Symmetry structure of integrable hyperbolic third order equations

Abstract

We explore the application of generating symmetries, i.e. symmetries that depend on a parameter, to integrable hyperbolic third order equations, and in particular to consistent pairs of such equations as introduced by Adler and Shabat (AS). Our main result is that different infinite hierarchies of symmetries for these equations can arise from a single generating symmetry by expansion about different values of the parameter. We illustrate this, and study in depth the symmetry structure, for two examples. The first is an equation related to the potential KdV equation taken from AS. The second is a more general hyperbolic equation than the kind considered in AS. Both equations depend on a parameter, and when this parameter vanishes they become part of a consistent pair. When this happens, the nature of the expansions of the generating symmetries needed to derive the hierarchies also changes.

arXiv comment21 pages

References in corpus (12)

• The Camassa-Holm Equation: A Loop Group Approach
• Bäcklund transformations for the Camassa-Holm equation
• Bäcklund Transformations for the Boussinesq Equation and Merging Solitons
• The Gardner method for symmetries
• Towards the theory of integrable hyperbolic equations of third order
• The Klein-Gordon Equation and Differential Substitutions of the Form $v=\phi (u,u_x,u_y)$
• Unfamiliar Aspects of Bäcklund Transformations and an Associated Degasperis-Procesi Equation
• Vector hyperbolic equations on the sphere possessing integrable third order symmetries
• Four Symmetries of the KdV equation
• Computation of generating symmetries
• On steady motions of an ideal fibre-reinforced fluid in a curved stratum. Geometry and integrability
• Formal Integrals and Noether Operators of Nonlinear Hyperbolic Partial Differential Systems Admitting a Rich Set of Symmetries