On consistency of determinants on cubic lattices

On consis tency of determinan ts on cubic latt ices O. I. Mokho v Consider the square lattice Z 2 with v ertices at p oin ts with in teger-v alued co ordi- nates in R 2 = { ( x 1 , x 2 ) | x k ∈ R , k = 1 , 2 } and complex (or real) scalar fields u on the lattice Z 2 , u : Z 2 → C , that are defined b y their v alues u i 1 i 2 , u i 1 i 2 ∈ C , at each v ertex of the lattice with the co o rdinates ( i 1 , i 2 ), i k ∈ Z , k = 1 , 2. Consider a class of t w o- dimensional discrete equations on the lattice Z 2 for the field u that a r e defined b y functions Q ( x 1 , x 2 , x 3 , x 4 ) of four v ar ia bles with the help of the relations Q ( u ij , u i +1 ,j , u i,j +1 , u i +1 ,j +1 ) = 0 , i, j ∈ Z , (1) so that in eac h elementary 2 × 2 squar e of the lattic e Z 2 , that is, in eac h set of ve rtices of the lattice with co ordinates of the fo rm ( i, j ), ( i + 1 , j ), ( i, j + 1), ( i + 1 , j + 1), i, j ∈ Z , the v alue of the field u at one of v ertices of the square is define d b y the v alues of the field at three other ve rtices. In this case the field u on the lattice Z 2 is completely determined by fixing initial data, for example, on the axes of co ordinates of the lattice, u i 0 and u 0 j , i, j ∈ Z . A pa rticularly imp ortan t ro le is play ed by inte gr able nonline ar discr ete e quations . In [1]–[3] in tegrable discrete equations of the form (1) a re singled out b y the v ery natural condition of c o n sistency on cubic lattic es (see also [4]–[9]). Consider the cubic lattice Z 3 with v ertices at p oints with integer- v alued co ordinates in R 3 = { ( x 1 , x 2 , x 3 ) | x k ∈ R , k = 1 , 2 , 3 } and fix initial data u i 00 , u 0 j 0 and u 00 k , i, j, k ∈ Z , o n the axes of co ordinates of the cubic lattice. A t w o- dimensional discre te equation (1) is called c onsistent on the cubic lattic e if for initial data in general p osition the discrete equation (1) can b e imp osed in a consisten t w a y on all t w o- dimensional sublattices of the cubic lattice Z 3 at o nce (see [1]–[5]) . Classifications of discrete equations of t he form (1) that are consisten t on the cubic lattice hav e been studied in [4] and [8] under some additio nal restrictions, see also [9]. The equation defined b y determinan ts o f 2 × 2 matrices of v alues of the field at v ertices o f elemen tary 2 × 2 squares of the lattice Z 2 : u i,j +1 u i +1 ,j − u i +1 ,j +1 u ij = 0 , i, j ∈ Z , (2) is an example of suc h tw o-dimensional nonlinear in tegrable discrete equation consis- ten t on the c ubic lattice. The equation (2) is linear with resp ect to eac h v ariable 1 The work was completed with the fina nc ia l suppor t of the Russian F oundation for Ba sic Research (grant no. 08 -01-0 0054 ) and the Progr a mme for Supp ort of Le a ding Scientific Scho ols (gra nt no. NSh-1824.2 008.1 ). 1 and in v arian t with resp ect to the full s ymmetry group of s quare. The fixing of ar- bitrary nonzero initial data u i 0 and u 0 j , i, j ∈ Z , o n the a xes of co ordinates of the lattice Z 2 completely determines the field u on the lattice Z 2 satisfying the dis- crete equation (2), and the fixing of arbitrary nonzero initial data u i 00 , u 0 j 0 and u 00 k , i, j, k ∈ Z , o n the axes of co ordinates of the lattice Z 3 completely deter- mines the field u on the lat t ice Z 3 satisfying the discrete equation (2) on all t wo- dimensional sublattices of the cubic latt ice Z 3 . The in tegrabilit y (in the broad sense of the word) of the discrete equation (2) is ob vious since it can b e easily linearized: ln u i,j +1 + ln u i +1 ,j − ln u i +1 ,j +1 − ln u ij = 0 , i, j ∈ Z . In this work, w e consider the question o n consis tency on cubic lattices for discrete nonlinear equations defined b y determinan t s of matrices of higher orders (f or orders N > 2). The condition o f con- sistency on cubic lattices in the form as it w as defined a b o v e is not satisfied for these discrete equations if N > 2. W e prov e that an other, mo dified, condition of consis- tency on cubic lattices, whic h is prop osed in this w ork, is satisfied for determinan ts of matr ices o f arbitr ary orders. Consider a discre te equation on the lattice Z 2 defined b y a relatio n for v alues of the field u at ve rtices of the lattice Z 2 that fo rm elementary 3 × 3 squar es : Q ( u ij , . . . , u i + s,j + r , . . . , u i +2 ,j +2 ) = 0 , 0 ≤ s, r ≤ 2 , i, j ∈ Z . (3) The fixing of initia l data u i 0 , u i 1 , u 0 j and u 1 j , i, j ∈ Z , completely determines the field u on the lattice Z 2 satisfying the discrete equation (3 ) . Cons ider the cubic lattice Z 3 and the condition o f consistency on all tw o-dimensional sublattices of the cubic lattice Z 3 for the discrete equation (3). Initial data can b e sp ecified, for example, at the following v ertices of the lattice: u i 00 , u i 10 , u i 01 , u i 11 , u 0 j 0 , u 1 j 0 , u 0 j 1 , u 1 j 1 , u 00 k , u 10 k , u 01 k and u 11 k , i, j, k ∈ Z . In the cub e { ( i, j, k ) , 0 ≤ i, j, k ≤ 2 } , the v alues u 202 , u 212 , u 220 , u 221 , u 022 and u 122 are determined by the relations (3), and three relations m ust b e satisfied f o r the v alue u 222 on three faces of the cub e at once. Cons ider the discrete nonlinear equation on the la ttice Z 2 defined b y determinan ts o f the matrices of v alues of the field u a t v ertices of the lattice Z 2 that form eleme n tary 3 × 3 squares: u i,j +2 u i +1 ,j +1 u i +2 ,j + u i,j +1 u i +1 ,j u i +2 ,j +2 + u i,j u i +1 ,j +2 u i +2 ,j +1 − − u i,j u i +1 ,j +1 u i +2 ,j +2 − u i,j +2 u i +1 ,j u i +2 ,j +1 − u i,j +1 u i +1 ,j +2 u i +2 ,j = 0 , (4) where i, j ∈ Z . The equation (4) is linear with resp ect to eac h v ariable and in v arian t with r esp ect to the full symmetry group of the configuration o f v ertices of the lattice Z 2 that form elemen tary 3 × 3 squares. F or init ia l dat a in general p osition, the discrete equation ( 4) is not consisten t on t w o- dimensional sublattices of the cubic lattice Z 3 . Consider an other, mo dified, condition of consistency o n cubic lattices for the dis- crete nonlinear equation (4). W e shall require that the discrete equation (4 ) would b e satisfied not only on all tw o-dimensional sublattices of the cubic la ttice Z 3 , but 2 also on all unions of an y tw o inte rsecting t w o-dimensional sublattices of the cubic lattice Z 3 , that is, the corresp onding elemen tary 3 × 3 squares, on whic h the dis- crete equation (4) is considere d, can b e b ent at righ t angle passing fro m one of tw o- dimensional sublattices t o a no t her, for example, { ( i, 0 , 0 ) , ( i, 1 , 0) , ( i, 0 , 1) , i = 0 , 1 , 2 } , { (0 , j, 0) , (1 , j, 0) , (0 , j, 1) , j = 0 , 1 , 2 } and { (0 , 0 , k ) , (1 , 0 , k ) , (0 , 1 , k ) , k = 0 , 1 , 2 } . In this case, initial data can b e sp ecified, for example, a t the follo wing ve rtices of the cubic lattice Z 3 : u i 00 , u i 01 , u 2 j 0 , u 2 j 1 , u 10 k , u 20 k , u 110 and u 112 , i, j, k ∈ Z . W e shall also call the corresp onding discrete equations c onsistent on cubic l a ttic es . Then the follo wing t heorem holds. Theorem 1 . F or arbitr ary in itial data in gener al p osition the n o nline ar dis- cr ete e quation (4) c an b e s a tisfi e d in a c onsi s tent way on al l unions of a ny two two- dimensional sublattic es of the cubic lattic e Z 3 , that is, the discr ete nonline ar e q uation (4) is c onsi s tent on the cubic l a ttic e Z 3 . A similar prop erty of consistency on cubic lattices ho lds f or determinants of arbi- trary orders N . Theorem 2 . F or any given inte ger N , the discr ete no nline ar e quation define d b y determinants of the matric es of values of the field u at vertic es of the lattic e Z 2 that form elem entary N × N squar e s is c onsistent on the cubic lattic e Z 3 . Bibliograph y [1] F.W. Nijhoff, A.J. W alk er, Glasgow Math. J. 43A (2 001), 109–123 . [2] F.W. Nijhoff, Ph ys. L e tt. A 297 (2002 ), 49 – 58. [3] A.I. Bob enk o , Y u.B. Suris, Int. Math. R es . Notic es 11 (2 002), 573 –611. [4] V.E. Adler, A.I. Bob enk o, Y u.B. Suris, Comm. Math. Phys. 233 (2003), 513–543 . [5] A.I. Bob enk o , Y u.B. Suris, arXiv: math/ 0 504358. [6] A.I. Bob enk o, Y u.B. Suris, Usp ekh i Mat. Nauk 62 :1 (2 0 07), 3–50 ; English transl., A.I. Bo b enk o, Y u.B. Suris, Russian Math. Surveys 62 :1 ( 2007), 1-43. [7] A.P . V eselov , Usp ekhi Mat. Nauk 46 :5 (1991), 3–4 5. English t r a nsl., A.P . V eselo v, Ru ssian Math. Surveys 46 :5 (1991 ), 1- 51. [8] V.E. Adler, A.I. Bob enk o, Y u.B. Suris, arXiv: abs/07 05.1663. [9] S.P . Tsarev, T. W olf, L ett. Math. Ph ys. 84 :1 (2008), 31–39. Cen tre for Non-L inear Studies, L.D.Landau Institute for Theoretical Ph ysic s, Russian Academ y of Sciences; Departmen t of Geometry a nd T o p ology , F aculty of Mec hanics and Mathematics, M.V.Lomonoso v Mosco w State Unive rsit y; E-mail : mokho v@mi.ras.ru; mokho v@landau.ac.ru; mokho v@bk.ru 3