Solitary wave collisions for Whitham-Boussinesq systems

TL;DR

This paper investigates solitary wave solutions and their collisions in two Whitham-Boussinesq models, revealing different Lax-categorization behaviors and providing high-accuracy numerical methods for large amplitude waves.

Contribution

It introduces high-accuracy numerical methods for solitary waves in Whitham-Boussinesq systems and analyzes their collision behaviors and Lax-categorization properties.

Findings

Both systems satisfy geometric Lax-categorization of two-soliton collisions.

One system admits an algebraic Lax-categorization based on amplitude ratios.

The second system does not exhibit algebraic Lax-categorization.

Abstract

This work concerns soliton-type numerical solutions for two Whitham-Boussinesq-type models. Solitary waves are computed using an iterative Newton-type and continuation methods with high accuracy. The method allow us to compute solitary waves with large amplitude and speed close to the singular limit. These solitary waves are set as initial data and overtaking collisions are considered for both systems. We show that both system satisfy the geometric Lax-categorization of two-soliton collision. Numerical evidences indicate that one of the systems also admits an algebraic Lax-categorization based on the ratio of the initial solitary wave amplitudes with a different range from the one predicted by Lax. However, we show that such categorization is not possible for the second system.