Integrable models for shallow water with energy dependent spectral problems

Abstract

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependance on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup-Bousinesq equation. We formulate the inverse problem as a Riemann-Hilbert problem with a Z2 reduction group. The soliton solutions are explicitly obtained.

17 pages, 3 figures

References in corpus (3)

• On an integrable two-component Camassa-Holm shallow water system
• Long-Time Asymptotics for the Camassa-Holm Equation
• Two-component {CH} system: Inverse Scattering, Peakons and Geometry

Cited by in corpus (3)

• The Dynamics of Flat Surface Internal Geophysical Waves with Currents
• On invariant analysis and conservation laws for degenerate coupled multi-KdV equations for multiplicity l = 3
• Traveling Wave Solutions of Degenerate Coupled KdV Equation