Integrable models for shallow water with energy dependent spectral problems

TL;DR

This paper investigates inverse spectral problems for energy-dependent operators, deriving integrable equations like Kaup-Boussinesq, and explicitly constructs soliton solutions using Riemann-Hilbert methods.

Contribution

It introduces a novel inverse problem framework for energy-dependent spectral operators and explicitly solves for soliton solutions within this context.

Findings

Formulated inverse spectral problem as a Riemann-Hilbert problem with Z2 reduction.

Derived integrable equations including the Kaup-Boussinesq equation.

Explicitly obtained soliton solutions for the studied operators.

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.