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

Abstract

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.

Algebro-geometric solutions to the lattice potential modified Kadomtsev--Petviashvili equation

“…Based on a series of work [9,10,12,13,[48][49][50], we have summarized a framework of the approach for constructing algebro-geometric solutions to multidimensionally consistent systems. The approach has proved effective and this paper added one more important successful example.…”

“…The approach has proved effective and this paper added one more important successful example. Reviewing the series of work [9,10,12,13,[48][49][50] and the present paper, there are several related problems that are interesting and remain open. Let us raise them below.…”

“…For fully discrete equations, such as the ABS equations, algebro-geometry solutions have been derived from several equations in a series of papers [9,10,12,13,[48][49][50] by Cao, Xu and their collaborators, which establish an algebro-geometric approach to periodic solutions for discrete equations that are multi-dimensionally consistent. This approach can be sketched as the following three steps.…”

“…Ref. [50] solves H 1 , H 3 (0) and Q 1 (0) and the three associated continuous spectral problems are, respectively,…”

Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps

“…Its lattice version, i.e., the (Q1) 0 lattice, which first appeared in [5], can be used to define a discrete conformal map, by which the Riemann theta function solutions to (Q1) 0 are calculated [11,12]. In the previous papers [13,14], we constructed algebro-geometric solutions of the (Q1) 0 equation, using the method of symplectic maps arising from a nonlinearisation approach [15,16]. The present paper considers the δ-parameter extension of (Q1) 0 , which amounts to a significant departure from the δ = 0 case, since in a sense it 'lifts' the equation away from the KdV related lattice equations, cf e.g.…”