Integrable Models for Shallow Water with Energy Dependent Spectral Problems

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup-Boussinesq equation. We formulate the inverse problem as a Riemann-Hilbert problem with a Z 2 reduction group. The soliton solutions are explicitly obtained.

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