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On Kupershmidt’s extended equation of dispersive water waves. (English) Zbl 1412.35074

Summary: An extended equation of dispersive water waves, proposed by Kupershmidt, is considered. By an ansatz on eigenfunctions of its recursion operator, a linear spectral problem, which turns to be type of the energy dependent Schrödinger, is constructed for the system. As by-products, modified systems are presented. Furthermore, a new bi-Hamiltonian system is obtained and shown to be related to a three component generalization of the Camassa-Holm equation under a Miura-type transformation.

MSC:

35G55 Initial value problems for systems of nonlinear higher-order PDEs
35Q35 PDEs in connection with fluid mechanics
35P05 General topics in linear spectral theory for PDEs
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