Spectrum transformation and conservation laws of lattice potential KdV equation

Abstract

Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this letter we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are different from those in the known literatures. They are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.

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

“…Many of the infinite‐dimensional discrete integrable models that are supported by (in the sense that they can be reduced to) integrable symplectic maps have interesting properties: the existence of Lax pairs, Bäcklund transformations, symmetries and conservation laws, Hamiltonian structures, the construction of integrable algorithms, (elliptic) soliton solutions, finite genus solutions [see Refs. 10–23, and references therein].…”