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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 Z2 reduction group. The soliton solutions are explicitly obtained.
R. Ivanov and T. Lyons, Integrable models for shallow water with energy dependent spectral problems, Journal of Nonlinear Mathematical Physics, Vol. 19, Suppl. 1 (2012) 1240008 (17 pages) DOI: 10.1142/S1402925112400086