Solitons of shallow-water models from energy-dependent spectral problems

TL;DR

This paper explores soliton solutions of the Kaup-Boussinesq equation through the inverse scattering transform, constructing a Riemann-Hilbert problem for energy-dependent spectral problems to solve this hydrodynamic system.

Contribution

It introduces a novel approach to solving the Kaup-Boussinesq equation by formulating a Riemann-Hilbert problem for energy-dependent spectral problems.

Findings

Successfully constructs the Riemann-Hilbert problem for the system

Derives explicit soliton solutions for the Kaup-Boussinesq equation

Provides a framework for analyzing similar hydrodynamic systems

Abstract

The current work investigates the soliton solutions of the Kaup-Boussinesq equation using the Inverse Scattering Transform method. We outline the construction of the Riemann-Hilbert problem for a pair energy-dependent spectral problems for the system, which we then use to construct the solution of this hydrodynamic system.