Integrable models for shallow water with energy dependent spectral problems

Rossen Ivanov, Tony Lyons

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)


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 12 reduction group. The soliton solutions are explicitly obtained.

Original languageEnglish
Article number1240008
JournalJournal of Nonlinear Mathematical Physics
Issue numberSUPPL. 1
Publication statusPublished - Oct 2012
Externally publishedYes


  • Inverse scattering method
  • nonlinear evolution equations
  • solitons


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