Discrete Levy Transformations and Casorati Determinant Solutions of Quadrilateral Lattices

Sequences of discrete Levy and adjoint Levy transformations for the multidimensional quadrilateral lattices are studied. After a suitable number of iterations we show how all the relevant geometrical features of the transformed quadrilateral lattice can be expressed in terms of multi-Casorati determinants. As an example we dress the Cartesian lattice. On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China Supported by Beca para estancias temporales de doctores y tecnólogos extranjeros en España: SB95-A01722297 Partially supported by CICYT: proyecto PB95–0401 1 1. Recently it has been shown that multidimensional quadrilateral lattices are integrable [6] and a number of results about this system have been obtained. Let us mention the reduction mechanism based on the ∂̄ formalism [4], the multidimensional circular lattice [2], the relation with multicomponent KP through the Miwa transformation and geometrical meaning of the τ -function [5] and Darboux and more general transformations [14, 7]. Quadrilateral lattice equations, as a discrete integrable system, appeared for the first time in [1], however no geometrical understanding of this system can be found there. We should mention that the quadrilateral lattice has a continuum limit to conjugate nets [3, 8]. Since last century [12] it has been known that there is a transformation, called Levy transformation, that preserves the conjugacy character of the net. This transformation was iterated, for the bidimensional case, in [10]; and recently [13] we have used standard techniques in Soliton Theory to obtain closed formulae, in terms of multiWroński determinants, for all the relevant geometrical objects. The analog of this transformation, say discrete Levy, can be found, for the points of the lattice, in [1] and a detailed geometrical exposition of it is contained in [7]. The aim of this paper is to obtain similar results for the quadrilateral lattice as we did with conjugate nets, namely to iterate the discrete Levy transformation and its adjoint [7] to get closed formulae in terms of multiCasorati determinants for all the geometrical features of the transformed lattice. We should mention that in [14] it was obtained, for zero background, multi-Casorati determinant representations for quadrilateral lattices; however, the expressions for the tangent vectors and Lamé coefficients are much more involved than here and no closed expression is given for the points of the lattice. Notice that for the Hirota equation (discrete KP) Casorati determinant representations can be found in [15]. The layout of this letter is as follows. In the first section we remind the reader some basic facts of the quadrilateral lattices and the discrete Levy tarnsformation. Next, in §3 we give the main result of this letter that is extended to the adjoint case in §4. In §5 we briefly indicate how to write our formulae, expressions depending on discrete multi-Wroński determinants, in terms of multi-Casorati determinants. Finally, in §6 we analyze a simple example by applying our main result to the Cartesian lattice [14]. We conlude the letter with an Appendix containg the proof of a lemma in §3. 2. A Multidimensional Quadrilateral Lattice (MQL) [6] is anN -dimensional 2 (N ≥ 2) lattice: x : Z → R, M ≥ N, n := (n1, . . . , nN) ∈ Z N , D ≥ N such that each elementary quadrilateral of it is planar. It can be shown that this condition can be rewritten as the following discrete Laplace equation ∆i∆jx = (TiAij)∆ix+ (TjAji)∆jx , i 6= j, i, j = 1, . . . , N, (1) where x ∈ R is an arbitrary point of the lattice, Aij are N(N − 1) real functions of n, Tj is the translation operator in the j-th variable: Tj(f(n1, . . . , nj , . . . , nN )) = f(n1, . . . , nj + 1, . . . , nN) and ∆j = Tj − 1 is the corresponding difference operator. The following nonlinear constraints must hold as compatibility conditions in order to have planarity in each pair of directions: ∆kAij = (TjAjk)Aij + (TkAkj)Aik − (TkAij)Aik, i 6= j 6= k 6= i, (2) which characterize completely all the MQL’s. Equations (1) can be written as first order systems [6], for this we introduce functions Hi, i = 1, . . . , N , defined by ∆jHi = AijHi. (3) Then (1) reads ∆jX i = (TjQij)Xj , i 6= j, (4) where the scalar functions Qij and the D-dimensional vectors X i are defined by the equations ∆iHj = QijTiHi, i 6= j, (5) ∆ix = (TiHi)X i, (6) whose compatibility gives the equations ∆jQik = (TjQijk)Qjk, i 6= j 6= k 6= i. (7) Equations (2) (or (7)) are the multidimensional quadrilateral lattice equations. 3 Given a solution ξj of ∆kξj = (TkQjk)ξk, for each of the N possible directions of the lattice there is a corresponding discrete Levy transformation that reads for the i-th case: x[1] = x− Ω(ξ,H) ξi X i,