Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation

Abstract

Abstract The main subject of the paper is to give a survey and to present new methods on how integrability results (i.e. results for symmetry groups, inverse scattering formulations, action-angle transformations and the like) can be transferred from one equation to others in case the equations are NOT related by Backlund transformations. As a main example the so-called Camassa-Holm equation is chosen for which the relevant results are obtained by having a look on the Korteweg de vries (KdV) equation. The Camassa-Holm equation turns out to be a different-factorization equation of the KdV, it describes shallow water waves and reconciles the properties which were known for different orders of shallow water wave approximations. We follow here an old method already marginally mentioned in Fuchssteiner and Fokas (1981), and Fuchssteiner (1983) and recently applied by others (Olver and Rosenau, 1995). The method allows an immediate recovery of the recursion operator for the Camassa-Holm equation from the invariance structure of the KdV, although both equations are not related by Backlund transformations. However, in addition, and different from other approaches, from there by use of the squared eigenfunction relation for the KdV equation, the Lax pair formulation for the different-factorization equation is derived. For the example under consideration it is, of course, the one obtained in Camassa and Holm (1993). Since the methods proposed here can be transferred to any compatible factorization of recursion operators its application, even in the special case which was chosen for illustration, leads to a large class of integrable equations among which the Camassa-Holm equation can be found as well as a three-parameter family of generalizations of the equation. The advantage of the general approach to the Lax pair presentation is that direct transformations between action and angle variables are obtained. So, using this Lax pair formulation, as a novel result, a direct transformation between action- and angle-variables for the Camassa-Holm equation is derived. Further novel results in the paper are: a hodograph link back from a Backlund transformation of the Camassa-Holm equation to a particular member of the KdV-hierarchy, additional symmetries, and the construction of the conformal algebra for the hierarchy of the Camassa-Holm equation. The methods involved include: hereditary operators, bi-hamiltonian formulations, nilpotent flows and scaling symmetries.