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.

Discrete Hirota reductions associated with the lattice KdV equation

“…preserves the spectral curve det(M(λ) − ν 1) = 0 (28) for the monodromy matrix M(λ) = M 0 obtained by setting m = 0 in (26), namely…”

“…It is known that the algebro-geometric solutions of the discrete Hirota equation are given in terms of the Fay trisecant identity for an arbitrary algebraic curve [26]. It would be interesting to use the above spectral coordinates on the hyperelliptic curves (28) to derive explicit for-mulae for the solutions of the iterated maps corresponding to the lattice KdV travelling wave reductions, as has been done for solutions of the discrete potential KdV equation in [28].…”

Discrete Hirota reductions associated with the lattice KdV equation

“…It is known that the algebro-geometric solutions of the discrete Hirota equation are given in terms of the Fay trisecant identity for an arbitrary algebraic curve [26]. It would be interesting to use the above spectral coordinates on the hyperelliptic curves (28) to derive explicit formulae for the solutions of the iterated maps corresponding to the lattice KdV travelling wave reductions, as has been done for solutions of the discrete potential KdV equation in [28].…”

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

“…For δ = 0 this equation first appeared in the classification of [1], but the case δ = 0 (which we will denote by (Q1) 0 ) first appeared in [2] where, due to the appearance of the canonical cross ratio of four variables, it was identified as a lattice version of the Schwarzian Korteweg-de Vries (KdV) equation. In the previous papers, [3,4], we constructed algebro-geometric solutions of the (Q1) 0 equation, using the method of symplectic maps arising from a nonlinearisation approach [5,6]. The present paper considers the δ-parameter extension of that equation, 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.…”