Spectrum transformation and conservation laws of the lattice potential KdV equation

TL;DR

This paper derives infinitely many new conserved quantities for the lattice potential KdV equation by analyzing scattering data and Jost solutions, revealing insights into its integrable structure with nonzero backgrounds.

Contribution

It introduces a novel method to obtain conserved densities for the lattice potential KdV equation using a discrete Riccati equation and conformal mapping, which differ from previous results.

Findings

Derived infinitely many conserved quantities

Conserved densities asymptotic to zero at infinity

New approach using conformal map and Riccati 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 Schr\"odinger 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.