Multiscale expansion and integrability properties of the lattice potential KdV equation

TL;DR

This paper uses discrete multiscale expansion to analyze the integrability of the lattice potential KdV equation, revealing its connection to the nonlinear Schrödinger hierarchy through Lax pairs and symmetries.

Contribution

It demonstrates that the lattice potential KdV equation's multiscale expansion reproduces the NLS hierarchy, linking discrete integrability structures to continuous models.

Findings

The secularity conditions lead to the nonlinear Schrödinger equation.

The Lax pair corresponds to the Zakharov-Shabat spectral problem.

Symmetries form a hierarchy of point and generalized symmetries.

Abstract

We apply the discrete multiscale expansion to the Lax pair and to the first few symmetries of the lattice potential Korteweg-de Vries equation. From these calculations we show that, like the lowest order secularity conditions give a nonlinear Schroedinger equation, the Lax pair gives at the same order the Zakharov and Shabat spectral problem and the symmetries the hierarchy of point and generalized symmetries of the nonlinear Schroedinger equation.