The Gardner method for 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.

References in corpus (3)

• Conformal Properties and Baecklund Transform for the Associated Camassa-Holm Equation
• Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method
• Infinitely many conservation laws for the discrete KdV equation

Cited by in corpus (7)

• Bäcklund transformations for the Camassa-Holm equation
• Bäcklund Transformations for the Boussinesq Equation and Merging Solitons
• On symmetries and conservation laws of a Gardner equation involving arbitrary functions
• Four Symmetries of the KdV equation
• Unfamiliar Aspects of Bäcklund Transformations and an Associated Degasperis-Procesi Equation
• Computation of generating symmetries
• Symmetry structure of integrable hyperbolic third order equations