A completeness study on a class of discrete, two by two Lax pairs

A COMPLETENESS STUD Y O N A CLASS OF DISCRETE, 2 × 2 LAX P AIRS MIKE HA Y Abstract. W e propose a method by whic h to examine all possi ble partial difference Lax pairs that consist of tw o 2 × 2 discrete linear problems, where the matrices cont ain one separable term in eac h entry . W e thereb y derive new, higher-order v ersions of t he lattice sine-Gordon and lat tice modified KdV equations, while sho wing that there can b e no other partial difference equations associated with this type of Lax pair. 1. Introduction Nonlinear integrable lattice equations provide a natur al discre te extension of cla s- sically integrable systems. An ex tens ively studied example is the la ttice mo dified Korteweg-de V ries equation, refer red to a s LMK dV: LMKdV : x l +1 ,m +1 = x l,m  p x l +1 ,m − r x l,m +1   p x l,m +1 − r x l +1 ,m  where x l,m is the dep endent v ariable, l a nd m a r e the discrete indep endent v a riables, and p and r ar e pa rameters. This equa tion provides a n integrable discr e te version of the well-known mo dified Korteweg-de V rie s equation. Note that, in the pa st, the par ameters p and r ha ve b een ta ken to b e constant [1], a lthough singular ity confinement has provided a wa y to de-a utonomize L MK dV while maintaining its int egrability , by allowing p and r to depend o n l and m in a s pecific wa y [2]. The int egrability of such equations lies in the fact that they ca n b e s o lved through an asso ciated linear problem called a La x pair. Lax pairs can app ea r in many guises, the type that we ar e exclusively concerned with in this pap er consist of a pair o f linear pr oblems written: θ ( l + 1 , m ) = L ( l , m ) θ ( l , m ) θ ( l , m + 1) = M ( l, m ) θ ( l , m ) . (1.1) where θ ( l , m ) is a tw o-comp onent vector and L ( l , m ) a nd M ( l, m ) a re 2 × 2 matrices. These linear problems are des crib ed by L a nd M , which are referre d to as the La x matrices. If we were e x amining a n explicitly deter mined Lax pair for LMKdV for example, the v a riable x l,m , alo ng with the para meters p a nd r , would b e found to reside within L and M . The easily der ived compa tibility condition on this La x pa ir is L ( l , m + 1) M ( l , m ) = M ( l + 1 , m ) L ( l , m ) and it is throug h this co mpatibility condition that we arrive at the integrable non- linear equation asso c iated with the Lax pair . Despite the ex istence of a La x pa ir often b eing used as the definition o f integra- bilit y for a g iven equa tion [3], there ha ve bee n few studies tha t so ught to find or categoriz e no nlinear equations that used Lax pairs a s their s tarting p oint. O f those studies tha t did b egin with Lax pairs, most chose a for m of the Lax pair a priori , that is an assumption was made co ncerning the dep endence o f the linea r systems on the sp e ctral parameter, thus limiting the p oss ible r e s ults. 1 2 MIKE H A Y The present s tudy b egins with La x pairs that ar e 2 × 2, where each entry of the Lax ma trices contains only o ne separ able term. The Lax pairs are otherwis e general in that no assumptions are made as to the explicit dep endence of any quantities within the Lax matrices on the lattice v a riables, l and m , or on the s pec tral v ariable n . By one s e parable term we mean that each e n try contains a term that can b e split into a pr o duct of tw o pa rts, o ne that dep ends o n the lattice v ariables, and another that dep ends only on the sp ectral v ariable. F o r example, in the 11 entry of the L matrix, we write the sepa rable ter m a ( l , m ) A ( n ). Both a and A may co n tain m ultiple terms themselves, say a = P i a i , A = P j A j , howev e r, a ll ter ms within a m ust multiply all those within A , the other e ntries their own similar term. So, we could have an L matrix L =  [ a 1 ( l, m ) + . . . a M ( l, m )][ A 1 ( n ) + . . . + A N ( n )] b ( l , m ) B ( n ) c ( l , m ) C ( n ) d ( l, m ) D ( n )  where we have written the 11 entry in the e xpanded form and left the other entries abbreviated to sav e space, no other terms can b e added into any ent ry . M must also contain just one separable ter m in each entry , although these terms ar e indep endent of thos e in L . The reas o n for limiting the Lax pa irs to those that are 2 × 2 with o ne separ a ble term in each entry is tw o fold. Firstly , La x pa ir s with more terms typically lead to equations of higher order, as ca n b e seen from hierar chies of equa tions with Lax pairs [4, 5], ther e fo re we constr ain our study to the lower order equations b y limiting the n um b er o f terms. Secondly , we limit the num b e r of terms present in the compatibility condition and thus re nder it less complica ted to examine all of the combinations of ter ms that can a rise there. A combination of terms in the compatibility condition defines a system o f eq uations that we subs e quent ly solve, in a manner that preserves its full gene r ality , up to a p oint where a nonlinear evolution equation is apparent, or it ha s been shown that the sy stem c annot be asso ciated with a no nlinear equation. T esting all c o mbin ations o f terms, we thereby survey the complete set of Lax pairs of the type descr ibe d. In fact, of a ll the po ten tial Lax pa irs identified by this metho d, only tw o lead to int eresting evolution eq uations. These ar e hig her order v arieties of the lattice sine- Gordon (LSG) and the LMKdV eq ua tions, which can b e found in section 1.1. The remaining systems ar e sho wn to b e trivia l, ov erdetermined or underdetermined. As noted in [6], this suggests a co nnection b etw een the sing ula rity confinement metho d and the existence of a Lax pair. As we do not make a ny a ssumptions ab out the explicit dep endence on the sp ec- tral par a meter, we s how that a particula r no nlinear equa tion may have many Lax pairs, all dep ending o n the sp ectra l parameter in different wa ys. The effect that this freedom has on the pro cess of inv erse sca ttering is, a s yet, unclear. This pap er is or ganized as follows: s ection 1.1 presents the ma jor results, those being the higher order versions of LMKdV and LSG, as well a s a statement o f the completeness theorem. The metho d of ident ifying and a na lyzing the viable Lax pairs is la id out in section 2, where a representative list o f all the Lax pair s ident ified can b e found. Section 3 explains how the hig her order LSG and LMK dV equations are derived from the general form of their Lax pair s and section 4 provides examples that describ e why most Lax pair s found in sectio n 2 lead to triv ia l systems. A conclusion rounds out the pap er. 1.1. Res ults. Note that all difference equations in the rema inder of this pape r will utilize the notation ¯ x = x ( l + 1 , m ) , ˆ x = x ( l, m + 1) . 2 × 2 LAX P AIRS 3 As one of the tw o main re sults of the pap er, we present tw o new integrable nonlinear partial difference equa tio ns. The first eq uation, LSG 2 : ρ σ ˆ x x + λ 1 µ 1 ˆ ¯ x ˆ y = σ ρ ˆ ¯ x ¯ x + λ 2 µ 2 x ¯ y (1.2a) σ ρ ˆ ¯ y ˆ y + λ 2 µ 2 ˆ xy = ρ σ ¯ y y + λ 1 µ 1 ¯ x ˆ ¯ y (1.2b) is referr ed to as LSG 2 bec ause it is second o rder in ea ch o f the lattice dimensions, and b ecause setting x = y returns the familiar LSG equation, in a non-autonomo us form. Her e x = x ( l, m ) a nd y = y ( l, m ) are the dep endent v ariables, λ i = λ i ( l ) and µ i = µ i ( m ) are parameter functions, as a r e ρ = λ ( − 1) m 3 and σ = µ ( − 1) l 3 . Similarly , the equa tion LMKdV 2 : λ 1 σ ˆ ¯ x ˆ x + µ 2 ρ y ˆ y = λ 2 σ y ¯ y + ρµ 1 ˆ ¯ x ¯ x (1.3a) ρµ 1 ˆ x ˆ ¯ y + λ 2 σ x ˆ y = µ 2 ρ x ¯ y + λ 1 σ ¯ x ˆ ¯ y (1.3b) where the terms are as for LSG 2 , is referred to as LMKdV 2 , ag ain bec ause setting x = y bring s ab out LMKdV. Note that the LMK dV so attained is of a more ge ne r al form than the most common v ar iety lis ted in the b eginning of the intro ductio n. The s e c ond ma in re s ult comes in the form of a theor em that we state as follows: Theorem 1. The system of e quations that arise via the c omp atibility c ondition of any 2 × 2 L ax p air (1.1) with one nonzer o, sep ar able term in e ach entry of e ach matrix (as describ e d in the intro duction) is either trivial, under determine d, over determine d, or c an b e r e duc e d to one of LSG 2 or LMKdV 2 . The pro of of theor e m 1 lies in consider ing a ll of the p ossible sets of equations that can aris e fr om the compatibility condition of such Lax pa irs, and so lving those sets o f equatio ns in a wa y that retains their full free do m. This pro o f o ccupies the remainder of the pap er. The Lax pairs asso ciated with LSG 2 and LMKdV 2 resp ectively are listed b elow, these Lax pairs are derived in section 3. The Lax pair for L SG 2 is L =  F 1 /ρ F 2 λ 1 ¯ x F 2 λ 2 /x F 1 ρ ¯ y /y  (1.4a) M =  F 2 ˆ x/ ( σ x ) F 1 /y F 1 µ 1 ˆ y F 2 σ  (1.4b) While the Lax pair for LMKdV 2 is L =  F 1 λ 1 ¯ x/x F 2 ρ/y F 2 ¯ y /ρ F 1 λ 2  (1.5a) M =  F 1 µ 1 ˆ x/x F 2 ¯ y /σ F 2 σ ¯ y F 1 µ 2  (1.5b) where F i = F i ( n ) ar e arbitra ry functions of the s pec tral v a riable n with the c ondi- tion that F 1 6 = k F 2 , where k is a consta nt. 2. Method of identifying potential Lax p airs F rom Lax pairs o f the type we consider her e, the co mpatibilit y conditio n pro duces a s e t of equations, each equation b eing due to one of the linea rly independent sp ectral terms that arises in s o me entry . Studies that hav e se a rched for integrable systems by b eginning with a Lax pair, whether an isosp e ctral or isomono dromy Lax pair o r o therwise, typically assume some dep endence of the Lax pair on the sp ectral 4 MIKE H A Y parameter, then solve the compatibility condition for the evolution equation [8, 9, 11, 10]. Mo st often a p olynomia l or rational dep endence o n the s pec tr al v aria ble is used [13 ], but any type of explicit dep endence could b e in vestigated, for example W eierstra ss elliptic functions. Example 1 . Consider the fol lowing L and M matric es L =  a℘ ′ 4 b℘ c d℘ ′  M =  α ( ℘ 2 + 1) β ℘ ′ γ ℘ ′ δ ℘  wher e ℘ is t he Weierstr ass el liptic function in the s p e ctr al p ar ameter n only, and al l other qu antities ar e fun ct ions of b oth the lattic e variables l and m . F r om the c omp atibili ty c ondition, b LM = M L , and n oting that ℘ ′ 2 = 1 4 ℘ 3 − g 2 ℘ − g 3 wher e g i = constant , we find t he fol lowing e quations in the 12 ent ry 12 ℘ 3 : ˆ aβ = b ¯ α + d ¯ β ℘ 2 : ˆ bδ = 0 ℘ : ˆ aβ = d ¯ β − b ¯ α 4 g 2 ℘ 0 : ˆ aβ = d ¯ β (2.1) wher e we have sep ar ate d out the e quations c oming fr om differ ent or ders of the sp e c- tr al p ar ameter. Clearly , this choice of L and M do es not yield an interesting evolution equatio n through their compatibility , this example w as instead chosen b ecause it illustrates an imp or tant p oint. Multiplying to gether tw o functions of the sp ectral par ameter can pro duce n umerous orders that ma y or may not b e propo rtional to other sp ectr al term pro ducts in the compa tibilit y condition. F ro m example 1, the sp ectra l term m ultiplying ℘ 2 did not ‘match up’ with other terms in the 12 entry at that order , which brought ab out the equation ˆ bδ = 0, forcing so me ter m to b e zer o. How ever, the other sp ectra l terms did turn out mor e mea ningful equations, hig hlighting the need to choo se the dep endence o n the sp ectral parameter carefully s o that none of the resulting equations for c e a ny lattice terms to be zero. The inclusion of a zero lattice term do es not preclude the exis tence of a n in- teresting ev olution equatio n. How ever, we are essentially classifying La x pair s by the n um be r of terms in their ent ries and the class o f L ax pa irs pr esently under insp e ction contains one sepa rable ter m in each entry of their 2 × 2 matrices . If any of those ter ms were forced to zero then the resulting La x pa ir would actually co me under a different categor y in the pres en t fr a mework. The most gener al form of a 2 × 2 Lax pair with exa ctly one separable ter m in each entry of the L and M matrices is: L =  aA bB cC dD  M =  α Λ β Ξ γ Γ δ ∆  Where low er case s repr esent lattice terms and upp er cases represent sp ectr al terms. The co mpatibilit y condition is b LM = M L , o f which we initially concentrate on the 12 entry . ˆ aβ A Ξ + ˆ bδ B ∆ = b ¯ αB Λ + d ¯ β D Ξ (2.2) A t this stage we a re only concer ned with the v a rious linea rly indep endent sp ectral terms that app ear. These will determine the set o f equa tio ns that come o ut of the compatibility condition, which are s ubsequently solved to find the corr esp onding 2 × 2 LAX P AIRS 5 evolution equation. That b eing the case, there are four q ua nt ities to contend with in this entry , A Ξ, B ∆, B Λ and D Ξ. Any of these four pro ducts ca n lead to m ultiple, linearly indep endent sp ectra l terms , all of which must match up with at least one other spectra l term fr o m one of the three remaining pro ducts in this ent ry . If there exists some sp e c tral term that do es no t match up with a sp ectral term from another pro duct, then the la ttice term that mu ltiplies it will have to b e zero, which is for bidden. Let us la be l the terms in each pro duct as follows A Ξ = P i F A Ξ i , B ∆ = P i F B ∆ i , etc. All the sp ectral terms that o cc ur in the 12 entry of the co mpatibilit y condition can be sorted into four groups ac c o rding to the lattice terms that they multiply . 12 (ˆ aβ ) ( ˆ bδ ) | ( b ¯ α ) ( d ¯ β ) F A Ξ 1 F B ∆ 1 | F B Λ 1 F D Ξ 1 F A Ξ 2 F B ∆ 2 | F B Λ 2 F D Ξ 2 . . . . . . | . . . . . . (2.3) Where the line separa ting the four g roups ma rks the p os itio n of the equals sign in the asso cia ted lattice term equations. O rganizing the ter ms fr o m ex a mple 1 in this wa y leads to 12 (ˆ aβ ) ( ˆ bδ ) | ( b ¯ α ) ( d ¯ β ) ℘ 3 ℘ 2 | ℘ 3 ℘ 3 ℘ | ℘ ℘ ℘ 0 | ℘ 0 The present study will utilize the word ‘g roup’ in the g eneral Englis h sens e , our groups refer to collections of sp ectral terms that multiply the sa me lattice terms in an e ntry of the co mpatibility co ndition. W e hav e already seen that a ll terms in a ll groups m ust b e prop or tional to a term from at least one o f the other thre e gr oups in this e n try . Conversely , where there are sp ectra l terms that are pro po rtional to others from o ther gro ups, a n equation rela ting the cor resp onding la ttice terms will th us b e defined. Still with the a b ove exa mple, that La x pair ha s a dep endence on the sp ectra l parameter s uch that the groups multiplying ˆ aβ , b ¯ α and d ¯ β all contain the sp e ctral term ℘ 3 , for which the corresp o nding equation in the lattice terms is ˆ aβ = b ¯ α + d ¯ β . A t this s ta ge, the num ber of p ossible sets of prop or tional sp ectral terms, and therefore the num ber of po ssible combinations of equations yielded by the compati- bilit y condition, is unmanageably larg e . W e r equire further considera tions to bring the problem under control. 2.1. Links and equiv alen t equations at diffe rent orders. The following defi- nition applies to sets of sp ectral terms within an entry of the co mpatibilit y condition that bring ab out the equations used to find the evolution equation. The word pr o- p ortional is used in the sense that t w o terms F 1 and F 2 are pro po rtional if F 1 = kF 2 for some finite constant k . Definition 1. A li nk is a set of pr op ortional sp e ctr al terms in the same matrix entry of the c omp atibility c ondition. A set of two pr op ort ional terms is a single link, thr e e terms a double link, and four terms a triple link. Naturally , the s pec tral terms that compr is e a link must each r eside in a differ ent group of sp ectral ter ms w ithin an entry . If ther e are tw o o r more sp ectral terms that a r e prop ortiona l to one a nother within the same gro up, they ar e simply a dded together to make one term. Since each gro up of sp ectral terms multiplies the same lattice term, one may s pea k o f either links b etw een the gro ups o f sp ectral ter ms or links be tw een lattice terms, with the same meaning . The ab ov e definition c a ptures 6 MIKE H A Y the idea that the entries of the compatibility condition give r ise to different lattice term equations a t different or ders in the sp ectr al term, without app ealing p ow ers of so me ba s ic function. With the employmen t o f pr op ortional ter ms comes the p ossibility of constants of pr op ortionality , which we b egin to dea l with here. F act 1. If ther e exist two distinct single links b et we en the same two gr oups of sp e ctr al terms, then the c orr esp onding c onstants of pr op ortionality must b e e qual. The pro of of this fact is e le men tary: s ay that one s ing le link is formed by a llowing F A Ξ 1 ∝ F D Ξ 1 , where neither of thes e terms is prop ortiona l to a n y o ther spec tral term that arises in the 1 2 entry , and the other single link corr esp onds to F A Ξ 2 ∝ F D Ξ 2 , w he r e ag ain these terms link with no others. By including some co nstants of pro p o rtionality , k 1 and k 2 resp ectively , we can wr ite down the eq uations that corres p o nd to these tw o s ingle links, those b eing ˆ a β = k 1 d ¯ β and ˆ aβ = k 2 d ¯ β . Since bo th equations must hold, and none o f the lattice terms can b e zero, it is clear tha t we must hav e k 1 = k 2 . This simple fact prov es to b e rather imp or tant b ecaus e it ensures that all links betw een the same tw o groups can b e bundled to gether. F urther , all the sp ectra l terms, in so me gr oup, that corr esp ond to those sing le links with one other gro up, can be tre ated as a single spec tr al term. So, wher e F A Ξ 1 and F D Ξ 1 formed one s ing le link and F A Ξ 2 and F D Ξ 2 formed a nother b etw ee n the sa me t wo groups, w e can lump together F A Ξ 1 + F A Ξ 2 = G 1 and b e sure that it links with F D Ξ 1 + F D Ξ 2 = k G 1 . Still with the 12 en try , if there exist m ultiple double links b etw een the same three la ttice terms, ˆ aβ , b ¯ α a nd d ¯ β say , then the lattice term equa tions tha t result from those links can b e written K   ˆ aβ b ¯ α d ¯ β   = 0 (2.4) Where K is a matrix of the constants of prop ortio na lit y b etw een the v ario us sp ectral terms, nor malized so that ea ch entry of the first column o f K is unity . Each row of K corr e s po nds to a double link . If K is such that equation (2.4) is over- determined or uniquely s olv able, then any L a x pair p osses sing the corr esp onding links is inconsistent or c o nt ains a zero lattice term. T her efore, we need not consider more than t wo do uble links b etw een the s a me three s p ectr al terms, although there may b e multiplicit y within those tw o double link s . By the s a me arg ument we ca n allow a maximum of three different triple links b etw een the same four lattice terms in an entry of the compatibility condition. 2.2. Link sym b olism . The abundance of Lax pa irs that need to be chec ked ne- cessitates the introduction of a sho rthand, which will be base d on their link s . The off-diagona l entries b oth contain four groups of sp ectral terms, each of which mul- tiplies a single pro duct of lattice terms. The 12 en try contains the sp ectral term pro ducts A Ξ, B ∆, B Λ, a nd D Ξ asso ciated with the lattice ter m pro ducts ˆ aβ , ˆ bδ , b ¯ α and d ¯ β res p ectively . F or the s horthand, we alwa ys set o ut the s pectr al terms in the same wa y on the pag e A Ξ B Λ B ∆ D Ξ Each link can b e represented by lines b etw een the quantities that a re pro po rtional to ea ch other , and s o we will use the symbols listed in table 1 to represe nt the combinations o f links in the 12 entry , wher e F A Ξ is some term from the gr oup of sp ectral terms formed by taking the pr o duct A Ξ, and other terms ar e simila rly lab eled. 2 × 2 LAX P AIRS 7 Symbol Links F A Ξ ∝ F D Ξ , F B ∆ ∝ F B Λ F A Ξ ∝ F B ∆ , F B Λ ∝ F D Ξ F A Ξ ∝ F B Λ , F B ∆ ∝ F D Ξ F A Ξ ∝ F D Ξ ∝ F B ∆ ∝ F B Λ F A Ξ ∝ F B ∆ ∝ F B Λ i , F B Λ j ∝ F D Ξ . . . . . . T able 1. Symbo ls used to represent the link combinations in the off-diagona l entries. Note that F B Λ i 6 = kF B Λ j , k a constant The 21 entry is simila r to the 12 e n try in that it po ssesses four distinct pro ducts of sp ectral terms. The s ame sy m b ols listed in table 1 ar e used a gain for the 21 ent ry , with clea r meaning given that the sp ectral pro ducts ar e set out as follows C Λ A Γ D Γ C ∆ How ever, the diagonal entries are s lightly differe n t b ecaus e each diag onal entry contains a pro duct that o ccurs twice: A Λ o ccurs twice in the 1 1 entry and D ∆ t wice in 22 . This automatically cause s the a sso ciated lattice terms to b e paired in their r esp ective entries and, as such, A Λ a nd D ∆ need not b e linked with another sp ectral term to pr even t a zer o la ttice term. This b eing the case, there are rea lly only three sp ectral term pro ducts to consider in b o th of the diago na l en tries, one of which need not b e linked to the o ther tw o, and our sym b o ls reflect that. Positioning the sp ectral term pro ducts as follows A Λ B Γ C Ξ W e symbolize the links as indicated in table 2, where the symbol ‘ ’ is used to represent the rep eated sp ectra l term pro ducts. Symbol Links F A Λ ∝ F B Γ ∝ F C Ξ F A Λ alone, F B Γ ∝ F C Ξ F A Λ i ∝ F B Γ and F A Λ j ∝ F C Ξ separately . . . . . . T able 2. Symbo ls used to represent the link combinations in the diagonal entries of the compatibility condition. Note that F A Λ i 6 = k F A Λ j , k a co nstant 2.3. Which Lax pairs nee d to b e chec k ed? All link co mbin ations that do not force any lattice terms to b e zer o are chec k ed systematica lly . The pro cedure for doing this runs as follows: (1) Begin with the 12 entry , a s sume that only s ingle links ex is t there, and list all s ingle link combinations that pro duce different lattice term eq uations. 8 MIKE H A Y (2) Constr uct the links found in the previous step by choo sing pr op ortional sets o f sp ectral terms in the appro priate g r oups. (3) Mov e to the 21 entry , no ting the sp ectral term constr uctions from the pre- vious step, a nd identif y all the viable link c ombinations in this entry . (4) Rep eat the previo us step in the diag onal entries. After identifying all link com binations with single links in the 12 en try , we rep eat the entire pro cess assuming that do uble and p ossible single links exist there, and then r ep e a t once mo re with triple, and p os sibly double and sing le links in the 12 ent ry . Finally , the co rresp onding set of lattice term equa tio ns for ea ch L a x pair m ust b e ana lyzed to find the resulting evolution equation. This analysis needs to be conducted in a manner that preserves the full freedom of the system, as describ ed in section 3. 2.3.1. Single links in the 12 en try. The fo ur gr oups of spectr al terms in the 12 entry are A Ξ B Λ B ∆ D Ξ Each group must b e linked to ano ther s o, using single links, the g roup A Ξ must be linked to at lea st one o f the three other gr oups, B Λ, B ∆, or D Ξ. F o r arguments sake, s ay that there exists a single link b etw een A Ξ and B Λ. A Ξ B Λ B ∆ D Ξ That leav es b oth gro ups B ∆ and D Ξ requiring links and, since we are only co n- cerned with single links at the mo men t, these t wo gro ups of spe c tral terms c an b e linked to each o ther, or to one o f A Ξ or B Λ, in distinct, single links . How ever, it can be shown tha t if any g roup in either o ff-dia gonal en try p ossesses single link s betw een it and tw o other gr oups, the r esulting Lax pair is a sso ciated with a tr ivial evolution equation (see prop osition 1 be low). Hence, given the first link b etw ee n A Ξ a nd B Λ, the only other single link that needs to be considered is b etw een B ∆ and D Ξ. There fo re, table 3 lists the o nly single link combinations in the 12 entry that require further ana ly sis. 12 A Ξ B Λ B ∆ D Ξ T able 3. Sing le link combinations in the 12 entry Prop ositio n 1. I f t her e exist two single links b et we en some lattic e term and two others in an off-diagonal en t ry, t hen the r esulting evolution e quation is trivial. The pro of o f Pr op osition 1 lies simply in chec king all the p ossible link combina- tions that meet the criterio n. In that sense there is nothing sp ecial ab out La x pa irs of this type, they are only tr e ated separately b ecaus e there a re ma n y s uch cases, so including them with the other s would unneces sarily le ng then the argument. Before links in the 21 ent ry are examined, we must select sp ectral ter ms that engender the links alre a dy chosen in the 12 ent ry . W e s hall pro ceed with the analysis under the as s umption that single links b etw een A Ξ and B Λ, and b etw een B ∆ a nd D Ξ, altho ug h the other co mbin ations can b e dea lt with in the same way . 2 × 2 LAX P AIRS 9 The sp ecified links are cons tructed by choo sing sp ectra l ter ms A = 1 Ξ ( F 1 + . . . ) (2.5a) D = 1 Ξ ( F 2 + . . . ) (2.5b) Λ = 1 B ( F 1 + . . . ) (2.5c) ∆ = 1 B ( F 2 + . . . ) (2.5d) where F i = F i ( n ) and F 1 6 = kF 2 , k a constant. It is understo o d that while ther e is ro om for o ther sp ectral terms in the ex pr ession A = 1 Ξ ( F 1 + . . . ), there ca nnot b e a term pr o p o rtional to F 2 / Ξ in A , a s this would cause the single link co r resp onding to F 2 to beco me a double link. In this w ay the desired links are co nstructed, plus we hav e allowed for additional links should they b e appropr iate or required later. Readers may note the o mis sion o f any constants of prop ortiona lit y in the ab ov e, how ev er, the consta n ts that co uld have b een wr itten at this p oint can all b e absorb ed into the lattice terms that they multiply . T urning our attention to the 21 entry of the compatibility condition, ˆ cαC Λ + ˆ dγ D Γ = a ¯ γ A Γ + c ¯ δ C ∆, the following sp ectral terms app ear 21 (ˆ c α ) ( ˆ dγ ) | ( a ¯ γ ) ( c ¯ δ ) C B F 1 Γ Ξ F 2 | Γ Ξ F 1 C B F 2 . . . . . . | . . . . . . (2.6) Since every sp ectral term must b e prop o r tional to a nother in the same entry , C B F 1 m ust b e prop or tional to Γ Ξ F 1 or Γ Ξ F 2 . Clear ly , C B F 1 cannot be prop ortio nal to C B F 2 , nor ca n it link with some other term that we are yet to define, as this would introduce s ingle links b etw een some lattice ter m and tw o others, the situation excluded by prop osition 1. Mor eov er, we can exclude the case with C B F 1 ∝ Γ Ξ F 2 bec ause this also requir es the remaining s pec tral ter ms to b e prop ortiona l to one another, i.e. C B F 2 ∝ Γ Ξ F 1 . These tw o co nditions on the s pec tr al ter ms imply that F 1 ∝ F 2 , contradicting a previous as sumption. Hence, the links c hosen in the 12 entry leav e only one choice for the links in the 2 1 entry o f the compatibilit y condition: C B F 1 ∝ Γ Ξ F 1 and C B F 2 ∝ Γ Ξ F 2 , which can be written mor e succinctly as C Ξ = B Γ 12 A Ξ B Λ 21 C Λ A Γ ⇒ B ∆ D Ξ D Γ C ∆ The ter ms arising in the diago nal entries ar e listed b elow 11 | (ˆ a α − a ¯ α ) F 2 1 / ( B Ξ) | ( c ¯ β − ˆ bγ ) B Γ | 22 | ( ˆ dδ − d ¯ δ ) F 2 2 / ( B Ξ) | ( b ¯ γ − ˆ cβ ) B Γ | (2.7) Note that the terms in eq uation (2.7) ar e set out slightly differently to tho se in table 2 b ecause we hav e already deter mined that B Γ ∝ C Ξ. The sp ectral terms in the dia gonal entries, in this cas e , do no t necess arily hav e to link with other s b ecause they ar e mult iplied by mo re than one lattice ter m, as 10 MIKE H A Y indicated in equation (2.7) where the lattice terms a ppea r in parentheses to the left of the sp ectral terms they m ultiply . A lone sp ectra l term in the diagonal entries, given the links a lready constr uc ted in the off-diag onal entries, will no t bring a bo ut a ze r o lattice term. Co nsequently , ther e exist three p ossible link co mbinations for the diago nal e ntries: B Γ ∝ F 2 1 / ( B Ξ), B Γ ∝ F 2 2 / ( B Ξ) or B Γ is prop or tional to neither F 2 2 / ( B Ξ) nor F 2 1 / ( B Ξ). All three choices lea d to trivial ev olution equations and we shall co nt inue the analys is under the assumption that B Γ ∝ F 2 1 / ( B Ξ). A link combination has now been chosen in each of the e ntries o f the compatibility condition, these are shown in table 4 12 A Ξ B Λ B ∆ D Ξ 21 C Λ D Γ A Γ C ∆ 11 A Λ B Γ C Ξ 22 D ∆ B Γ C Ξ T able 4. An ex ample o f the links that define a Lax pair Gauge transfor ma tions can b e us ed to remove the dep endence on so me of the sp ectral terms and, a s such, we exp ect some redundancy . The links constructed ab ov e are achiev ed by setting the v alues o f the sp ectral terms to A = F 1 , Λ = F 1 B = 1 , Ξ = 1 C = F 2 1 , Γ = F 2 1 D = F 2 , ∆ = F 2 where F 1 and F 2 are any functions of the sp ectral v a riable n , such tha t F 1 6 = k F 2 , k = constant. Note that o ne of F 1 or F 2 may itse lf be a cons ta n t. No cons tant s of prop ortiona lity a re r equired as thes e can b e absor bed into lattice ter ms in this case. The same links ar e repro duced by any suite of spec tr al terms that s atisfies the following c onditions, broug ht ab out by the links des crib ed a bove A = F 1 / Ξ , Λ = F 1 /B C = F 2 1 / Ξ , Γ = F 2 1 /B D = F 2 / Ξ , ∆ = F 2 /B The r esulting s et of equatio ns that a re pro duce d by the compa tibilit y co ndition for this Lax pair are written in (2.8), although they can be read from the links given in table 4. ˆ aα − a ¯ α = c ¯ β − ˆ bγ ˆ dδ − d ¯ δ = 0 b ¯ γ − ˆ cβ = 0 ˆ aβ = b ¯ α ˆ bδ = d ¯ β ˆ cα = a ¯ γ ˆ dγ = c ¯ δ (2.8) This r ounds the des cription of the La x pair s with only single links in the off- diagonal entries, in practice one would contin ue by analy zing equa tions (2.8) to find the asso cia ted evolution equation, which in this cas e is trivial. On the p ossibility of including extra sp ectr a l terms to augment the links used her e, see equation (2 .5), we note that fewer terms allow greater freedo m and that any a dditional links could 2 × 2 LAX P AIRS 11 only le ad to a more constr ained sy stem, one that cer tainly co uld not sustain an int eresting evolution equation consider ing that the less constra ine d exa mple her e leads to a trivial r esult. Also , the alternative links that cause ther e to be one equation in the 22 e ntry and tw o eq uations in the 11 entry of the compatibilit y condition, see after eq ua tion (2.7), lead to the sa me evolution equations found here as the Lax pair is symmetric in that sens e. 2.3.2. Double links in the 1 2 ent ry. Here link combinations that consist of double and p ossibly single link s in the 12 entry a re inv e s tigated. There are mo re p ossi- bilities in this c la ss than when only consider ing single links, howev e r the num b er is reduced b y noting that so me s ets of link combinations are equiv alent from the per sp ective of the lattice term equations . F or exa mple the following pair of link combinations in the 12 entry are clea r ly equiv alen t. 12 A Ξ B Λ B ∆ D Ξ ↔ Also, all link combinations that po ssess tw o do uble links in the 12 entry a re nearly equiv alent, enough to co nsider them a ll together . The equiv alence is b ecause the lattice term eq uations corr esp onding to any tw o double links in this entry ca n be manipulated so that the they are the s a me as those from any other combination of t wo double links. The difference that may ar ise comes from the diagonal entries, where the pa rticular pair of double links chosen in the 1 2 entry ca n affect the v a riety of link combinations p oss ible. How e ver, the difference is not sufficient to alter the ov erall outcome that these systems are overdetermined. The complete list of double link combinations in the 12 entry is listed in table 5 12 A Ξ B Λ B ∆ D Ξ T able 5. All double link combinations in the 12 entry that r equire analys is T o exemplify the metho d of constructio n o f Lax pairs with double links in the 12 ent ry , w e cho ose link combination in table 5 where there is a double link betw een A Ξ, B ∆ and B Λ and a single link b etw een B ∆ and D Ξ. Spe c tral terms that generate this choice of links ar e g iven in equatio n (2.9 ). A = F 1 / Ξ , Λ = F 1 /B ∆ = ( F 1 + k F 2 ) /B , D = F 2 / Ξ (2.9) where our usual nomenclature applies, i.e. F i = F i ( n ), k = constant. Using the expr essions fo und in the 12 entry , the terms that a r ise in the 21 entry are F 1 C /B , F 2 Γ / Ξ, F 1 Γ / Ξ and F 2 C /B , where F 1 C /B arises twice. It is conv enien t to r earra nge the terms found in the 21 entry into c olumns o f terms that ca nnot b e linked, this is done in equation (2.10). F 1 C /B F 1 Γ / Ξ F 2 Γ / Ξ ( F 1 + F 2 ) C /B F 1 Γ / Ξ F 1 C /B F 2 Γ / Ξ F 2 C /B (2.10) F 2 Γ / Ξ must link with F 2 C /B b ecause linking with F 1 C /B would lead to the contradictory F 1 ∝ F 2 , since tha t would a lso necessita te a link b etw een F 2 C /B and F 1 Γ / Ξ. Tha t leaves tw o po ssibilities: B Γ ∝ C Ξ, o r we ca n split F 2 C /B ∝ ( F 1 + F 2 )Γ / Ξ, noting that the multiplicit y of the ter m F 1 C /B means that it need not link to another in this entry . The second p o s sibility is ne g lected, thoug h, as it gives r ise to a z ero la ttice term in one o f the dia gonal en tries to b e considered 12 MIKE H A Y below. Thus, b y a pro cess of elimination, the links sele cted in the 12 ent ry leav e only one choice for the 21 entry , w hich is shown in equation (2.11) 12 A Ξ B Λ 21 C Λ A Γ ⇒ B ∆ D Ξ D Γ C ∆ (2.11) Mov e now to the 11 entry wher e the rele v ant sp ectra l term pro ducts are A Λ ∝ F 2 1 / ( B Ξ), a nd B Γ ∝ C Ξ, see equa tio n (2.12). W e choose to link these tw o sp ectral terms to get one eq uation in the 1 1 entry and find that there a re neces sarily tw o equations in the 22 entry . Note that the opposite c ase where there ar e tw o equations in the 11 entry a nd one in the 22 entry ca n also ex ist by splitting up B Γ into tw o terms, how e ver this will lead to the s ame e volution equation. 11 | (ˆ a α − a ¯ α ) F 2 1 / ( B Ξ) | ( c ¯ β − ˆ bγ ) B Γ | 22 F 1 F 2 / ( B Ξ) | ( ˆ dδ − d ¯ δ ) | ( b ¯ γ − ˆ cβ ) B Γ F 2 2 / ( B Ξ) | (2.12) Using o ur symbolism, the links that define this La x pa ir a re shown in ta ble 6. 12 A Ξ B Λ B ∆ D Ξ 21 C Λ D Γ A Γ C ∆ 11 A Λ B Γ C Ξ 22 D ∆ B Γ C Ξ T able 6. Link s for a La x pa ir with double links in the off-diag onal entries The sp ectra l term rela tions that must b e satisfied to br ing a bo ut the links in table 6 are given in (2.13). A = F 1 , Λ = F 1 B = 1 , Ξ = 1 C = F 2 1 , Γ = F 2 1 D = F 2 , ∆ = F 1 + k F 2 (2.13) where F 1 and F 2 are any functions of the sp ectral v a riable n , such tha t F 1 6 = k F 2 , k = co nstant. Notice that a constant app ears in the expres sion for ∆ in equation (2.13), since one of the co nstants of prop or tionality in this expressio n cannot b e absorb ed into the multiplying lattice term, δ . 2 × 2 LAX P AIRS 13 The lattice term equa tio ns that arise via the compa tibility condition from this Lax pair are shown in equation (2.14). ˆ aα − a ¯ α = c ¯ β − ˆ bγ ˆ dδ − d ¯ δ = 0 b ¯ γ − ˆ cβ = 0 ˆ aβ + ˆ bδ = b ¯ α ˆ bδ = k d ¯ β ˆ cα = d ¯ γ + c ¯ δ ˆ aγ = k c ¯ δ (2.14) W e thu s conclude the description o f the for mation of Lax pairs with at least o ne double link in the 12 entry . Natur ally there are other Lax pair s o f this type but they are formed in a similar manner to that de s crib ed her e. 2.3.3. A triple link in the 12 entry. Las tly , link combinations including triple, and po ssibly double and single links in the 12 entry must b e considered. It is not difficult to see that the only p ossibility that needs to b e investigated is that with one triple link b etw een all four lattice terms in the 12 ent ry , a ny additional links in this entry constrain the problem to o heavily and lea d to tr ivial evolution eq uations only . A triple link in the 12 entry of the compa tibility condition can b e for med by setting the sp ectral terms to those in equa tion (2.1 5) b elow. A = F 1 / Ξ , Λ = F 1 /B ∆ = F 1 /B , D = F 1 / Ξ 12 A Ξ B Λ B ∆ D Ξ (2.15) Given the v alues in (2.15), the r esulting s p ectr al ter ms that a pp ea r in the 21 ent ry are as shown in equation (2.1 6). 21 C Λ A Γ D Γ C ∆ F 1 C /B F 1 Γ / Ξ F 1 Γ / Ξ F 1 C /B (2.16) W e therefor e hav e tw o p oss ibilities in the 21 entry depe nding on weather or not B Γ ∝ C Ξ. If B Γ is prop or tional C Ξ then there is another triple link in the 21 ent ry , o ther wise there is a pair o f sing le links. The latter case yields the link s g iven in item 4 o f table 7 and leads to a triv ial ev olution equation, while the former yields Lax pairs tha t include one for the LMK dV 2 system. W e co nt inue the analysis here assuming B Γ ∝ C Ξ, in this case the spectra l terms in the diagona l entries are shown in equatio n (2.1 7). 11 | (ˆ a α − a ¯ α ) F 2 1 / ( B Ξ) | ( c ¯ β − ˆ bγ ) B Γ | 22 | ( ˆ dδ − d ¯ δ ) F 2 1 / ( B Ξ) | ( b ¯ γ − ˆ cβ ) B Γ | (2.17) Again, tw o poss ibilities pr esent themselves, this time depending on whether F 2 1 / ( B Ξ) ∝ B Γ. The case wher e the prop or tionality do es not hold provides a L ax pair for the LMKdV 2 system, which is discussed in s ection 3. When F 2 1 / ( B Ξ) ∝ B Γ do e s hold, a unique s ituation unfolds where each e ntry of the compa tibilit y condition contains only one equation. This sp e cial c ase is discussed in sectio n 4.4. 14 MIKE H A Y 2.4. List of l i nk com binations. W e are now in a p os itio n to tabulate the results. T able 7 contains a represe n tative selection of all p os sible link combinations for 2 × 2 Lax pairs with a single, sepa rable term in each entry o f the L a nd M matr ices. There are still o ther link combinations that were analyzed but do not app ear in table 7 b eca us e they ar e equiv alent to a combination that do es app ear, or b e c ause it is clear that the corresp onding Lax pair ca nnot yield a n interesting evolution equation since a similar , less constra ined combination of link s is listed as trivial or ov er-determined. 12 A Ξ B Λ B ∆ D Ξ 21 C Λ D Γ A Γ C ∆ 11 A Λ B Γ C Ξ 22 D ∆ B Γ C Ξ Evolution E qn 1 LSG 2 (1.2) 2 LMKdV 2 (1.3) 3 T rivia l 4 T rivia l 5 Zero ter m 6 T rivia l 7 Underdetermined 8 T rivia l 9 T rivia l 10 Overdetermined 11 T rivia l 12 Overdetermined 13 Overdetermined 14 Overdetermined 15 Spec ia l case T able 7. List o f link combinations used to constr uc t p ossible Lax pair s Items 1 and 2 in table 7 are a nalyzed thor oughly in section 3. The other repre s en- tative link combinations are dealt with in section 4. 3. Deriv a tion of higher order LSG and L MKdV systems While fourteen p otentially viable types of La x pa irs were identified in section 2, only t wo t yp es lead to non-trivia l, w ell determined evolution equatio ns. The t wo systems th us found ar e LMKdV 2 and LSG 2 and this section describ es the deriv atio n o f these systems from the general for m o f their Lax pairs. It is impor tant 2 × 2 LAX P AIRS 15 to note that no freedo m in the lattice terms is lost through this pro cess , all v alues that the lattice terms ta ke are dictated b y the s e ts of equations effectuated by the compatibility co ndition. As such we conclude that the s y stems so derived (or equiv alent sy s tems) a re the most gener al o nes that can b e asso ciated with their Lax pairs. In fact, such ca lculations hav e b een perfor med many times b efore and yet neither LSG 2 or LMKdV 2 app ear to hav e b een published, despite coming from Lax pairs with simple for ms that hav e cer tainly already b een cons idered previously [6, 1 2]. Hence, it is necessar y to outline the metho d used to derive LSG 2 and LMKdV 2 in detail. 3.1. LSG 2 . F or LSG 2 the Lax pair used as a sta r ting p oint is L =  F 1 a F 2 b F 2 c F 1 d  (3.1a) M =  F 2 α F 1 β F 1 γ F 2 δ  (3.1b) Where a , b , c , d , α , β , γ and δ ar e a ll functions of b oth the lattice v ar iables l and m , r eferred to as lattice terms, and F 1 and F 2 depe nd on the sp ectral v ariable only , we r efer to these terms as sp ectral terms. It is p ossible to r emov e s ome of the la ttice terms using a g auge tra nsformation. While this would reduce the complex ity of the system, it is not clear at this p oint which of the lattice ter ms would b est b e removed. Exp erience shows that natu- ral transforma tions present themselves in the course of solving the compatibility condition and so we shall wait until la ter to remov e some lattice terms, keeping in mind that we exp ect so me freedo m to disapp ear from each of the Lax ma trices L and M . The co mpatibilit y condition for (3.1), b LM = M L , lea ds to the following sys tem of differenc e equations in the lattice terms ˆ aα + ˆ bγ = a ¯ α + c ¯ β (3.2a) ˆ dδ + ˆ cβ = d ¯ δ + b ¯ γ (3.2b) ˆ aβ = d ¯ β (3.2c) ˆ bδ = b ¯ α (3.2d) ˆ cα = c ¯ δ (3.2e) ˆ dγ = a ¯ γ (3.2f ) Some of equations (3.2) are linear and some nonlinea r. The linear equations can be solved ea sily when in the form k 1 ˆ φ − k 2 φ = k 3 ¯ ψ − k 4 ψ (3.3) where φ and ψ are lattice terms and k i constants. Equation (3.3) implies ψ = k 1 ˆ v − k 2 v + µ ( k 4 k 3 ) l φ = k 3 ¯ v − k 4 v + λ ( k 2 k 1 ) m where we have intro duced the new lattice ter m v = v ( l , m ), and λ = λ ( l ) a nd µ = µ ( m ) are cons tant s of int egra tion. The same fa ct also applies in a m ultiplicative sense, in particular b φ φ = ¯ ψ ψ ⇒ φ = λ ¯ v v , ψ = µ ˆ v v 16 MIKE H A Y W e now pro ceed to so lve the system (3.2). E q uations (3.2c) to (3.2 f) are linear and may solved in pairs as follows. Multiply equa tio ns (3.2c) by (3.2f) to find ˆ a ˆ d/ ( ad ) = ¯ β ¯ γ / ( β γ ), which is integrated for a = λ 2 0 ¯ v 5 / ( v 5 d ) β = µ 5 ˆ v 5 / ( v 5 γ ) where we hav e introduced λ 0 = λ 0 ( l ), µ 5 = µ 5 ( m ) and v 5 = v 5 ( l, m ). Now use equation (3.2c) again to find ¯ v 5 ¯ γ / ( v 5 γ ) = ˆ d ˆ λ 0 d λ 0 and integrate for d = λ 0 ρ ¯ v 2 /v 2 γ = µ 2 ˆ v 2 v 2 /v 5 where ρ = λ 3 ( l ) ( − 1) m . Substitute these v alues back into the expressio ns for a a nd β , and r eplace v 5 /v 2 7→ v 1 , and µ 5 /µ 2 7→ µ 1 resulting in a = λ 0 ρ ¯ v 1 v 1 (3.4a) d = λ 0 ρ ¯ v 2 v 2 (3.4b) β = µ 1 ˆ v 1 v 2 (3.4c) γ = µ 2 ˆ v 2 v 1 (3.4d) Perform similar c a lculations on equations (3.2d) and (3.2 e) to find b = λ 1 ¯ v 3 v 4 (3.5a) c = λ 2 ¯ v 4 v 3 (3.5b) α = µ 0 σ ˆ v 3 v 3 (3.5c) δ = µ 0 σ ˆ v 4 v 4 (3.5d) Where σ = µ 3 ( m ) ( − 1) l . When equations (3 .4) and (3.5) are subs tituted into (3 .2a) and (3.2 b) we find the following e quations resp ectively λ 0 ρ µ 0 σ ˆ ¯ v 1 ˆ v 3 ˆ v 1 v 3 + λ 1 µ 1 ˆ v 2 ˆ v 3 v 1 ˆ v 4 = λ 0 ρ µ 0 σ ¯ v 1 ˆ ¯ v 3 v 1 ¯ v 3 + λ 2 µ 2 ˆ ¯ v 1 ¯ v 4 ¯ v 2 v 3 (3.6a) λ 0 ρ µ 0 σ ˆ ¯ v 2 ˆ v 4 ˆ v 2 v 4 + λ 2 µ 2 ˆ ¯ v 4 ˆ v 1 ˆ v 3 v 2 = λ 0 ρ µ 0 σ ˆ v 2 ˆ ¯ v 4 v 2 ¯ v 4 + λ 1 µ 1 ˆ ¯ v 2 ¯ v 3 ¯ v 1 v 4 (3.6b) Multiplying (3.6 a) by v 1 / ˆ ¯ v 1 and (3.6 b) b y v 4 / ˆ ¯ v 4 indicates that certain v ariables alwa ys appear in combination. As such, we set v 1 , v 2 ≡ 1 without lo ss of generality , and r ename v 3 = x and v 4 = y . This is the manifest reductio n in fr eedom that was exp ected fro m the p ersp ective of g auge transfor ma tions. The par ameter functions similarly app ea r in ratios, hence we set λ 0 ≡ 1 and µ 0 ≡ 1 without lo ss of g enerality . Making the substitutions we arrive at a pair of nonlinear pa rtial difference equations in x and y , with ar bitrary non-a utonomous ter ms λ i ( l ) and µ i ( m ), which together 2 × 2 LAX P AIRS 17 form LSG 2 , the evolution equation asso cia ted with the Lax pair . ρ σ ˆ x x + λ 1 µ 1 ˆ ¯ x ˆ y = σ ρ ˆ ¯ x ¯ x + λ 2 µ 2 x ¯ y (3.7a) σ ρ ˆ ¯ y ˆ y + λ 2 µ 2 ˆ xy = ρ σ ¯ y y + λ 1 µ 1 ¯ x ˆ ¯ y (3.7b) This pair of equations can b e thought of as a higher or de r lattice sine-Gordon system b ecaus e the la ttice sine-Go r don equation (LSG) is retr ieved by setting y = x in either expressio n LSG : ˆ ¯ xx ( σ ρ − λ 1 µ 1 ¯ x ˆ x ) = ρ σ ¯ x ˆ x − λ 2 µ 2 The La x pa ir (1.4) for LSG 2 is obtained by substituting the c a lculated v a lues of the lattice terms back into (3.1). 3.2. LMKdV 2 . The the gener al form of the La x pair fo r LMKdV 2 is similar to that for LSG 2 , the only difference b eing that here b oth matrices exhibit the same depe ndence on the sp ectra l v ar iable, while the dep endence was antisymmetric in the previous case. L =  F 1 a F 2 b F 2 c F 1 d  (3.8) M =  F 1 α F 2 β F 2 γ F 1 δ  (3.9) As ab ov e, a , b , c , d , α , β , γ and δ are all functions of the b oth the lattice v ar iables l a nd m , F 1 and F 2 depe nd on the sp ectral v ariable only . The compa tibilit y condition leads to six equations coming from the different orders of the sp ectral v ar iable in each entry . ˆ aα = a ¯ α (3.10a) ˆ bγ = c ¯ β (3.10b) ˆ dδ = d ¯ δ (3.10c) ˆ cβ = b ¯ γ (3.10d) ˆ aβ + ˆ bδ = b ¯ α + d ¯ β (3.10e) ˆ cα + ˆ dγ = a ¯ γ + c ¯ δ (3.10f ) Equations (3.10a) and (3.10c) a r e integrated immediately , while (3.10 b) and (3.10d) are multiplied together and dealt with by a similar metho d to that used for the LSG 2 system of section 3.1 leading to the following r esults a = λ 1 ¯ v 1 /v 1 (3.11a) b = λ 0 ρ ¯ v 3 /v 4 (3.11b) c = λ 0 ¯ v 4 / ( ρv 3 ) (3.11c) d = λ 2 ¯ v 2 /v 2 (3.11d) α = µ 1 ˆ v 1 /v 1 (3.11e) β = µ 0 ˆ v 3 / ( σ v 4 ) (3.11f ) γ = µ 0 σ ˆ v 4 /v 3 (3.11g ) δ = µ 2 ˆ v 2 /v 2 (3.11h) Where v i = v i ( l, m ), λ i = λ i ( l ), µ i = µ i ( m ), ρ = λ ( − 1) m 3 and σ = µ ( − 1) l 3 . Substitut- ing these v alues into the tw o rema ining eq ua tions, (3 .10e) and (3.1 0f), shows that 18 MIKE H A Y terms consistently app ear in ra tios, as they did with the LSG 2 system. W e choos e to s e t v 2 ≡ v 3 ≡ 1, λ 0 ≡ µ 0 ≡ 1, v 1 = x and v 4 = y and ar rive at LMKdV 2 : λ 1 σ ˆ ¯ x ˆ x + µ 2 ρ y ˆ y = λ 2 σ y ¯ y + ρµ 1 ˆ ¯ x ¯ x (3.12) ρµ 1 ˆ x ˆ ¯ y + λ 2 σ x ˆ y = µ 2 ρ x ¯ y + λ 1 σ ¯ x ˆ y (3.13) 4. How most link combina tions lead to bad evolution equa tions Here we explain ho w most of the p otentially via ble Lax pairs fail to pro duce int eresting evolution equa tions. The topic is split into four pa rts dealing with Lax pairs that le ad to trivial, over-determined, under-determined evolution equatio ns and the spe cial case of item 15 in table 7. 4.1. Lax pairs that yield only trivial evolution equations. This section is per tinent to items 3, 4, 6, 8 , 9 and 11 in table 7, those Lax pairs who ’s co mpa tibilit y conditions can be solved to a p oint where only linear equatio ns remain, or the equations can b e reduced to firs t or der equatio n in o ne lattice direction only . The simplest r oute to tr iviality is to have a Lax pair with a set of eq uations that are all linear, a s p er item 3 in table 7. The compatibility co nditio n leads to a set of eig ht linear equations in the eight initial lattice terms. ˆ aα = a ¯ α, ˆ bγ = c ¯ β ˆ dδ = d ¯ δ , ˆ cβ = n ¯ γ ˆ aβ = d ¯ β , ˆ bδ = b ¯ α ˆ dγ = a ¯ γ , ˆ cα = c ¯ δ (4.1) Equations (4.1) c a n b e solved easily using tech niques describ ed in section 3, with the res ult of a simple, linear evolution equation. Ho wev er, it is not neces sary to conduct such analys is on this sys tem since all equatio ns (4.1) are linear a nd they can not b e exp ected to pro duce a nonlinear evolution equation. F or this reason, other exa mples of link combinations that pro duce only linear equations hav e b een omitted from table 7. Item n umber 8 fro m ta ble 7 is a n example of a La x pair that lea ds to a trivial evolution equation in a more co mplex way . The equatio ns that come out of its compatibility condition ar e ˆ aα = a ¯ α (4.2a) ˆ bγ = c ¯ β (4.2 b) ˆ dδ + ˆ cβ = d ¯ δ + b ¯ γ (4.2c) ˆ aβ + ˆ bδ = b ¯ α (4.2 d) k b ¯ α = − d ¯ β (4.2e) ˆ cα = a ¯ γ + c ¯ δ (4.2f ) k ˆ cα = − ˆ dγ (4.2g ) Where k is a constant. This system of equations is solved as follows: integrate (4.2a) for a = λ ¯ v /v , α = µ ˆ v /v , intro ducing λ = λ ( l ), µ = µ ( m ) and v = v ( l , m ). Multiply equations (4.2 b) and (4.2g), then divide the pr o duct b y (4.2e) and use the v alues calcula ted for a a nd α to find ˆ b ˆ c ˆ v bcv = ˆ d ˆ ¯ v d ¯ v (4.3) 2 × 2 LAX P AIRS 19 Equation (4 .3) is int egra ted for b , which is s ubstituted into equa tions (4.2e) a nd (4.2b) y ielding the following v a lues b = λ 2 d ¯ v cv ¯ β = − k µλ 2 ˆ ¯ v cv γ = − k µ ˆ c ˆ v ˆ dv There ar e now thr ee equations left to solve, (4.2c ), (4.2d) and (4.2 f), where it is found that c and v alwa ys appea r as a pro duct which suggests we introduce u = cv . Equation (4.2c) is then used to write ¯ δ = µ ˆ u u + k µλ ˆ ¯ u u ˆ ¯ d Finally , we intro duce x = u d/u and achiev e the final tw o eq uations in x λ 2 ( ˆ x − x ) = k ( λ 2 λ − λ λ 2 ) (4.4a) ˆ x − x + k ( λ − λ 2 ) = k x ˆ ¯ x ( µλ − λ 2 ) (4.4b) Equations (4.4) is an ov erdetermined s et of tw o equations in the one v aria ble, x , which ca nnot hop e to yield an interesting evolution equation. This is b ecause (4.4a) can be used to remove the dep endence of x on m ( m is the indep endent v ariable of the ‘ ˆ ’ directio n), leaving x with a fir st order dep endence on l in (4.4b) at b est. 4.2. Overdetermined Lax pairs. Here items 10 and 12 through 1 4 in table 7 are dealt with. Such Lax pair s hav e compatibility c o nditions that bo il do wn to more equations than ther e are fre e lattice terms. T he s e are no t ne c e ssarily trivial, as the solution to one equation may solve ano ther as well, that is it may b e pos sible to make one or more equatio ns re dundant . Or one equation may b e a co mpati- ble similarity condition fo r a nother equation as expla ined in [23]. Some systems of this type , where any hop e o f suppo rting an int eresting evolution equation has bee n quas he d, hav e a lready b een consider e d in section 4.1. The rema ining ov er- determined s ystems are co nsidered her e , how ev er we do not attempt to resolve the issue of whether these systems s upp or t int eresting evolution equatio ns, we simply list them as b eing overdetermined. The most int eresting ins tances o f this type o f system a rise fro m Lax pair s with t wo double links in the off-diago nal entries, those represented by item 14 from table 7. All s uch sy stems, with minimum constra int on the la ttice terms in the diago nal ent ries of the compatibility condition, lead to the same evolution equa tio ns. An example of a Lax pair with tw o do uble links in the o ff-diagonal entries is L =  F 1 a b ( F 2 1 + k 4 F 1 F 2 ) c F 2 d  (4.5a) M =  ( F 1 + k 2 F 2 ) α β ( F 2 1 + k 3 F 1 F 2 ) γ ( F 1 + k 1 F 2 ) δ  (4.5b) where lower case letters except k i are la ttice terms, k i are constants of prop ortion- ality and F i are sp ectral ter ms with F 1 not pro po rtional to F 2 . This particular Lax pair is e s pec ially int eresting b ecause the solutio n of its c o mpatibility condition inv o lves in tegrating bo th additive and mult iplicative linear difference equations, as describ ed near equation (3.3) in sec tion 3. The fina l evolution equations achiev ed are relatively complica ted and nonlinea r, how ever there a re t wo equations for one v aria ble and it rema ins to be seen whether they can b e re conciled. The following outlines how to solve the compatibility co ndition. 20 MIKE H A Y In the compatibility condition, b LM = M L , the 11 e n try dictates k 2 = k 3 = k 4 . Redefine F 2 7→ k 2 F 2 to remove k 2 from everywhere ex c e pt the 22 entry of L where we set k 0 = 1 /K 2 and th us arrive at the following equations from the v arious orders of the c o mpatibility condition ˆ aα + ˆ bγ = a ¯ α + c ¯ β (4.6a) ˆ c ¯ β = b ¯ γ (4.6 b) ˆ dδ = d ¯ δ (4.6c) ˆ aβ + ˆ bδ = b ¯ α (4.6d) k 1 ˆ bδ = b ¯ α + k 0 d ¯ β (4.6e) ˆ cα = a ¯ γ + c ¯ δ (4.6f ) ˆ cα + k 0 ˆ dγ = k 1 c ¯ δ (4.6g ) Int egrate (4.6c) for d = λ ¯ v /v , δ = µ ˆ v /v , introducing v = v ( l, m ), λ = λ ( l ) and µ = µ ( m ), and use equa tio n (4.6b) to see that β = b ¯ γ / ˆ c . Now define s = c/ ¯ v , t = γ / ˆ v and u = bv so that β = b ¯ t/ ˆ s (4.7a) (4.6g) ⇒ α = 1 ˆ s ( k 1 µs − k 0 λt ) (4.7b) (4.6f) ⇒ a = 1 ¯ t (( k 1 − 1) µs − k 0 λt ) (4.7c) Substituting these (4.7a) v alues into equa tio n (4.6e) and rearr anging leads to k 1 \  u ¯ s λ ¯ λ  − k 1  u ¯ s λ ¯ λ  = k 0  u ¯ t λµ  − k 0  u ¯ t λµ  (4.8) which is an a dditive linea r difference equation that ca n b e integrated to find u ¯ s λ ¯ λ = ¯ w − w + λ 2 (4.9a) u ¯ t λµ = ˆ w − w + µ 2 (4.9b) The tw o equations that rema in, equations (4.6a) and (4.6 d) res p ectively , are written in terms of s , t and u ˆ ¯ s ˆ ¯ t (( k − 1) ˆ µ − k 0 λ ˆ t ˆ s )( k 1 µ s t − k 0 λ ) + ˆ u ˆ ¯ s = (( k − 1) µ s t − k 0 λ )( k 1 µ − k 0 ¯ λ ¯ t ¯ s ) + ¯ u ¯ ¯ t s t (4.10a) ˆ ¯ s ˆ ¯ t (( k − 1) ˆ µ − k 0 λ ˆ t ˆ s ) + µ ˆ u ˆ ¯ s u ¯ t = k 1 µ ¯ s ¯ t − k 0 ¯ λ (4.1 0b) Make a further change of v aria bles x = u ¯ t , y = u ¯ s , to co mpletely re mov e u from equations (4.10) and (4.9). Equatio n (4.10b) b ecomes µ ˆ ¯ y ¯ x + ˆ ¯ y ˆ ¯ x (( k 1 − 1) ˆ µ − k 0 λ ˆ x ˆ y ) = k 1 µ ¯ y ¯ x − k 0 ¯ λ (4.11) while equation (4.10 b) b ecomes ˆ ¯ y ˆ ¯ x (( k 1 − 1) ˆ µ − k 0 λ ˆ x ˆ y )( k 1 µ y x − k 0 λ ) + ˆ ¯ y = (( k 1 − 1) µ y x − k 0 λ )( k 1 µ − k 0 ¯ λ ¯ x ¯ y ) + ¯ ¯ x y x (4.12) where x and y ar e b oth wr itten in ter ms of w a ccording to ¯ x = λµ k 0 ( ˆ w − w + µ 2 ) (4.13a) ¯ y = λ ¯ λ k 1 ( ¯ w − w − λ 2 ) ( 4.13b) 2 × 2 LAX P AIRS 21 Hence, there ar e tw o complica ted, no nlinear e q uations fo r the one v ar iable w , (4.1 1) and (4.12). The s e tw o equa tions may or may no t be reconcila ble, one w ay that they co uld b e r econciled is if one eq uation was shown to b e a compatible similarity constraint fo r the other [22, 23] 4.3. Underdetermine d Lax pai rs. There are Lax pairs who’s compatibilit y con- dition can b e solved completely , while still leaving at lea st o ne lattice term free. In these cases there is no genuine evolution equation, althoug h the freedom inheren t in the system could to cause it to app ear as thoug h there was. In fac t, any evolution equation, trivial, integrable or even chaotic, co uld b e fals e ly r epresented by such a Lax pair . One such case is item 7 on table 7 that has as its compatibility condition ˆ aα + ˆ bγ = a ¯ α + c ¯ β (4.14a) ˆ dδ + ˆ cβ = d ¯ δ + b ¯ γ (4.14b) ˆ aβ = − ˆ bδ (4.14c) b ¯ α = − d ¯ β (4.14d) ˆ cα = − ˆ dγ (4.14e) a ¯ γ = − c ¯ δ (4.14f ) Here is a roa dmap to the solution: m ultiply equa tion (4 .14c) by (4.14 f) and divide by (4 .1 4d) a nd (4.14e) to find an express io n that can b e integrated for d = λ 1 ¯ v cb/ ( v a ) γ = µ 1 ˆ v αδ / ( v β ) Substituting these v alues for d a nd γ into the ratio of equations (4.14d) and (4.14f) shows that v = λ 1 µ 1 ˆ ¯ v , which indicates that v must b e se pa rable in to a pro duct such as v = λ 2 ( l ) µ 2 ( m ), wher e λ 1 = λ 2 / ¯ λ 2 and µ 1 = µ 2 / ˆ µ 2 . With the ab ov e v alues included and no further integration r equired, equation (4.14c) is used to find δ while (4.14 d) offers c . In summary: d = − b ¯ α/ ¯ β (4.15a) γ = − ˆ aα/ ˆ b (4.15b) δ = − ˆ aβ / ˆ b (4 .1 5c) c = − a ¯ α/ ¯ β (4.15d) The key feature is that, with (4.15), equations (4.14 a) and (4.14b) ar e automati- cally satisfied, furnishing us with no mo re constraints on the remaining lattice terms a , b , α and β . Therefor e, equations (4.1 5) are the evolution equa tions, but these are not uniquely determined and there is freedom enough to write a ny equa tion at all into this set, including equations tha t are k nown to b e not integrable. The undeniable co nclusion is that any Lax pair o f this type is false b ecaus e one would exp ect that the informa tion to be g leaned a bo ut the solutio n to any evolution equa - tion asso ciated with this La x pair must b e underdetermined like equa tion (4.1 5). Note further that the level of freedom left in the system is exactly that which can be remov ed by gauge tr a nsformations, so, in essence, this systems compa tibilit y is simply determined by the v alues of the para meter functions and no evolution equation exists. 4.4. A sp ecial case. Item 15 from table 7 is a sp ecial c a se tha t s ees a ll lattice terms linked in a ll four entries of the compatibility condition. Thus, the co mpati- bilit y condition supplies only nonlinear equatio ns, none of which can b e ex plicitly 22 MIKE H A Y int egrated, and s o no parameter functions present themselves a s co nstants of inte- gration. This is a n unusual situation but it is no t necessa ry to solve the s ystem bec ause of the following arg uments rega r ding the sp ectra l dep endence. T o form the links that define this L ax pa ir we re quire the fo llowing conditio ns , or an equiv alent set, o n the sp ectr al ter ms. A = B Γ / Λ Ξ = Λ 2 / Γ C = B Γ 2 / Λ 2 ∆ = Λ D = B Γ / Λ (4.16) Equation (4.17) shows the g eneral for m of the Lax pair that p ossesses the links in question is L = B F 1  a b/F 1 cF 1 d  (4.17a) M = Λ  α β / F 1 γ F 1 δ  (4.17b) where F 1 ( n ) = Γ / Λ. The prefac tors in equa tio n (4.17) are obsolete beca use they cancel in the compatibilit y condition. This leaves the L a x pa ir with the sa me depe ndence on just one s pec tr al term in b oth the L and M matr ices. As such, a ga uge transformatio n c an b e used to completely remove the dep endence on the sp ectral v ar iable from the linear pro blem, implying that equation (4.17) is actually not a Lax pair at all. 5. conclusion All 2 × 2 Lax pairs, with one separa ble term in the four ent ries of each matrix, hav e b een consider ed through the v arious combinations o f terms pos sible in their compatibility conditions. It has b een shown that the o nly non-trivial evolution equations that can b e supp o rted are the higher or der generalizations of the LMK dV and LSG equations, with the p ossible ex ception of over-determined sys tems that may yet be consistent, see s ection 4.2. The results of the present work supp or t the po stulated connection b etw een Lax pair s a nd sing ula rity confinement [6]. There is an imp ortant question in where the present work sits in r elation to studies in multid imensional consistency , o r c onsistency aro und a cub e (CAC), tha t provide a metho d of sear chin g for integrable partia l difference equa tio ns where a Lax pair can b e derived a s a bi-pro duct of the procedur e [20, 25]. The present work adds to multidimensional consistency studies b y removing the r estriction that the Lax matrices must be symmetric, a consequence of using the same Q equation in all three directions of the cub e [1 4], and also leads to non-auto no mous equations, where m ultidimensional co ns istency studies have only considered autonomous systems. Non-autonomy is vital to reductions o f the type used in [15] and [16], that lead to nonlinear ordina ry difference equatio ns a nd Lax pairs fo r them. In addition, while it has bee n s hown that (CA C) ensur es the e xistence of a L a x pair, the conv erse is not necessarily tr ue, which in itself indicates a need for the present completeness study . F uture studies using the techniques explained here should investigate Lax pa irs with non-s e parable terms o r with more terms in each matrix entry , and p ossibly aim for a genera l alg orithm for dealing with an arbitrar y num b er of ter ms in ea ch ent ry o f the Lax matr ic e s. Other types, s uc h as differ ent ial difference Lax pair s, should be considere d, as s hould pur ely contin uous Lax pair s where there is already a v a st b o dy of knowledge with which to compare results. F urther, this technique is easily adaptable to isomono dr omy Lax pairs. 2 × 2 LAX P AIRS 23 There is some conflict ab out whether the para meters of the Q 4 equation, in the ABS scheme, must lie o n elliptic curves or no t [17, 18, 19]. The Lax pair for Q 4 , found via m ultidimensional consistency , is for a v ersion of the equa tion wher e the autonomous parameters are r estricted to elliptic c ur ves [20]. An explo ration in to the links present in that Lax pair might reso lve the issue by providing a La x pa ir for Q 4 with non-autonomo us, a nd p o ssibly free, parameters . References [1] F.W. Ni jhoff, and H.W. Cap el, The discrete Kor tew eg-de V ri es equation, Acta Appl. Math. , 39 (1-3) (1995) 133–158 [2] V. P apageorgiou, B. Grammaticos, and A. 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Sak ai, A q -analog of the Garnier system, F unkcial. Ekvac. 48 (2005) 273–297. School of Ma thema tics and S t a tistics F07, The University of Sy dney, NS W 2006, Aust ralia E-mail addr ess : mhay@mail.usy d.edu.au