Solutions to the Kaup--Broer System and Its 2+1 Dimensional Integrable Generalization via the Dressing Method

TL;DR

This paper develops a dressing method for solving the Kaup--Broer system and its 2+1 dimensional generalization, enabling explicit construction of soliton and primitive solutions, including numerical counter-propagating shockwaves.

Contribution

It introduces a nonlocal dbar problem dressing method for all Kaup--Broer system classes and their 2+1 dimensional extension, providing a unified solution framework.

Findings

Computed N-soliton solutions explicitly.

Derived dressing functions for finite gap solutions.

Numerically simulated counter-propagating dispersive shockwaves.

Abstract

In this paper we formulate the nonlocal dbar problem dressing method of Manakov and Zakharov [28, 29, 27] for the 4 scaling classes of the 1+1 dimensional Kaup--Broer system [7, 13]. The method for the 1+1 dimensional Kaup--Broer systems are reductions of a method for a complex valued 2+1 dimensional completely integrable partial differential equation first introduced in [23]. This method allows computation of solutions to all cases of the Kaup--Broer system. We then consider the case of non-capillary waves with usual gravitational forcing, and use the dressing method to compute N-soliton solutions and more general solutions in the closure of the N-soliton solutions in the topology of uniform convergence in compact sets called primitive solutions. These more general solutions are an analogue of the solutions derived in [11, 30, 31] for the KdV equation. We derive dressing functions for finite gap solutions. We compute counter propagating dispersive shockwave type solutions numerically.