Infinitely many conservation laws for the discrete KdV equation

TL;DR

This paper introduces a new method using the discrete Gardner transformation to generate infinitely many conservation laws for the discrete KdV equation and proves their nontriviality and distinction from previous methods.

Contribution

It presents an alternative construction of conservation laws via the Gardner transformation and confirms their nontriviality, clarifying their relationship with earlier conservation laws.

Findings

New conservation laws constructed using Gardner transformation

Confirmed the conservation laws are distinct and nontrivial

Unified understanding through continuum limit analysis

Abstract

In \cite{RH3} Rasin and Hydon suggested a way to construct an infinite number of conservation laws for the discrete KdV equation (dKdV), by repeated application of a certain symmetry to a known conservation law. It was not decided, however, whether the resulting conservation laws were distinct and nontrivial. In this paper we obtain the following results: (1) We give an alternative method to construct an infinite number of conservation laws using a discrete version of the Gardner transformation. (2) We give a direct proof that the Rasin-Hydon conservation laws are indeed distinct and nontrivial. (3) We consider a continuum limit in which the dKdV equation becomes a first-order eikonal equation. In this limit the two sets of conservation laws become the same, and are evidently distinct and nontrivial. This proves the nontriviality of the conservation laws constructed by the Gardner method, and gives an alternate proof of the nontriviality of the conservation laws constructed by the Rasin-Hydon method.