) =  ζ , ξ = p R ( ζ )  , ( τ p )  ζ ) = ( ζ , ξ = − p R ( ζ )  . F rom the Lax m atrix (3.4), w e get the zeros of the oﬀ-diagonal ent ries, which are exactly the elliptic v ariables µ 2 j , ν 2 j , L 12 ( λ ) = − λ < Ap, p > m ( ζ ) α ( ζ ) , m ( ζ ) = Π g j =1 ( ζ − µ 2 j ) , L 21 ( λ ) = λ < Aq , q > n ( ζ ) α ( ζ ) , n ( ζ ) = Π g j =1 ( ζ − ν 2 j ) , (3.10) in terms of w hic h the corresp ond ing q u asi-Ab el-Jacobi v ariables and Ab el-Jacobi v ariables read ~ φ ′ = g X k =1 Z p ( ν 2 k ) p 0 ~ ω ′ , ~ φ = C ~ φ ′ = A ( g X k =1 p ( ν 2 k )) , ~ ψ ′ = g X k =1 Z p ( µ 2 k ) p 0 ~ ω ′ , ~ ψ = C ~ ψ ′ = A ( g X k =1 p ( µ 2 k )) , (3.11) 23 where ~ φ ′ = ( φ ′ 1 , · · · , φ ′ g ) T , ~ ψ ′ = ( ψ ′ 1 , · · · , ψ ′ g ) T , and ~ ω ′ = ( ω ′ 1 , · · · , ω ′ g ) T , ω ′ l = ζ g − l d ζ / 2 p R ( ζ ) (1 ≤ l ≤ g ). It turn s out that (1.5) can b e n onlinearised to create a completely integ rable Hamilto- nian system p ossessing integ rals F 0 , F 1 . . . , F N − 1 [45]. The latter is d eﬁned by the canon- ical equations ∂ x   p j q j   =   − ∂ H 1 /∂ q j ∂ H 1 /∂ p j   =   α 2 j / 2 − α j < Ap, p > α j < Aq , q > − α 2 j / 2     p j q j   , (1 ≤ j ≤ N ) , (3.12) where H 1 = F 1 / 2 is the ﬁr st member in the expression of square r o ot H λ satisfying − 4 F λ = ( − 4 H λ ) 2 , H λ = − 1 4 + ∞ X j =1 H j λ − 2 j . (3.13) The n onlinearisation pr o cedure exp lained ab ov e also plays an imp ortant r ole in solving the (2+1)-dimensional d eriv ativ e T o da equation by algebra-ge ometric tec hn ique [45], while the nonlinearisation of the discrete sp ectral problem (1.6 ) can lead to n ew theta function solutions f or lpmKdV equation (1.4). 3.2 An integrable symplectic map F ollo wing the conclusion of Section 3.1, we no w construct the in tegrable symplectic map arising f r om the N copies of the d iscr ete sp ectral p r oblem (1.6),   ˜ p j ˜ q j   = ( α 2 j − β 2 ) − 1 / 2 D ( β ) ( α j ; a )   p j q j   , ( j = 1 , . . . , N ) . ( 3.14) According to the p ro cedure of Section 2.2, w e d iscuss th e discrete Lax equation in the present case. Th e Lax matrix (3.4) can b e rewritten as L ( λ ; p, q ) = ( 1 2 − < p, q > ) σ 3 + λ 2 N X j =1 ( ε j λ − α j + δ j λ + α j ) , where δ j = σ 3 ε j σ 3 satisfying ˜ δ j D ( β ) ( − α j ) = D ( β ) ( − α j ) δ j . Thr ough dir ect calculatio ns , w e get L ( λ ; ˜ p, ˜ q ) D ( β ) ( λ ; a ) − D ( β ) ( λ ; a ) L ( λ ; p, q ) = − β P ( β ) ( a ; p, q )i σ 2 , (3.15) where aP ( β ) ( a ; p, q ) = a ( < ˜ p , ˜ q > + < p , q > − 1) = a 2 L 12 ( β ) − 2 aL 11 ( β ) − L 21 ( β ) . (3.16) 24 Th u s , the constraint on a is deriv ed b y solving the quadratic equ ation aP ( β ) ( a ; p, q ) = 0, a = f β ( p, q ) = − 1 β Q β ( Ap, p )  1 / 2 + Q β ( A 2 p, q ) ± p R ( β 2 ) 2 α ( β 2 )  . (3.17) Moreo v er, β a giv es tw o v alues of a single-v alued meromorph ic fun ction on the cur v e R giv en b y (3.9), A ( p ) = − 1 Q β ( Ap, p )  1 / 2 + Q β ( A 2 p, q ) + ξ 2 α ( β 2 )  , at the p oin ts p ( β 2 ) and ( τ p )( β 2 ), r esp ectiv ely . The constrain t (3.17) leads to the nonlinear map S β :   ˜ p ˜ q   = ( A 2 − β 2 ) − 1 / 2   aAp + β q a − 1 Aq + β p        a = f β ( p,q ) . (3.18) Prop osition 3.1. The map S β of (3.18) is symp lectic and Liouville integ rable, i.e., S ∗ β (d p ∧ d q ) = d p ∧ d q , and F 0 , F 1 . . . , F N − 1 giv en by equation (3.7) satisﬁes S ∗ β F j = F j . Pr o of. S ubstituting (3.17) in to (3.15), we obtain L ( λ ; ˜ p , ˜ q ) D ( β ) ( λ ; a ) − D ( β ) ( λ ; a ) L ( λ ; p, q ) = 0 . (3.19) Th u s det L ( λ ; ˜ p, ˜ q ) = det L ( λ ; p, q ) by taking th e determinant, which imp lies S ∗ β F j = F j . The symp lectic prop ert y is conﬁ r med by the exp ression S ∗ β (d p ∧ d q ) − d p ∧ d q = N X j =1 (d ˜ p j ∧ d ˜ q j − d p j ∧ d q j ) = 1 2 a d a ∧ d P ( β ) ( a ; p, q ) . (3.20) whic h is d eriv ed f r om equation (3.14).  W e no w deﬁne the d iscr ete orbit  p ( m ) , q ( m )  = S m β ( p 0 , q 0 ). This is more discernib le if we r eform ulate the p oten tials a ( m ) = a m and u ( m ) = u m as a ( m ) = f β  p ( m ) , q ( m )  = ( S m β ) ∗ f β ( p 0 , q 0 ) , ˜ u/u = a, or u m +1 /u m = a m . (3.21) On th e S m β -ﬂo w, equation (3.19) is rewr itten as L m +1 ( λ ) D ( β ) m ( λ ) = D ( β ) m ( λ ) L m ( λ ) , (3.22) where L m ( λ ) = L ( λ ; p ( m ) , q ( m )) , D ( β ) m ( λ ) = D ( β ) m ( λ ; a m ). 25 Then by equation (3.11), th e Ab el-Jacobi v ariables in the Jacobi v ariet y J ( R ) = C g / T can b e d eﬁned as ~ φ ( m ) = A  g X j =1 p ( ν 2 j ( m ))  = g X j =1 Z p ( ν 2 j ( m )) p 0 ~ ω , ~ ψ ( m ) = A  g X j =1 p ( µ 2 j ( m ))  = g X j =1 Z p ( µ 2 j ( m )) p 0 ~ ω . (3.23) Let us no w in tro du ce the discrete sp ectral problem with p otent ial a m h β ( m + 1 , λ ) = D ( β ) m ( λ ) h β ( m, λ ) , (3.24) and let M β ( m, λ ) b e solution matrix w ith M β (0 , λ ) = I . Obviously , M β ( m, λ ) = D ( β ) m − 1 ( λ ) D ( β ) m − 2 ( λ ) . . . D ( β ) 0 ( λ ) , L m ( λ ) M β ( m, λ ) = M β ( m, λ ) L 0 ( λ ) , (3.25) where d et M β ( m, λ ) = ( λ 2 − β 2 ) m . As λ → ∞ , we ha ve the asymp totic b eha viour M β ( m, λ ) =   O ( λ m ) O ( λ m − 1 ) O ( λ m − 1 ) O ( λ m )   . (3.26) Solving th e eigen v alues of the linear map L m ( λ ), ρ ± λ = ± ρ λ = ± p −F λ = ± p R ( ζ ) / 2 α ( ζ ) , ρ λ = 1 / 2 + O ( λ − 2 ) , ( λ → ∞ ) . (3.27) the asso ciated eigenfunctions satisfy h β , ± ( m + 1 , λ ) = D ( β ) m ( λ ) h β , ± ( m, λ ) , (3.28) h β , ± ( m, λ ) =   h (1) β , ± ( m, λ ) h (2) β , ± ( m, λ )   = M β ( m, λ )   c ± λ 1   . (3.29) W e now study the common eigen v ectors h β , ± ( m, λ ) for op erators L m ( λ ) and D ( β ) m ( λ ) with the h elp of the Bak er-Akhiezer functions expressed b y the theta functions of hyper elliptic Riemann su r face deﬁned b y the curve R . Since h β , ± (0 , λ ) = ( c ± λ , 1) T , the relev ant entries c ± λ are give n by c ± λ = L 11 0 ( λ ) ± ρ λ L 21 0 ( λ ) = − L 12 0 ( λ ) L 11 0 ( λ ) ∓ ρ λ , c + λ c − λ = − L 12 0 ( λ ) L 21 0 ( λ ) , (3.30) whic h as λ → ∞ , b eha v e as c + λ = λ < Aq , q > | 0 (1 + O ( λ − 2 )) , c − λ = < Ap, p > | 0 λ (1 + O ( λ − 2 )) . (3.31) 26 F urth er m ore, λc + λ and λc − λ are the v alues of a meromorphic function on R , C ( p ) = − ζ < ( ζ − A 2 ) − 1 Ap 0 , p 0 > − 1 / 2 − < ( ζ − A 2 ) − 1 A 2 p 0 , q 0 > + ξ / 2 α ( ζ ) , at the p oints p ( λ 2 ) and ( τ p )( λ 2 ), resp ectiv ely . Quite similarly as in Section 2, relying on equations (3.10), (3.25), (3.29) and (3.30) w e ha v e in this case the follo wing f ormulas: h (1) β , + ( m, λ ) · h (1) β , − ( m, λ ) = − L 12 m ( λ ) L 21 0 ( λ ) ( ζ − β 2 ) m = < Ap, p > | m < Aq , q > | 0 ( ζ − β 2 ) m g Y j =1 ζ − µ 2 j ( m ) ζ − ν 2 j (0) , h (2) β , + ( m, λ ) · h (2) β , − ( m, λ ) = L 21 m ( λ ) L 21 0 ( λ ) ( ζ − β 2 ) m = < Aq , q > | m < Aq , q > | 0 ( ζ − β 2 ) m g Y j =1 ζ − ν 2 j ( m ) ζ − ν 2 j (0) . (3.32) As λ → ∞ , from equations (3.26), (3.29) and (3.31 ) we ﬁ nd their asymptptic b eha viours, h (1) β , + ( m, λ ) = u m < Aq , q > | 0 u 0 λ m +1 + O ( λ m − 1 ) , h (1) β , − ( m, λ ) = O ( λ m − 1 ) , h (2) β , + ( m, λ ) = O ( λ m ) , h (2) β , − ( m, λ ) = u 0 u m λ m + O ( λ m − 2 ) . (3.33) T o get w ell-deﬁned meromorp hic fu nctions on R , we separate the t w o cases of o dd and ev en m , i.e., m = 2 k − 1 , 2 k , then put equation (3.29) in the f orm h (1) β , ± (2 k − 1 , λ ) = λc ± λ [ λ − 1 M 11 β (2 k − 1 , λ )] + M 12 β (2 k − 1 , λ ) , λh (2) β , ± (2 k − 1 , λ ) = λc ± λ M 21 β (2 k − 1 , λ ) + λM 22 β (2 k − 1 , λ ) , λh (1) β , ± (2 k, λ ) = λc ± λ M 11 β (2 k, λ ) + λM 12 β (2 k, λ ) , h (2) β , ± (2 k, λ ) = λc ± λ [ λ − 1 M 21 β (2 k, λ )] + M 22 β (2 k, λ ) . (3.34) Apart from λc ± λ , th e remaining fu nctions M ij β app earing on the r igh t-hand sides are p oly- nomials of the argument ζ = λ 2 . Thus, four meromorphic functions on R can b e obtained, with the v alues at p and τ p given as h (1) β (2 k − 1 , p ( λ 2 )) = h (1) β , + (2 k − 1 , λ ) , h (1) β (2 k − 1 , τ p ( λ 2 )) = h (1) β , − (2 k − 1 , λ ) , h (2) β (2 k − 1 , p ( λ 2 )) = λh (2) β , + (2 k − 1 , λ ) , h (2) β (2 k − 1 , τ p ( λ 2 )) = λh (2) β , − (2 k − 1 , λ ) , h (1) β (2 k, p ( λ 2 )) = λh (1) β , + (2 k, λ ) , h (1) β (2 k, τ p ( λ 2 )) = λh (1) β , − (2 k, λ ) , h (2) β (2 k, p ( λ 2 )) = h (2) β , + (2 k, λ ) , h (2) β (2 k, τ p ( λ 2 )) = h (2) β , − (2 k, λ ) . (3.35) 27 Then by using equation (3.32), w e get h (1) β (2 k − 1 , p ( λ 2 )) h (1) β (2 k − 1 , τ p ( λ 2 )) = < Ap, p > | 2 k − 1 < Aq , q > | 0 ( ζ − β 2 ) 2 k − 1 g Y j =1 ζ − µ 2 j (2 k − 1) ζ − ν 2 j (0) , h (2) β (2 k − 1 , p ( λ 2 )) h (2) β (2 k − 1 , τ p ( λ 2 )) = < Aq , q > | 2 k − 1 < Aq , q > | 0 ζ ( ζ − β 2 ) 2 k − 1 g Y j =1 ζ − ν 2 j (2 k − 1) ζ − ν 2 j (0) , h (1) β (2 k, p ( λ 2 )) h (1) β (2 k, τ p ( λ 2 )) = < Ap, p > | 2 k < Aq , q > | 0 ζ ( ζ − β 2 ) 2 k g Y j =1 ζ − µ 2 j (2 k ) ζ − ν 2 j (0) , h (2) β (2 k, p ( λ 2 )) h (2) β (2 k, τ p ( λ 2 )) = < Aq , q > | 2 k < Aq , q > | 0 ( ζ − β 2 ) 2 k g Y j =1 ζ − ν 2 j (2 k ) ζ − ν 2 j (0) . (3.36) Note th at at the branch p oint 0 , w e hav e the lo cal co ordinate λ . Thus by usin g equa- tions (3.33) and (3.36), we obtain divisors for the f ou r meromorphic fu nctions h (1) β (2 k − 1 , p ) , h (2) β (2 k − 1 , p ) , h (1) β (2 k, p ) and h (2) β (2 k, p ), resp ectiv ely: Div( h (1) β (2 k − 1 , p )) = g X j =1  p ( µ 2 j (2 k − 1)) − p ( ν 2 j (0))  + (2 k − 1) p ( β 2 ) − k ∞ + − ( k − 1) ∞ − , Div( h (2) β (2 k − 1 , p )) = g X j =1  p ( ν 2 j (2 k − 1)) − p ( ν 2 j (0))  + { p (0) } + (2 k − 1) p ( β 2 ) − k ∞ + − k ∞ − , Div( h (1) β (2 k, p )) = g X j =1  p ( µ 2 j (2 k )) − p ( ν 2 j (0))  + { p (0) } + 2 k p ( β 2 ) − ( k + 1) ∞ + − k ∞ − , Div( h (2) β (2 k, p )) = g X j =1  p ( ν 2 j (2 k )) − p ( ν 2 j (0))  + 2 k p ( β 2 ) − k ∞ + − k ∞ − . (3.37) Similarly as pro ved in S ection 2.3, we put the ab o ve resu lts in the Jacobi v ariet y J ( R ), and arrive at the ev olution form ula for Ab el-Jacobi v ariables (3.23), ~ ψ (2 k − 1) ≡ ~ φ (0) + (2 k − 1) ~ Ω − β + k ~ Ω , (mo d T ) , ~ φ (2 k − 1) ≡ ~ φ (0) + (2 k − 1) ~ Ω − β + k ~ Ω + ~ Ω − 0 , (mo d T ) , ~ ψ (2 k ) ≡ ~ φ (0) + 2 k ~ Ω − β + ( k + 1) ~ Ω + ~ Ω − 0 , (mo d T ) , ~ φ (2 k ) ≡ ~ φ (0) + 2 k ~ Ω − β + k ~ Ω , (mo d T ) , (3.38) where ~ Ω − β = R ∞ − p ( β 2 ) ~ ω , ~ Ω − 0 = R ∞ − p (0) ~ ω , and ~ Ω = R ∞ + ∞ − ~ ω . As a result, the theta fu n ction 28 expressions of Bak er-Akhiezer functions h ( l ) β ( m, p )) , ( l = 1 , 2), read h (1) β (2 k − 1 , p ) = θ [ − A ( p ) + ~ ψ (2 k − 1) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ ψ (2 k − 1) + ~ K ] · · u 2 k − 1 < Aq , q > | 0 u 0 · 1 ( r + β ) k · e (1 − k ) R ∞ + p ω [ p ( β 2 ) , ∞ − ]+ k R p p 0 ω [ p ( β 2 ) , ∞ + ] , h (2) β (2 k − 1 , p ) = θ [ − A ( p ) + ~ φ (2 k − 1) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k − 1) + ~ K ] · · u 0 u 2 k − 1 · 1 ( r − β ) k − 1 r − 0 · e − k R ∞ − p ω [ p ( β 2 ) , ∞ + ]+ R p p 0 ( k − 1) ω [ p ( β 2 ) , ∞ − ]+ ω [ p (0) , ∞ − ] , h (1) β (2 k, p ) = θ [ − A ( p ) + ~ ψ (2 k ) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ ψ (2 k ) + ~ K ] · · u 2 k < Aq , q > | 0 u 0 · 1 ( r + β ) k r + 0 · e − k R ∞ + p ω [ p ( β 2 ) , ∞ − ]+ R p p 0 k ω [ p ( β 2 ) , ∞ + ]+ ω [ p (0) , ∞ + ] , h (2) β (2 k, p ) = θ [ − A ( p ) + ~ φ (2 k ) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k ) + ~ K ] · · u 0 u 2 k · 1 ( r − β ) k · e − k R ∞ − p ω [ p ( β 2 ) , ∞ + ]+ k R p p 0 ω [ p ( β 2 ) , ∞ − ] , (3.39) where r + 0 = lim p →∞ + 1 ζ ( p ) e R p p 0 ω [ p (0) , ∞ + ] , r − 0 = lim p →∞ − 1 ζ ( p ) e R p p 0 ω [ p (0) , ∞ − ] , r + β = lim p →∞ + 1 ζ ( p ) e R p p 0 ω [ p ( β 2 ) , ∞ + ] , r − β = lim p →∞ − 1 ζ ( p ) e R p p 0 ω [ p ( β 2 ) , ∞ − ] . In this case, w e meet a pr ob lem th at the discr ete p oten tials are in th e expression (3.39). Ho w ev er, when we deduce th e formulas for the p otentia ls u ( m ) and a ( m ), the p roblem is solv ed with the help of th e relation in (3.21). Th is relation arises from the parametrizati on for constructing the Lax pair of lp mKdV equation (1.4). According to Section 2.3, we no w in ve rse the discrete p oten tials b y u sing the ab o v e resu lts. Prop osition 3.2. Th e p oten tials u ( m ) and a ( m ), deﬁned b y equation (3.21 ), h a v e explicit ev olution form ulas along th e S m β -ﬂo w , resp ectiv ely u ( m ) = u ( δ m ) · θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( m )] · θ [ ~ K ( δ m ) + ~ Ω] θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( δ m )] · θ [ ~ K ( m ) + ~ Ω] · · e m − δ m 2 [ δ m R β +( − 1) m R 0 β ] , (3.40) 29 a ( m ) =( a (0)) ( − 1) m · θ [(1 − δ m +1 ) ~ Ω − ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( m + 1)] · θ [ ~ K ( δ m +1 ) + ~ Ω] θ [(1 − δ m +1 ) ~ Ω − ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( δ m +1 )] · θ [ ~ K ( m + 1) + ~ Ω] · · θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( δ m )] · θ [ ~ K ( m ) + ~ Ω] θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( m )] · θ [ ~ K ( δ m ) + ~ Ω] · · e 1 2 [ m ( − 1) m + δ m ] R β + m ( − 1) m +1 R 0 β , (3.41) where δ j is equ al to 0 and 1 for even and o dd j resp ectiv ely , and ~ K ( m ) = ~ φ ( m ) + ~ K + Z p 0 ∞ + ~ ω , R β = ln r + β r ± β r − β r ∓ β , R 0 β = ( Z p ( β 2 ) p 0 ω [ p (0) , ∞ + ] + ω [ p (0) , ∞ − ]) · ln r + β β 2 r ∓ β r + 0 r − 0 , r ± β = e R ∞ + p 0 ω [ p ( β 2 ) , ∞ − ] , r ∓ β = e R ∞ − p 0 ω [ p ( β 2 ) , ∞ + ] . (3.42) Pr o of. By equation (3.28), we hav e      h (1) β (2 k, p ) = ζ a 2 k − 1 h (1) β (2 k − 1 , p ) + β h (2) β (2 k − 1 , p ) , h (2) β (2 k, p ) = β h (1) β (2 k − 1 , p ) + a − 1 2 k − 1 h (2) β (2 k − 1 , p ) , (3.43) and      h (1) β (2 k + 1 , p ) = a 2 k h (1) β (2 k, p ) + β h (2) β (2 k, p ) , h (2) β (2 k + 1 , p ) = β h (1) β (2 k, p ) + ζ a − 1 2 k h (2) β (2 k, p ) , (3.44) where p = p ( ζ ) , ζ = λ 2 . According to (3.37), the order of th e zero p ( β 2 ) of h ( l ) β ( m, p )) , ( l = 1 , 2), is equal to m . Thus, from th e ab ov e equ ations we get a 2 k − 1 = lim λ → β − h (2) β (2 k − 1 , p ( λ 2 )) β h (1) β (2 k − 1 , p ( λ 2 )) , a 2 k = lim λ → β − β h (2) β (2 k, p ( λ 2 )) h (1) β (2 k, p ( λ 2 )) . By usin g equations (3.38) and (3.39), w e obtain th e follo wing r elation in terms of theta functions b et w een u m and a m a 2 k − 1 = θ [2 k ~ Ω − β + ( k + 1) ~ Ω + ~ Ω − 0 + ~ K (0)] θ [2 k ~ Ω − β + ( k + 1) ~ Ω + ~ K (0)] · θ [ ~ Ω + ~ K (0)] θ [(2 k − 1) ~ Ω − β + ( k + 1) ~ Ω + ~ Ω − 0 + ~ K (0)] · · θ [(2 k − 1) ~ Ω − β + k ~ Ω + ~ K (0)] θ [ ~ K (0)] · < Aq , q > | 0 u 2 0 ( − β ) u 2 2 k − 1 · ( r + β ) k ( r ± β ) k − 1 ( r − β ) k − 1 ( r ∓ β ) k r − 0 · e R p ( β 2 ) p 0 ω [ p (0) , ∞ − ] , a 2 k = θ [(2 k + 1) ~ Ω − β + ( k + 1) ~ Ω + ~ K (0)] θ [(2 k + 1) ~ Ω − β + ( k + 2) ~ Ω + ~ Ω − 0 + ~ K (0)] · θ [ ~ Ω + ~ K (0)] θ [2 k ~ Ω − β + ( k + 1) ~ Ω + ~ K (0)] · · θ [2 k ~ Ω − β + ( k + 1) ~ Ω + ~ Ω − 0 + ~ K (0)] θ [ ~ K (0)] · ( − β ) < Aq , q > | 0 u 2 0 u 2 2 k · ( r + β r ± β ) k r + 0 ( r − β r ∓ β ) k · e − R p ( β 2 ) p 0 ω [ p (0) , ∞ + ] . (3.45) 30 Note that (3.21) giv es another relation b et w een them, i.e. a 2 k − 1 = u 2 k /u 2 k − 1 , a 2 k = u 2 k + 1 /u 2 k , which imp lies a 2 k a 2 k − 1 = u 2 k − 1 u 2 k + 1 u 2 2 k , a 2 k + 1 a 2 k = u 2 k u 2 k + 2 u 2 2 k + 1 . (3.46) Substituting (3.45) into (3.46), we obtain the cen tral result (3.40) for the s olution by induction and some calculations. Then by using (3.21), equation (3.41) is obtained as w ell.  No w the theta function expression for th e discrete p oten tial u ( m ) is wr itten do wn . By whic h w e will discuss the exp licit solutions to th e lpmKd V equation (1.4) thr ough the comm utativit y of discrete ﬂo ws. 3.3 Th e ﬁnite gen us solutions to the lpmKdV equation T aking n o w an y tw o distinct lattice p arameters β 2 1 , β 2 2 , the in tegrable symplectic maps S β 1 and S β 2 share the same Liouville set of integ rals, the confo cal p olynomials, th ere- fore, the resu lting discr ete p hase ﬂo ws, i.e., S m β 1 - and S n β 2 -ﬂo w commute. Th u s a well- deﬁned function  p ( m, n ) , q ( m, n )  is obtained, and b y equation (3.21) th e j -th comp onent ( p j ( m, n ) , q j ( m, n )) solves t wo copies of equation (3.14 ) with β = β 1 , β 2 sim ultaneously in the case of λ = α j ,   ˜ p j ˜ q j   = ( α 2 j − β 2 1 ) − 1 / 2 D ( β 1 ) ( α j ; ˜ u/u )   p j q j   , (3.47)   ¯ p j ¯ q j   = ( α 2 j − β 2 2 ) − 1 / 2 D ( β 2 ) ( α j ; ¯ u/u )   p j q j   . (3.48) The compatibilit y of equations (3.47) and (3.48 ) is pro vided b y the commutat ivity of the S m β 1 - and S n β 2 -ﬂo w exp ressed by ¯ D ( β 1 ) D ( β 2 ) = ˜ D ( β 2 ) D ( β 1 ) . T hen from equation (3.3), the ev olution of the function u ( m ) giv en by equation (3.40) along the discrete ﬂo ws yields Prop osition 3.3. The lpm KdV equation (1.4) h as ﬁnite genus solutions as u ( m, n ) = u ( δ m , δ n ) · θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( m, n )] θ [(1 − δ n ) ~ Ω + ( δ n +1 − δ n ) ~ Ω − 0 + ~ K ( δ m , δ n )] · · θ [(1 − δ n ) ~ Ω + ( δ n +1 − δ n ) ~ Ω − 0 + ~ K ( δ m , n )] · θ [ ~ K ( δ m , δ n ) + ~ Ω] θ [(1 − δ m ) ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 + ~ K ( δ m , n )] · θ [ ~ K ( m, n ) + ~ Ω] · · e m − δ m 2 [ δ m R β 1 +( − 1) m R 0 β 1 ]+ n − δ n 2 [ δ n R β 2 +( − 1) n R 0 β 2 ] , (3.49) where ~ K ( m, n ) = ~ φ ( m, n )+ ~ K + R p 0 ∞ + ~ ω , ~ φ ( m, n ) = ~ φ (0 , 0)+ m ~ Ω − β 1 + n ~ Ω − β 2 + m + n + δ m + δ n 2 ~ Ω+ ( δ m + δ n ) ~ Ω − 0 , and R β k , R 0 β k are give n by equ ation (3.42) with β = β k , k = 1 , 2. 31 Up to n ow, the lpmK dV equation has b een resolve d via new integ rable symp lectic maps generated by a ﬁnite dimensional in tegrable Hamiltonian system asso ciate d with the K aup-New ell p roblem, wh ic h is diﬀerent from the r esults in [41]. 4 The lattice Sc h w arzian KdV equation Let us n ow study the lSKdV equation. T h e asso ciat ed contin uous sp ectral p roblem (1.8 ) and d iscr ete s p ectral pr ob lem (1.9 ) carries t w o p oten tials resp ectiv ely , similar to the case of the lpK dV equation, while the w a y to deal with them are s omewh at diﬀerent fr om the other cases. Nonetheless, from the earlier sections, it is eviden t that the d iscrete Lax equation p la ys an essential role, and that is also tru e in the present case. W e n ote in passin g that the lSK dV (1.7) ﬁrst app eared in [35] as a sp ecial parameter limit of a sligh tly more general equation, the NQC equation derive d in [48] in the cont ext of the direct lin earisation metho d. The latter quad r lateral lattice equ ation, that is equiv alen t to the (Q3) δ =0 equation of the ABS list [24] foun d more recen tly , also giv es rise discrete v ersion of the V olterra-Kac-v an Mo erb eke equation in sp ecia l parameter and con tin uu m limits. In the present con text, follo wing the computations of the p revious sections, w e are naturally concerned with the Lax matrix, for whic h we tak e the one of [49] (u p to a factor − 2 λ ), w h ic h corresp onds to the ﬁn ite-dimensional Hamiltonian sys tems for the Kac-v an Mo erb eke h ierarc h y , L ( λ ; p, q ) =   λ/ 2 + λQ λ ( p, q ) − < p, q > − Q λ ( Ap, p ) 1 + Q λ ( Aq , q ) − λ/ 2 − λQ λ ( p, q )   , (4.1) where Q λ ( ξ , η ) = < ( λ 2 − A 2 ) − 1 ξ , η > , A = diag( α 1 , . . . , α N ) w ith α 2 1 , . . . , α 2 N pairwise distinct and non-zero. Moreo ve r, we ha ve the follo wing linear map from (1.9 ):   ˜ p j ˜ q j   = ( α 2 j − β 2 ) − 1 / 2 D ( β ) ( α j ; a, s )   p j q j   , ( j = 1 , . . . , N ) . (4.2) No w w e compute Υ △ = L ( λ ; ˜ p , ˜ q ) D ( β ) ( λ ; a, s ) − D ( β ) ( λ ; a, s ) L ( λ ; p, q ) , (4.3) 32 with the en tries Υ 11 = − β s − 1 < ˜ p, ˜ q > − β s + a ( < ˜ p, ˜ q > − < p, q > ) , Υ 12 = λβ s − λa − 1 < ˜ p, ˜ q > + λa < p, q >, Υ 21 = λa − λa − 1 − λβ s − 1 , Υ 22 = β s − 1 < p, q > + β s − a − 1 ( < ˜ p, ˜ q > − < p, q > ) . By u sing Υ 21 , we choose the f orm ula s = β / ( a − a − 1 ) . (4.4) On one h and, (4.4 ) guaran tees the realization of an asso ciated integ rable symp lectic map; On the other hand, imp osing (4.4) , the sp ectral prob lem (1.9 ) can b e wr itten in the form ˜ χ = ( λ 2 − β 2 ) − 1 / 2 D ( β ) ( λ ; a ) χ, D ( β ) ( λ ; a ) =    λa β 2 a − a − 1 a − a − 1 λa − 1    . (4.5) No w the n umber of the d iscrete p oten tials is r educed to one, same as the lpmKdV situation. Inspired by the construction of the Lax pair for lp mKdV equation (1.4) (see S ection 3), w e ﬁnd the Lax p air for lSK dV equation (1.7), ˜ χ = D ( β 1 ) ( λ ; ˜ z /z , β 1 z ˜ z / ( ˜ z 2 − z 2 )) χ, ¯ χ = D ( β 2 ) ( λ ; ¯ z /z , β 2 z ¯ z / ( ¯ z 2 − z 2 )) χ. (4.6) Indeed, by direct calculat ion, w e get ¯ D ( β 1 ) D ( β 2 ) − ˜ D ( β 2 ) D ( β 1 ) =     ˜ ¯ z z ( ˜ ¯ z 2 − ¯ z 2 )( ˜ ¯ z 2 − ˜ z 2 ) λz ˜ ¯ z ( ˜ z 2 + ¯ z 2 − ˜ ¯ z 2 − z 2 ) ( ˜ z 2 − z 2 )( ¯ z 2 − z 2 )( ˜ ¯ z 2 − ˜ z 2 )( ˜ ¯ z 2 − ¯ z 2 ) 0 − z ˜ ¯ z ( ˜ z 2 − z 2 )( ¯ z 2 − z 2 )     Ξ , (4.7) where Ξ = β 2 1 ( ˜ ¯ z 2 − ˜ z 2 )( ¯ z 2 − z 2 ) − β 2 2 ( ˜ ¯ z 2 − ¯ z 2 )( ˜ z 2 − z 2 ), and hence the discrete zero curv ature equation ¯ D ( β 1 ) D ( β 2 ) − ˜ D ( β 2 ) D ( β 1 ) = 0 implies Ξ = 0. The lSKdV equation (1.7) can b e deduced by letting u = z 2 . F ollo wing the pr o cedure ap p lied in the p receding s ections, we n o w treat the lSKd V equation in a similar wa y . 4.1 The in tegrable Hamiltonian system Based on the Lax matrix (4.1), we no w exhibit an in tegrable Hamiltonian system for further calculations. Th e follo wing fundamental Poisson br ac k et relation links (4.1) to a 33 classical r -matrix stucture, { L ( λ ) ⊗ , L ( µ ) } = [ r ( λ, µ ) , L 1 ( λ )] + [ r ′ ( λ, µ ) , L 2 ( µ )] , r = 2 λ 2 − µ 2 P µλ + σ 3 ⊗ σ + , r ′ = 2 λ 2 − µ 2 P λµ − σ 3 ⊗ σ + , P λµ =         λ 0 0 0 0 0 µ 0 0 µ 0 0 0 0 0 λ         . (4.8) The asso ciated generating f unction reads: F λ = − λ 2 (1 / 4 + Q 2 λ ( p, q ) + Q λ ( p, q ))+ < p, q > (1 + Q λ ( Aq , q )) + Q λ ( Ap, p )(1 + Q λ ( Aq , q )) , and the corresp onding ev olution along the t λ -ﬂo w for the Lax matrices reads: d L ( µ ) / d t λ = [ W ( λ, µ ) , L ( µ )] , W ( λ, µ ) = 2 µ λ 2 − µ 2 L ( λ ) +  2 L 11 ( λ ) λ + µ − L 21 ( λ )  σ 3 . (4.9) Considering n o w the p o w er series expression, F λ = − ζ 4 + ∞ X j =1 F j ζ − j , ζ = λ 2 , (4.10) it yields t w o types of ob jects: a) N smo oth fun ctions { F j ( p, q ) , 1 ≤ j ≤ N } inv olutiv e with eac h other, F 1 = − < p, q > 2 − < A 2 p, q > + < Ap, p > + < p, q >< Aq , q >, F j = − < A 2 j p, q > + < A 2 j − 1 p, p > + < p, q > < A 2 j − 1 q , q > − X k + l +1= j ; k ,l ≥ 0 < A 2 k p, q > < A 2 l p, q > + X k + l +2= j ; k ,l ≥ 0 < A 2 k + 1 p, p >< A 2 l +1 q , q >, ( j ≥ 2) . b) square ro ot H λ satisfying − 4 λ 2 F λ = (1 + 4 H λ ) 2 , H λ = ∞ X j =1 H j ζ − j − 1 , ( 4.11) where H 1 = − 1 2 F 1 , w h ose corresp ond ing Hamiltonian s ystem is ∂ x   p j q j   =   − ∂ H 1 /∂ q j ∂ H 1 /∂ p j   =    − α 2 j / 2+ < p, q > + < Aq , q > 2 α j < p, q > − α j α 2 j / 2 − < p, q > − < Aq , q > 2      p j q j   , (4.12) 34 (1 ≤ j ≤ N ). Comp aring with equation (1.8), w e select the constrain t ( v , w ) = ( < p, q >, < Aq , q > / 2) . (4.13) In this sense, (4.12) is the nonlinearisation of (1.8 ). Consider th e fractional expression F λ = − 1 4 R ( ζ ) ζ α 2 ( ζ ) , R ( ζ ) = ζ α ( ζ ) N +1 Y j =1 ( ζ − ζ j ) , α ( ζ ) = N Y j =1 ( ζ − α 2 j ) , (4.14) then a curv e R : ξ 2 = R ( ζ ), with genus g = N , is obtained. The curve R has t wo inﬁnities ∞ + , ∞ − , and branch p oin ts ζ j , α 2 j , 0 . And the general p oin ts on R are p ( ζ ) =  ζ , ξ = p R ( ζ )  , ( τ p )  ζ ) = ( ζ , ξ = − p R ( ζ )  , ζ ∈ C . In tro d ucing the corr esp onding elliptic co ordinates µ 2 j , ν 2 j : L 12 ( λ ) = − < p, q > m ( ζ ) α ( ζ ) , m ( ζ ) = Π N j =1 ( ζ − µ 2 j ) , L 21 ( λ ) = n ( ζ ) α ( ζ ) , n ( ζ ) = Π N j =1 ( ζ − ν 2 j ) , (4.15) the qu asi-Ab el-Jacobi and Ab el-Jacobi v ariables are deﬁn ed resp ectiv ely as ~ φ ′ = ( φ ′ 1 , · · · , φ ′ g ) T = g X k =1 Z p ( ν 2 k ) p 0 ~ ω ′ , ~ φ = C ~ φ ′ = A ( g X k =1 p ( ν 2 k )) , ~ ψ ′ = ( ψ ′ 1 , · · · , ψ ′ g ) T = g X k =1 Z p ( µ 2 k ) p 0 ~ ω ′ , ~ ψ = C ~ ψ ′ = A ( g X k =1 p ( µ 2 k )) , (4.16) where ~ ω ′ = ( ω ′ 1 , · · · , ω ′ g ) T , ω ′ j = ζ g − j d ζ / (2 p R ( ζ )). Let us consider one of the ent ries in (4.9), namely d L 12 ( µ ) / d t λ = 2( W 11 ( λ, µ ) L 12 ( µ ) − W 12 ( λ, µ ) L 11 ( µ )) , and setting µ = µ k , then w e get the Dubro vin t yp e equations 1 2 q R ( µ 2 k ) d( µ 2 k ) d t λ = 1 α ( ζ ) m ( ζ ) ( ζ − µ 2 k ) m ′ ( µ 2 k ) , (1 ≤ k ≤ g ) (4.1 7) from wh ic h we ha ve { ψ ′ l , F λ } = d ψ ′ l d t λ = 1 α ( ζ ) ζ g − l , (1 ≤ l ≤ g ) . Hence, ∞ X j =1 { ψ ′ l , F j } ζ − j = − ∞ X j = l A j − l ζ − j , (4 .18) 35 where A 0 = 1, A j − l = 0 ( j < l ). Thus, we conclude  { ψ ′ l , F j }  g × g =             1 A 1 A 2 . . . A g − 1 1 A 1 . . . A g − 2 . . . . . . . . . . . . A 1 1             . whic h imp lies F 1 , . . . , F N are fun ctionally indep enden t on the ph ase space N = ( R 2 N , d p ∧ d q ). The Liouville in tegrabilit y for the Hamiltonian system (4.12) is established. W e w ill pro ceed by constructing in tegrable symplectic maps for lSKdV equation. 4.2 The in tegrable symplectic map With the help of the formula (4.4), th e map (4.2) can b e written in the form   ˜ p j ˜ q j   = 1 q α 2 j − β 2 D ( β ) ( α j ; a )   p j q j   , (1 ≤ j ≤ N ) , (4.19) where D ( β ) is giv en in (4.5). Moreo v er, equ ation (4.3) b ecomes Υ △ = L ( λ ; ˜ p , ˜ q ) D ( β ) ( λ ; a ) − D ( β ) ( λ ; a ) L ( λ ; p, q ) , (4.2 0) with the comp onents Υ 11 = a − 1 < ˜ p, ˜ q > − a < p, q > − β 2 a − a − 1 , Υ 12 = − λ Υ 11 , Υ 21 = 0 , Υ 22 = − Υ 11 . Here we also us e Υ for short. Th en b y equ ations (4.1) and (4.19), w e get < ˜ p, ˜ q > = ( a 2 − 1) L 12 ( β ) + a 2 < p, q > + β 2 a 2 − 1 (1 − L 21 ( β )) + 2 β ( β 2 − L 11 ( β )) . (4.21) Substituting it into Υ 11 , we obtain Υ = P ( β ) ( a ; p, q ) a 3 − a   1 − λ 0 − 1   , (4.22) 36 where P ( β ) ( a ; p, q ) = ( a 2 − 1) 2 L 12 ( β ) − 2 β ( a 2 − 1) L 11 ( β ) − β 2 L 21 ( β ) , (4.23 ) whic h is a quadratic p olynomial with resp ect to a 2 − 1. Th e ro ots to th e quadratic equation P ( β ) ( a ; p, q ) = 0 are giv en by a 2 − 1 = − 1 < p, q > + Q β ( Ap, p )  β 2 (1 / 2 + Q β ( p, q )) ± p R ( β 2 ) 2 α ( β 2 )  , (4.24) whic h are the v alues of a w ell-deﬁned meromorph ic function on R , D ( p ) = − 1 < p, q > + Q β ( Ap, p )  β 2 (1 / 2 + Q β ( p, q )) + ξ 2 α ( β 2 )  , at the p oin ts p ( β 2 ) and ( τ p )( β 2 ), resp ectiv ely . Equation (4.24) provides th e constraint on the discrete p oten tial a , den oting it a = f β ( p, q ). Thus, w e obtain the follo wing nonlinear map from the linear map (4.19): S β :   ˜ p ˜ q   = ( A 2 − β 2 ) − 1 / 2    aAp + β 2 q a − a − 1 ( a − a − 1 ) p + a − 1 Aq         a = f β ( p,q ) . (4.25) W e assert that th e map S β is an in tegrable symplectic map sharing the same set of in tegrals { F j ( p, q ) , 1 ≤ j ≤ N } as the Hamiltonian system (4.12). In fact, under th e constrain t (4.24) we h a v e L ( λ ; ˜ p , ˜ q ) D ( β ) ( λ ; a ) = D ( β ) ( λ ; a ) L ( λ ; p, q ) , (4.26) b y equations (4.20) and (4.22 ), wh ic h imp lies S ∗ β ◦ F j ( p, q ) = F j ( p, q ) , 1 ≤ j ≤ N . The sym plectic p rop erty for S β , i.e., S ∗ β (d p ∧ d q ) = d p ∧ d q relies on the follo wing f orm ula: N X j =1 (d ˜ p j ∧ d ˜ q j − d p j ∧ d q j ) = 1 a ( a 2 − 1) 2 d P ( β ) ( a ; p, q ) ∧ d a. (4.27) As a consequence, a d iscrete S m β -ﬂo w can b e set up by setting  p ( m ) , q ( m )  = S m β ( p 0 , q 0 ), with ( p 0 , q 0 ) as an initial p oint. By equations (4.6) and (4.24), w e denote the corresp ond ing p oten tials as a ( m ) = a m = z m +1 /z m , u ( m ) = u m = z 2 m , (4.28) whic h lead to the d iscrete sp ectral p r oblem h β ( m + 1 , λ ) = D ( β ) m ( λ ) h β ( m, λ ) , (4.29) where D ( β ) m ( λ ) = D ( β ) ( λ ; a m ) , wh ose fund amen tal solution matrix M β ( m, λ ) satisﬁes M β ( m + 1 , λ ) = D ( β ) m ( λ ) M β ( m, λ ) , M β (0 , λ ) = I . (4.3 0) 37 Hence, follo wing this by iteration, we get the solution as a matrix pro du ct chai n M β ( m, λ ) = D ( β ) m − 1 ( λ ) D ( β ) m − 2 ( λ ) . . . D ( β ) 0 ( λ ) , (4.31) whic h implies d et M β ( m, λ ) = ( λ 2 − β 2 ) m , and as λ → ∞ , M β ( m, λ ) =    z m z 0 λ m + O ( λ m − 2 ) O ( λ m − 1 ) O ( λ m − 1 ) z 0 z m λ m + O ( λ m − 2 )    . (4.32) F urth er m ore, fr om the compatibilit y r elation (4.26) along the S m β -ﬂo w L m +1 ( λ ) D ( β ) m ( λ ) = D ( β ) m ( λ ) L m ( λ ) , (4.33) where L m ( λ ) = L ( λ ; p ( m ) , q ( m )), and equation (4.31), we obtain L m ( λ ) M β ( m, λ ) = M β ( m, λ ) L 0 ( λ ) , (4.34) whic h is helpful to deriv e th e relev ant formulas b elow for zeros and p oles of the corre- sp ond in g meromorp hic fun ctions. In order to pro ceed we need some pr op erties of the linear op erator L m ( λ ) with v al- ues in the solution space of equation, (4.29). T hrough dir ect calculation, w e obtain the eigen v alues of the op erator as follo ws: ρ ± λ = ± ρ λ = ± p −F λ = ± p R ( ζ ) / 2 λα ( ζ ) , (4.35) ρ λ = λ 2 (1 + O ( ζ − 2 )) , ( λ → ∞ ) , (4.36) together w ith the asso ciated eigenfunctions satisfying h β , ± ( m + 1 , λ ) = D ( β ) m ( λ ) h β , ± ( m, λ ) , (4.37) h β , ± ( m, λ ) =   h (1) β , ± ( m, λ ) h (2) β , ± ( m, λ )   = M β ( m, λ )   c ± λ 1   , (4.38)  L m ( λ ) − ρ ± λ  h β , ± ( m, λ ) = 0 . (4.39) Let m = 0 in equations (4.38 ) and (4.39) , then c ± λ = L 11 0 ( λ ) ± ρ λ L 21 0 ( λ ) = − L 12 0 ( λ ) L 11 0 ( λ ) ∓ ρ λ , c + λ c − λ = − L 12 0 ( λ ) L 21 0 ( λ ) , (4.40) As λ → ∞ , we h a v e c + λ = λ (1 + O ( ζ − 1 )) , c − λ = < p 0 , q 0 > λ − 1 (1 + O ( ζ − 1 )) . (4.41) 38 Moreo v er, λc + λ and λc − λ are the v alues of a meromorph ic function on R constru cted by (4.14), C ( p ) = ζ / 2 + ζ < ( ζ − A 2 ) − 1 p 0 , q 0 > + ξ / 2 α ( ζ ) 1+ < ( ζ − A 2 ) − 1 Aq 0 , q 0 > , at the p oints p ( λ 2 ) and ( τ p )( λ 2 ), resp ectiv ely . Based on the resu lts ab o v e, we now prepare some f orm ulas to discuss the common eigen v ectors h β , ± ( m, λ ) on the lev el of Riemann surface th eory . Throu gh some calculations, w e ha v e h (1) β , + ( m, λ ) · h (1) β , − ( m, λ ) = < p, q > | m ( ζ − β 2 ) m N Y j =1 ζ − µ 2 j ( m ) ζ − ν 2 j (0) , h (2) β , + ( m, λ ) · h (2) β , − ( m, λ ) =( ζ − β 2 ) m N Y j =1 ζ − ν 2 j ( m ) ζ − ν 2 j (0) , (4.42) b y using equations (4.15), (4.31), (4.34), (4.38) and (4.40). Similarly as the pr evious sections, the follo wing asymptotic b eha viours ( λ → ∞ ) are also obtained via equ ations (4.32), (4.38), (4.41): h (1) β , + ( m, λ ) = z m z 0 λ m +1 + O ( λ m − 1 ) , h (1) β , − ( m, λ ) = O ( λ m − 1 ) , h (2) β , + ( m, λ ) = O ( λ m ) , h (2) β , − ( m, λ ) = z 0 z m λ m + O ( λ m − 2 ) . (4.43) T ec hn ically , separating out th e t w o cases: m = 2 k − 1 , 2 k , from equation (4.38) w e get h (1) β , ± (2 k − 1 , λ ) = λc ± λ [ λ − 1 M 11 β (2 k − 1 , λ )] + M 12 β (2 k − 1 , λ ) , λh (2) β , ± (2 k − 1 , λ ) = λc ± λ M 21 β (2 k − 1 , λ ) + λM 22 β (2 k − 1 , λ ) , λh (1) β , ± (2 k, λ ) = λc ± λ M 11 β (2 k, λ ) + λM 12 β (2 k, λ ) , h (2) β , ± (2 k, λ ) = λc ± λ [ λ − 1 M 21 β (2 k, λ )] + M 22 β (2 k, λ ) , (4.44) then four meromorph ic fun ctions on R can b e constructed, with the v alues at p and τ p as h (1) β (2 k − 1 , p ( λ 2 )) = h (1) β , + (2 k − 1 , λ ) , h (1) β (2 k − 1 , τ p ( λ 2 )) = h (1) β , − (2 k − 1 , λ ) , h (2) β (2 k − 1 , p ( λ 2 )) = λh (2) β , + (2 k − 1 , λ ) , h (2) β (2 k − 1 , τ p ( λ 2 )) = λh (2) β , − (2 k − 1 , λ ) , h (1) β (2 k, p ( λ 2 )) = λh (1) β , + (2 k, λ ) , h (1) β (2 k, τ p ( λ 2 )) = λh (1) β , − (2 k, λ ) , h (2) β (2 k, p ( λ 2 )) = h (2) β , + (2 k, λ ) , h (2) β (2 k, τ p ( λ 2 )) = h (2) β , − (2 k, λ ) . (4.45) 39 F rom the formulas (4.42), w e ha v e h (1) β (2 k − 1 , p ( λ 2 )) h (1) β (2 k − 1 , τ p ( λ 2 )) = < p, q > | 2 k − 1 ( ζ − β 2 ) 2 k − 1 N Y j =1 ζ − µ 2 j (2 k − 1) ζ − ν 2 j (0) , h (2) β (2 k − 1 , p ( λ 2 )) h (2) β (2 k − 1 , τ p ( λ 2 )) = ζ ( ζ − β 2 ) 2 k − 1 N Y j =1 ζ − ν 2 j (2 k − 1) ζ − ν 2 j (0) , h (1) β (2 k, p ( λ 2 )) h (1) β (2 k, τ p ( λ 2 )) = < p, q > | 2 k ζ ( ζ − β 2 ) 2 k N Y j =1 ζ − µ 2 j (2 k ) ζ − ν 2 j (0) , h (2) β (2 k, p ( λ 2 )) h (2) β (2 k, τ p ( λ 2 )) = ( ζ − β 2 ) 2 k N Y j =1 ζ − ν 2 j (2 k ) ζ − ν 2 j (0) , (4.46) No w the zeros and p oles for the meromorp hic functions h ( l ) β ( m, p )) , ( l = 1 , 2) can b e ob- tained by equations (4.43) and (4.46). Th is results in to the f ollo win g expr essions of the divisors for h ( l ) β ( m, p )) , ( l = 1 , 2): Div( h (1) β (2 k − 1 , p )) = g X j =1  p ( µ 2 j (2 k − 1)) − p ( ν 2 j (0))  + (2 k − 1) p ( β 2 ) − k ∞ + − ( k − 1) ∞ − , Div( h (2) β (2 k − 1 , p )) = g X j =1  p ( ν 2 j (2 k − 1)) − p ( ν 2 j (0))  + { 0 } + (2 k − 1) p ( β 2 ) − k ∞ + − k ∞ − , Div( h (1) β (2 k, p )) = g X j =1  p ( µ 2 j (2 k )) − p ( ν 2 j (0))  + { 0 } + 2 k p ( β 2 ) − ( k + 1) ∞ + − k ∞ − , Div( h (2) β (2 k, p )) = g X j =1  p ( ν 2 j (2 k )) − p ( ν 2 j (0))  + 2 k p ( β 2 ) − k ∞ + − k ∞ − . (4.47) W e no w view th e ab o v e formula (4.47) from the Ab el-Jacobi v ariables. T hen, the asso ci- ated S m β -ﬂo w is linearized on the Jacobi v ariet y J ( R ) as ~ ψ (2 k − 1) ≡ ~ φ (0) + (2 k − 1) ~ Ω − β + k ~ Ω , (mo d T ) , ~ φ (2 k − 1) ≡ ~ φ (0) + (2 k − 1) ~ Ω − β + k ~ Ω + ~ Ω − 0 , (mo d T ) , ~ ψ (2 k ) ≡ ~ φ (0) + 2 k ~ Ω − β + ( k + 1) ~ Ω + ~ Ω − 0 , (mo d T ) , ~ φ (2 k ) ≡ ~ φ (0) + 2 k ~ Ω − β + k ~ Ω , (mo d T ) , (4.48) where ~ Ω − β = R ∞ − p ( β 2 ) ~ ω , ~ Ω − 0 = R ∞ − 0 ~ ω , and ~ Ω = R ∞ + ∞ − ~ ω . W e can now w rite d o wn Bak er- 40 Akhiezer fu nctions h ( l ) β ( m, p )) , ( l = 1 , 2) in terms of theta fu nctions, h (1) β (2 k − 1 , p ) = θ [ − A ( p ) + ~ ψ (2 k − 1) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ ψ (2 k − 1) + ~ K ] · · z 2 k − 1 z 0 · 1 ( r + β ) k · e (1 − k ) R ∞ + p ω [ p ( β 2 ) , ∞ − ]+ k R p p 0 ω [ p ( β 2 ) , ∞ + ] , h (2) β (2 k − 1 , p ) = θ [ − A ( p ) + ~ φ (2 k − 1) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k − 1) + ~ K ] · · z 0 z 2 k − 1 · 1 ( r − β ) k − 1 r − 0 · e − k R ∞ − p ω [ p ( β 2 ) , ∞ + ]+ R p p 0 ( k − 1) ω [ p ( β 2 ) , ∞ − ]+ ω [ 0 , ∞ − ] , h (1) β (2 k, p ) = θ [ − A ( p ) + ~ ψ (2 k ) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ ψ (2 k ) + ~ K ] · · z 2 k z 0 · 1 ( r + β ) k r + 0 · e − k R ∞ + p ω [ p ( β 2 ) , ∞ − ]+ R p p 0 k ω [ p ( β 2 ) , ∞ + ]+ ω [ 0 , ∞ + ] , h (2) β (2 k, p ) = θ [ − A ( p ) + ~ φ (2 k ) + ~ K ] θ [ − A ( p ) + ~ φ (0) + ~ K ] · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k ) + ~ K ] · · z 0 z 2 k · 1 ( r − β ) k · e − k R ∞ − p ω [ p ( β 2 ) , ∞ + ]+ k R p p 0 ω [ p ( β 2 ) , ∞ − ] , (4.49) where r + 0 = lim p →∞ + 1 ζ ( p ) e R p p 0 ω [ 0 , ∞ + ] , r − 0 = lim p →∞ − 1 ζ ( p ) e R p p 0 ω [ 0 , ∞ − ] , r + β = lim p →∞ + 1 ζ ( p ) e R p p 0 ω [ p ( β 2 ) , ∞ + ] , r − β = lim p →∞ − 1 ζ ( p ) e R p p 0 ω [ p ( β 2 ) , ∞ − ] . (4.50) With the help of the expression (4.49), we no w calculate th e d iscrete p oten tial whic h leads to the ﬁn ite gen us solutions for lSKd V equation (1.7). Prop osition 4.1. Th e discrete p oten tial u ( m ), deﬁned by (4.28), satisﬁes the recurs ive relation, u ( m ) − u ( m + 1) = u (0) · θ [ ~ Ω + ~ K (0)] θ [ ~ K (0)] · θ [ δ m ~ Ω + ( δ m − δ m +1 ) ~ Ω − 0 + ~ K ( m + 1)] θ [( δ m + δ m +1 ) ~ Ω + ~ K ( m + 1)] · · θ [ δ m +1 ~ Ω − ( δ m − δ m +1 ) ~ Ω − 0 + ~ K ( m )] θ [ ~ Ω + ~ K ( m )] · ( β 2 ) δ m +1 · ( r + β ) ( m + δ m ) / 2 ( r − β ) ( m − δ m ) / 2 · ( r + 0 ) δ m +1 ( r − 0 ) δ m · · e R ∞ + p 0 m − δ m 2 ω [ p ( β 2 ) , ∞ − ] − R ∞ − p 0 m + δ m 2 ω [ p ( β 2 ) , ∞ + ] − δ m +1 R p p 0 ω [ 0 , ∞ + ]+ δ m R p p 0 ω [ 0 , ∞ − ] , (4.51) where ~ K ( m ) = ~ φ ( m ) + ~ K + R p 0 ∞ + ~ ω , and δ j is equal to 0 and 1 for ev en and o dd j r esp ectiv ely . 41 Pr o of. F rom equation (4.37), we obtain h (1) β (2 k + 1 , p ) = a 2 k h (1) β (2 k, p ) + β 2 a 2 k − a − 1 2 k h (2) β (2 k, p ) , (4 .52) h (1) β (2 k + 2 , p ) = ζ a 2 k + 1 h (1) β (2 k + 1 , p ) + β 2 a 2 k + 1 − a − 1 2 k + 1 h (2) β (2 k + 1 , p ) , (4.53 ) whic h implies h (1) β (2 k + 1 , p ) h (1) β (2 k, p ) = a 2 k + β 2 a 2 k − a − 1 2 k h (2) β (2 k, p ) h (1) β (2 k, p ) , (4.5 4) h (1) β (2 k + 2 , p ) h (1) β (2 k + 1 , p ) = ζ a 2 k + 1 + β 2 a 2 k + 1 − a − 1 2 k + 1 h (2) β (2 k + 1 , p ) h (1) β (2 k + 1 , p ) , (4.55) where p = p ( ζ ) , ζ = λ 2 . Let λ → β , we hav e a 2 2 k + β 2 lim λ → β h (2) β (2 k, p ( λ 2 )) h (1) β (2 k, p ( λ 2 )) = 1 , (4.56) a 2 2 k + 1 + lim λ → β h (2) β (2 k + 1 , p ( λ 2 )) h (1) β (2 k + 1 , p ( λ 2 )) = 1 , (4.57) according to the d ivisors giv en by (4.47). Then sub stituting (4.28) and the theta function expressions (4.49) int o (4.56) and (4.57), resp ectiv ely , we get u (2 k ) − u (2 k + 1) = u (0) · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] · θ [ − A ( p ( β 2 )) + ~ φ (2 k ) + ~ K ] θ [ − A ( p ( β 2 )) + ~ ψ (2 k ) + ~ K ] · · θ [ − A ( ∞ + ) + ~ ψ (2 k ) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k ) + ~ K ] · β 2 · r + 0 ·  r + β r − β  k · · e k R ∞ + p 0 ω [ p ( β 2 ) , ∞ − ] − k R ∞ − p 0 ω [ p ( β 2 ) , ∞ + ] − R p p 0 ω [ 0 , ∞ + ] , (4.58) and u (2 k + 1) − u (2 k + 2) = u (0) · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] · θ [ − A ( p ( β 2 )) + ~ φ (2 k + 1) + ~ K ] θ [ − A ( p ( β 2 )) + ~ ψ (2 k + 1) + ~ K ] · · θ [ − A ( ∞ + ) + ~ ψ (2 k + 1) + ~ K ] θ [ − A ( ∞ − ) + ~ φ (2 k + 1) + ~ K ] · 1 r − 0 · ( r + β ) k +1 ( r − β ) k · · e k R ∞ + p 0 ω [ p ( β 2 ) , ∞ − ] − ( k +1) R ∞ − p 0 ω [ p ( β 2 ) , ∞ + ]+ R p p 0 ω [ 0 , ∞ − ] , (4.59) 42 whic h giv e rise to the un iﬁed form u ( m ) − u ( m + 1) = u (0) · θ [ − A ( ∞ − ) + ~ φ (0) + ~ K ] θ [ − A ( ∞ + ) + ~ φ (0) + ~ K ] · θ [ − A ( p ( β 2 )) + ~ φ ( m ) + ~ K ] θ [ − A ( p ( β 2 )) + ~ ψ ( m ) + ~ K ] · · θ [ − A ( ∞ + ) + ~ ψ ( m ) + ~ K ] θ [ − A ( ∞ − ) + ~ φ ( m ) + ~ K ] · ( β 2 ) δ m +1 · ( r + β ) ( m + δ m ) / 2 ( r − β ) ( m − δ m ) / 2 · ( r + 0 ) δ m +1 ( r − 0 ) δ m · · e R ∞ + p 0 m − δ m 2 ω [ p ( β 2 ) , ∞ − ] − R ∞ − p 0 m + δ m 2 ω [ p ( β 2 ) , ∞ + ] − δ m +1 R p p 0 ω [ 0 , ∞ + ]+ δ m R p p 0 ω [ 0 , ∞ − ] . (4.60) Then b y us ing f orm ulas − A ( p ( β 2 )) = ~ Ω − β + ~ Ω + R p 0 ∞ + ~ ω and ~ ψ ( m ) = ~ φ ( m ) + δ m +1 ~ Ω + ( δ m +1 − δ m ) ~ Ω − 0 deduced by equation (4.48 ), equation (4.51) is pro ved.  4.3 Th e ﬁnite gen us solutions to the lSKdV equation According to the method s us ed in the preceding sections, we n o w ha v e tw o in tegrable sym- plectic maps S β 1 , S β 2 b y imp osing th e lattice parameter β tw o v alues β 1 , β 2 resp ectiv ely . Then b y iteration, S m β 1 - and S n β 2 -ﬂo w comm uting with eac h other are obtained as wel l. As a resu lt, from r ecursiv e relation (4.51 ) we obtain the solutions for the lSK d V equation. Prop osition 4.2. The ﬁ nite gen us solutions for the lSKdV equ ation (1.7) satisfy u ( m, n ) − u ( m + 1 , n ) = u (0 , n ) · θ [ ~ Ω + ~ K (0 , n )] θ [ ~ K (0 , n )] · θ [ δ m ~ Ω + ( δ m − δ m +1 ) ~ Ω − 0 + ~ K ( m + 1 , n )] θ [( δ m + δ m +1 ) ~ Ω + ~ K ( m + 1 , n )] · · θ [ δ m +1 ~ Ω − ( δ m − δ m +1 ) ~ Ω − 0 + ~ K ( m, n )] θ [ ~ Ω + ~ K ( m, n )] · ( β 2 1 ) δ m +1 · · ( r + β 1 ) ( m + δ m ) / 2 ( r − β 1 ) ( m − δ m ) / 2 · ( r + 0 ) δ m +1 ( r − 0 ) δ m · · e R ∞ + p 0 m − δ m 2 ω [ p ( β 2 1 ) , ∞ − ] − R ∞ − p 0 m + δ m 2 ω [ p ( β 2 1 ) , ∞ + ] − δ m +1 R p p 0 ω [ 0 , ∞ + ]+ δ m R p p 0 ω [ 0 , ∞ − ] , (4.61) where u (0 , n ) is giv en by u (0 , n ) − u (0 , n + 1) = u (0 , 0) · θ [ ~ Ω + ~ K (0 , 0)] θ [ ~ K (0 , 0)] · θ [ δ n ~ Ω + ( δ n − δ n +1 ) ~ Ω − 0 + ~ K (0 , n + 1)] θ [( δ n + δ n +1 ) ~ Ω + ~ K (0 , n + 1)] · · θ [ δ n +1 ~ Ω − ( δ n − δ n +1 ) ~ Ω − 0 + ~ K (0 , n )] θ [ ~ Ω + ~ K (0 , n )] · ( β 2 2 ) δ n +1 · ( r + β 2 ) ( n + δ n ) / 2 ( r − β 2 ) ( n − δ n ) / 2 · ( r + 0 ) δ n +1 ( r − 0 ) δ n · · e R ∞ + p 0 n − δ n 2 ω [ p ( β 2 2 ) , ∞ − ] − R ∞ − p 0 n + δ n 2 ω [ p ( β 2 2 ) , ∞ + ] − δ n +1 R p p 0 ω [ 0 , ∞ + ]+ δ n R p p 0 ω [ 0 , ∞ − ] , (4.62) 43 and ~ K ( m, n ) = ~ φ ( m, n )+ ~ K + R p 0 ∞ + ~ ω , ~ φ ( m, n ) = ~ φ (0 , 0)+ m ~ Ω − β 1 + n ~ Ω − β 2 + m + n + δ m + δ n 2 ~ Ω+ ( δ m + δ n ) ~ Ω − 0 . Besides, r + β j , r − β j are obtained by putting β = β j , j = 1 , 2 in equation (4.50) resp ectiv ely . Another w ay to obtain the analytic solution in terms of theta functions for equation (1.7) is calculating the p oten tial u ( m ) by the discrete integ ration u ( m ) = u (0) + m X j =1  u ( j ) − u ( j − 1)  , with the help of equation (4.51). The ev olution of u ( m ) along the corresp ond in g discrete ﬂo ws leads to the solutions as well. 5 Conclusion In this pap er, we exhib ited a new version of the algebro-ge ometric app roac h to deal with the partial d iﬀerence equations of K dV-t yp e, wh ic h is diﬀeren t fr om the existing r esults in the literatures [16, 41, 42]. W e h a v e p resen ted examples of int egrable s ymplectic maps and ﬁn ite gen us solutions for lattice Kd V-t yp e equations. In the lpK dV and lSKdV cases, there are t wo discr ete p oten tials, and w e n eed to imp ose constrain ts b etw een th em in order to construct the algebro-geo metric solutions usin g the tec hn ique of n onlinearisation. App lyin g the metho d and the constraint s, w e en d up with expressions for a single p oten tial f or the lSKdV equation as in the lpmKd V case. T hese cases share a similar algebro-geo metric structur e when constructing the explicit solutions in terms of theta functions. How ev er, in the lpKdV case, the Riemann surface is diﬀerent and the constraint is not enough to charact erize the solution. Hence, an alternativ e parametrization was constructed in ord er to s olv e the problem. In this pap er the discrete ve rsion of the Liouville-Arnol’d theorem, [2, 3], plays an essen tial role. W e p oint out in th e pr esen t cont ext that diﬀerent Liouville integ rable reductions can b e considered asso ciated w ith distinct Hamiltonian systems, leading all to solutions of one and the same partial d iﬀerence equation. A t this juncture, we wo uld lik e to p oin t out that No ether’s p rinciple for Hamiltonian systems tells u s that there is a corresp ondence b etw een integrals and symmetries. F urther- more, the inte grals of a Hamiltonian system form a Lie algebra with resp ect to th e P oisson brac k et, while the corresp ondin g ﬂo ws generate a Lie group. Therefore, we may conjecture 44 that th e algebraic structure b ehind the app roac h emplo ye d in our analysis could sh ed a ligh t on this phenomenon in discrete in tegrable systems. Ac kno wledgmen ts This wo rk is sup p orted by National Natural Science F ound ation of C hina (Gran t Nos. 11426 206; 11501521) , State Sc holarship F oun d of China (CSC No. 20190 7045035), and Graduate Stud en t Education Research F oun dation of Z hengzhou Unive rsity (Gran t No. YJSXWK C201913). W e would lik e to express m any thanks to Prof. Cewe n Cao and Prof. Da-jun Zh ang for h elpful d iscu ssions. References [1] Q u isp el G R W, Rob erts J A G and Thompson C J. Integrable mappin gs and soliton equations. Phys. L ett. A 126 :419-421, 1988. [2] V eselo v A P . Int egrable maps. R uss. M ath. Surv. 46 :3-45, 1991. [3] Bru sc hi M, Ragnisco O, Sant ini P M and T u G Z. Integrable symplectic maps. P hys. D 49(3) :273-29 4, 1991. [4] S uris Y u B. The Pr oblem of Inte gr able Discr etization: H amil ltonian A ppr o ach. (Basel: Birkh¨ auser), 2003 . [5] Belok olos E D, Bob enk o A I, Enolskii V Z, Its A R and Matv eev V B. Algebr o- ge ometric Appr o ach to Nonline ar Inte gr able Equations . (Berlin: Sp ringer), 1994. [6] C ao C W, Geng X G and W u Y T. F rom the sp ecial 2+1 T o da lattice to the Kadom tsev-P etviash vili equation. J. Phys. A : Math. Gen. 32 :8059-8 078, 1999. [7] Matv eev V B. 30 ye ars of ﬁnite-gap in tegration theory . Phil. T r ans. R. So c. A 366 :837- 875, 2008. [8] Gesztesy F, Holden H, Mic hor J and T esc hl G. Soliton E q uations and Ther e Al gebr o- ge ometric Solutions. V olume II: (1+1)-Dimensional Discr ete Mo dels . (Cam br idge: Cam bricge Univ ersit y Pr ess), 2009. 45 [9] Geng X G, W u L H and He G L. Quasi-p eriod ic solutions of n onlinear ev olution equa- tions asso ciated with a 3 × 3 matrix sp ectral problem. Stud. Appl. Math. 127(2) :107 - 140, 2011. [10] Nijhoﬀ F W. Lax pair for the Ad ler (lattice Krichev er-No vik o v) system. Phys. L e tt. A 297 :49-58, 2002. [11] Ma W X. Zhu Z N. Constructing nonlinear discrete integ rable Hamiltonian couplings. Comput. Math. Appl. 60(9) :2601-2608 , 2010. [12] Nijhoﬀ F W and Atkinson J. Elliptic N -sol iton solutions of ABS lattice equations. Int. Math. R es. Not. 2010 :3 837-3895, 2010. [13] Zh ang D J and Chen S T. S ymmetries for the Ablo witz-Ladik hierarc hy: part I I. In tegrable discrete nonlinear Schr¨ odinger equations and discrete AKNS hierarch y . Stud. Appl. M ath. 125(4) :419-443 , 2010. [14] Zhu Z N, Zhao H Q and Z hang F F. Exp licit solutions to an integ rable lattice. Stud. Appl. Math. 125(1) :55-6 7, 2010. [15] Xenitidis P . S ymmetries and conserv ation la ws of the ABS equations and corre- sp ond in g d iﬀeren tial diﬀerence equ ations of V olterra t yp e. J. Phys. A: M ath. The or. 44 :435 201(22pp), 2011. [16] Cao C W and Xu X X. A ﬁnite gen us solution of the H1 m o del. J. Phys. A: M ath. The or. 45 :055213 (13pp), 2012. [17] Zh ang D J, Zh ao S L and Nijhoﬀ F W. Direct linearization of extended lattice BSQ systems. Stud. Appl. M ath. 129(2) :220-2 48, 2012. [18] Hu X B, Liu Q P , Lou S Y, Qu C Z and Zhang Y J. Recent dev elopmen t in sy m metries and integ rability of diﬀerence equations. F r ont. Math. China 8( 5) : 999-1000, 2013. [19] Hietarin ta J , Joshi N and Nijh oﬀ F W. Discr ete Systems and Inte gr ability (Cam brid ge: Cam brid ge Univ ersit y Pr ess), 2016. [20] Lou S Y, Shi Y and Z h ang D J. Sp ectrum transformation and conserv ation la ws of lattice p oten tial KdV equation. F r ont. M ath. China 12 : 403-41 6, 2017. 46 [21] Zh ang D D and Zhang D J . O n decomp osition of the ABS lattice equations and related B¨ ackl un d transformations. J. Nonl. M ath. Phys. 25 :34-53, 2018. [22] Geng X G, W ei J and Zeng X. Algebro-geome tric integ ration of the mo diﬁed Belo v- Chaltikian lattice hierarch y . The or. Math. P hys. 199 :675-694 , 2019. [23] Ch ang X K, He Y, Hu X B, S un J Q and W eniger E J. Construction of new general- izations of Wynn’s epsilon and rho algorithm by solving ﬁ nite diﬀerence equations in the transformation order. N umer. Algorithms 83 :593-6 27, 2020. [24] Adler V E, Bob enko A I and S uris Y u B. Classiﬁcation of in tegrable equations on quad-graphs. T he consistency approac h. Comm. Math. P hys. 233 :513-543 , 2003. [25] Papag eorgiou V G, Nijhoﬀ F W and C ap el H W. Integrable mappings an d n on lin ear in tegrable lattice equations. Phys. L e tt. 147A :106-11 4, 1990. [26] Cap el H W, Nijhoﬀ F W and P apageorgio u V G. Complete integ rability of Lagrangian mappings and lattice s of Kd V t yp e. Phys. L ett. 155A :377-38 7, 1991. [27] Nijhoﬀ F W and En olskii V Z , In tegrable Mappin gs of Kd V t yp e and Hyp erelliptic Addition Theorems. Symmetries and Inte gr ability of Diﬀe r enc e Equations, V ol.255 (Cam bridge: C am bridge Un iv ersit y Pr ess) 64-78, 1999. [28] Dubr o vin B A, Matv eev V B and No vik o v S P . Non-linear equations of Korteweg de V ries t yp e, ﬁnite-zone linear op erators, and Ab elian v arietie s. Russ. Math. Surv. 31 :59-1 46, 1976. [29] Nijhoﬀ F W. Discrete Du b ro vin equations and separation of v ariables for d iscrete systems. Chaos Solitons F r actals 11(1) :19-28, 2000. [30] Alsallami S A M, Niesen J and Nijhoﬀ F W. Closed-form mo diﬁ ed Hamiltonians for in tegrable n um erical inte gration sc hemes. Nonline arity 31(11) :5110-5 146, 2018. [31] Sm ir no v F A. Dual Baxter equations and quan tization of the aﬃne Jacobian. J. Phys. A: Math. Gen. 33 :3385 -3405, 2000. [32] Bab elon O and T alon M. Riemann surfaces, sep aration of v ariables and classical and quan tum in tegrabilit y . P hys. L ett. A 312 :71 -77, 2003. 47 [33] Field C M and Nijhoﬀ F W. Quantum discrete Dubro vin equations. J. Phys. A: M ath. Gen. 37(33) :8065-8 087, 2004. [34] Alsallami S A M. Discr ete Inte gr able Systems and Ge ometric N umeric al Inte gr ation. (PhD thesis, Univ ersity of Leeds), 2018. [35] Nijhoﬀ F W and Cap el H W. Th e discrete Kortew eg-de V r ies equation. A cta Appl. Math. 39 :133-158 , 1995. [36] Levi D and Benguria R. B¨ ac klund transformations and n onlinear diﬀeren tial diﬀerence equations. Pr o c. Natl. A c ad. Sci. USA . 77 :5025-50 27, 1980. [37] Rogers C and Schief W K . B¨ acklund and D arb oux T r ansformatio ns: Ge ometry and Mo dern Applic ations in Soliton The ory. (Cam br idge: Cambridge Universit y Pr ess), 2002. [38] Li R M and Geng X G. O n a v ector long w a ve- sh ort wa ve-t yp e mo del. Stud. Appl. Math. 144(2) :164-184, 2020. [39] Cao C W. Nonlinearization of the Lax system for AKNS hierarch y . Sci. China Ser. A 33(5) :528-536, 1990. [40] Cao C W. A classical integrable system and the inv olutiv e representati on of solutions of the KdV equations. A cta Math. Sinic a: New Series 7 :216-223, 1991. [41] Cao C W and Zhang G Y. A ﬁnite gen us s olution of the Hirota equation via in tegrable symplectic maps. J. P hys. A: Math. The or. 45 :0952 03(25pp), 2012. [42] Cao C W and Zh ang G Y. Integ rab le symp lectic maps asso ciated with the ZS -AKNS sp ectral problem. J. P hys. A: Math. The or. 45 :26520 1(15pp), 2012. [43] Hirota R. Non-linear partial diﬀerence equations: I I I. Discrete sine-Gordon equation. J. Phys. So c. Jap an 43 :2079- 2086, 1977. [44] Bob enko A and Pink all U. Discrete surfaces with constan t n egativ e Gaussian curv a- ture and the Hirota equation. J. Diﬀ e r. Ge om. 43 :527- 611, 1996. [45] Cao C W and Y ang X. A (2+1)-dimensional deriv ativ e T o da equ ation in the cont ext of the Kau p -New ell sp ectral problem. J. P hys. A: M ath. The or. 41 :025203 (19pp), 2008. 48 [46] Hertric h-J er omin U, McIn tosh I, Norman P and Pedit F. P erio dic discrete conformal maps. J. r eine angew. Math. 534 :129-153 , 2001. [47] Xu X X, Cao C W and Zh ang G Y. Finite genus solutions to the lattice Sc hw arzian K ortew eg-de V ries equation. J. Nonline ar M ath. Phys. (accepted), arXiv:2004 .08689 v1 [nlin.SI] 18 Apr 2020. [48] Nijhoﬀ F W, Q uisp el G R W and Cap el H W. Direct linearizatio n of n onlinear diﬀerence-diﬀerence equations. Phys. L ett. A 97 :125-128, 1983. [49] Cao C W, Geng X G and W ang H Y. Algebro-geometric solution of th e 2+1 dimen - sional Burgers equation with a d iscrete v ariable. J . Math. Phys. 43 :621 -643, 2002. [50] Flasc hk a H. Relations b et w een in ﬁ nite-dimensional an d ﬁnite-dimensional isosp ec- tral equations. P r o c. RIMS Symp os. on Nonline ar Inte gr able Systems, Kyoto, Jap an (Singap ore: W orld Scien tiﬁc Publishing), 219-239, 1983 . [51] Griﬃths P and Harris J. Princi ples of Algebr aic Ge ometry . (New Y ork: Wiley), 1978. [52] Mumf ord D. T ata L e ctur es on Theta I . (Boston, MA: Birkh¨ auser), 1983. [53] F ark as H M and Kra I. Riema nn Surfac es . (New Y ork: Sp ringer), 1992. [54] Moser J. Geometry of qu ad r ics and sp ectral theory . Pr o c. Internat. Symp os. on Diﬀer- ential Ge ometry i n honor of S.- S. Chern, Be rke ley, California (New Y ork: Sprin ger), 147-1 88, 1979. [55] Moser J. Inte gr able Hamiltonian Systems and Sp e ctr al The ory . (Pisa: Accademia Nazionale d ei Lincei), 1981. [56] F addeev L D an d T akht a jan L A. Hamiltonian M etho ds in the The ory of Solitons . (Berlin: Sprin ger), 1987. [57] Bab elon O, Bernard D an d T alon M. Intr o duction to Classic al Inte gr able Systems . (Cam bridge: C am bridge Un iv ersit y Pr ess), 2003. [58] Gerdjiko v V S , Vilasi G and Y ano vski A B. Inte gr able Hamiltonian Hier ar chies . (Berlin: Sprin ger), 2008. [59] Abr ah am R and Marsden J E. F oundations of Me chanics . (London: Benjamin- Cummings), 1978. 49 [60] Arn old V I. Mathematic al Metho ds in Classic al Me chanics. (Berlin: Sprin ger), 1978 . [61] Arn old V I and Novik ov S P (Eds.). Dynamic al Systems IV . (Berlin: Sp ringer), 1990. [62] Lam´ e G. L e¸ cons su r les Co or donn ´ ee s Curvilignes e t leurs Diverses Applic ations . (P aris: Mallet-Bac h elier), 1859. [63] Dubr o vin B A. Periodic pr oblems f or the Kortew eg-de V ries equation in the class of ﬁnite band p oten tials. F unct. Anal. Appl. 9(3) :215-223, 1975. [64] Burchnall J L and Ch aundy T W. Comm utativ e ordinary diﬀerentia l op erators. Pr o c. L ond. Math. So c. s2-21(1) :420-440 , 1923. [65] Bak er H F. Note on the foregoing p ap er, “comm utativ e ordinary diﬀeren tial op era- tors,” by J. L. Burc hn all and J. W. Chau n dy . P r o c. R oy. So c. L ond., Ser. A 118 :584- 593, 1928. [66] Akhiezer N I. Con tinuous analogues of orthogonal p olynomials on a system of inte r- v als. Dokl. Akad. Nauk SSSR 141(2) :263-266, 1961. [67] Kr ichev er I M and No viko v S P . Holomorphic bun dles ov er algebraic curve s and non- linear equations. Russ. Math. Surv. 35(6) : 53-79, 1980. [68] Ch en K, Deng X, Lou S Y and Zhang D J. S olutions of nonlo cal equations reduced from the AKNS hierarch y . Stud. A ppl. Math. 141(1) :113 -141, 2018. [69] T o da M. The ory of N online ar L attic es . (Berlin: Sp ringer), 1981. [70] W ah lqu ist H D and Estabr o ok F B. B¨ ac klund transform ation for solutions of the Kortew eg-de V ries equation. P hys. R ev. L ett. 31 :1386-139 0, 1973. 50

Integrable symplectic maps associated with discrete Korteweg-de Vries-type equations

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment