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Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation. (English) Zbl 0900.35345
Summary: 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 Bäcklund transformations. As a main example the so-called Camassa-Holm equations chosen for which the relevant results are obtained by having a look on the Korteweg de Vries (KdV) equation. The Camassa-Holm equations 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 Bäcklund 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.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35-02Research monographs (partial differential equations)