On consistency of determinants on cubic lattices

arXiv:0809.2032v1 [nlin.SI] 11 Sep 2008 On consistency of determinants on cubic lattices O. I. Mokhov Consider the square lattice Z2 with vertices at points with integer-valued coordi- nates in R2 = {(x1, x2)| xk ∈R, k = 1, 2} and complex (or real) scalar fields u on the lattice Z2, u : Z2 →C, that are defined by their values ui1i2, ui1i2 ∈C, at each vertex of the lattice with the coordinates (i1, i2), ik ∈Z, k = 1, 2. Consider a class of two-dimensional discrete equations on the lattice Z2 for the field u that are defined by functions Q(x1, x2, x3, x4) of four variables with the help of the relations Q(uij, ui+1,j, ui,j+1, ui+1,j+1) = 0, i, j ∈Z, (1) so that in each elementary 2×2 square of the lattice Z2, that is, in each set of vertices of the lattice with coordinates of the form (i, j), (i + 1, j), (i, j + 1), (i + 1, j + 1), i, j ∈Z, the value of the field u at one of vertices of the square is defined by the values of the field at three other vertices. In this case the field u on the lattice Z2 is completely determined by fixing initial data, for example, on the axes of coordinates of the lattice, ui 0 and u0j, i, j ∈Z. A particularly important role is played by integrable nonlinear discrete equations. In [1]–[3] integrable discrete equations of the form (1) are singled out by the very natural condition of consistency on cubic lattices (see also [4]–[9]). Consider the cubic lattice Z3 with vertices at points with integer- valued coordinates in R3 = {(x1, x2, x3)| xk ∈R, k = 1, 2, 3} and fix initial data ui 00, u0j0 and u00k, i, j, k ∈Z, on the axes of coordinates of the cubic lattice. A two-dimensional discrete equation (1) is called consistent on the cubic lattice if for initial data in general position the discrete equation (1) can be imposed in a consistent way on all two-dimensional sublattices of the cubic lattice Z3 at once (see [1]–[5]). Classifications of discrete equations of the form (1) that are consistent on the cubic lattice have been studied in [4] and [8] under some additional restrictions, see also [9]. The equation defined by determinants of 2 × 2 matrices of values of the field at vertices of elementary 2 × 2 squares of the lattice Z2: ui,j+1ui+1,j −ui+1,j+1uij = 0, i, j ∈Z, (2) is an example of such two-dimensional nonlinear integrable discrete equation consis- tent on the cubic lattice. The equation (2) is linear with respect to each variable 1The work was completed with the financial support of the Russian Foundation for Basic Research (grant no. 08-01-00054) and the Programme for Support of Leading Scientific Schools (grant no. NSh-1824.2008.1). 1 and invariant with respect to the full symmetry group of square. The fixing of ar- bitrary nonzero initial data ui 0 and u0j, i, j ∈Z, on the axes of coordinates of the lattice Z2 completely determines the field u on the lattice Z2 satisfying the dis- crete equation (2), and the fixing of arbitrary nonzero initial data ui 00, u0j0 and u00k, i, j, k ∈Z, on the axes of coordinates of the lattice Z3 completely deter- mines the field u on the lattice Z3 satisfying the discrete equation (2) on all two- dimensional sublattices of the cubic lattice Z3. The integrability (in the broad sense of the word) of the discrete equation (2) is obvious since it can be easily linearized: ln ui,j+1 + ln ui+1,j −ln ui+1,j+1 −ln uij = 0, i, j ∈Z. In this work, we consider the question on consistency on cubic lattices for discrete nonlinear equations defined by determinants of matrices of higher orders (for orders N > 2). The condition of con- sistency on cubic lattices in the form as it was defined above is not satisfied for these discrete equations if N > 2. We prove that an other, modified, condition of consis- tency on cubic lattices, which is proposed in this work, is satisfied for determinants of matrices of arbitrary orders. Consider a discrete equation on the lattice Z2 defined by a relation for values of the field u at vertices of the lattice Z2 that form elementary 3 × 3 squares: Q(uij, . . . , ui+s,j+r, . . . , ui+2,j+2) = 0, 0 ≤s, r ≤2, i, j ∈Z. (3) The fixing of initial data ui 0, ui1, u0j and u1j, i, j ∈Z, completely determines the field u on the lattice Z2 satisfying the discrete equation (3). Consider the cubic lattice Z3 and the condition of consistency on all two-dimensional sublattices of the cubic lattice Z3 for the discrete equation (3). Initial data can be specified, for example, at the following vertices of the lattice: ui 00, ui10, ui 01, ui11, u0j0, u1j0, u0j1, u1j1, u00k, u10k, u01k and u11k, i, j, k ∈Z. In the cube {(i, j, k), 0 ≤i, j, k ≤2}, the values u202, u212, u220, u221, u022 and u122 are determined by the relations (3), and three relations must be satisfied for the value u222 on three faces of the cube at once. Consider the discrete nonlinear equation on the lattice Z2 defined by determinants of the matrices of values of the field u at vertices of the lattice Z2 that form elementary 3×3 squares: ui,j+2ui+1,j+1ui+2,j + ui,j+1ui+1,jui+2,j+2 + ui,jui+1,j+2ui+

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