The Gardner method for symmetries

TL;DR

This paper extends the Gardner method, traditionally used for conservation laws, to generate symmetries for integrable equations like KdV, Camassa-Holm, and Sine-Gordon, revealing their hierarchy and relations to other methods.

Contribution

It generalizes the Gardner method to produce symmetries, demonstrating its effectiveness for multiple integrable equations and exploring its connections with Lenard recursion and discrete analogs.

Findings

Generated infinite hierarchies of symmetries for KdV, Camassa-Holm, and Sine-Gordon equations.

Showed that the symmetries obtained commute.

Established relations between the Gardner method, Lenard recursion, and discrete symmetries.

Abstract

The Gardner method, traditionally used to generate conservation laws of integrable equations, is generalized to generate symmetries. The method is demonstrated for the KdV, Camassa-Holm and Sine-Gordon equations. The method involves identifying a symmetry which depends upon a parameter; expansion of this symmetry in a (formal) power series in the parameter then gives the usual infinite hierarchy of symmetries. We show that the obtained symmetries commute, discuss the relation of the Gardner method with Lenard recursion (both for generating symmetries and conservation laws), and also the connection between the symmetries of continuous integrable equations and their discrete analogs.