Overtaking solitary wave collisions for Whitham–Boussinesq systems

Abstract

This study focuses on solitary waves and their pairwise interactions within two fully dispersive and weakly nonlinear models known as Whitham–Boussinesq systems. Solitary waves are numerically computed using an iterative Newton-type method, incorporating continuation in wave amplitude and speed. These computed solitary waves are then used as initial data to study overtaking collisions in both systems. Our findings show that both systems satisfy the geometric Lax-categorisation for two-soliton collisions. Additionally, numerical evidence suggests that one of the systems admits an algebraic Lax-categorisation, though within a different range than that originally demonstrated by Lax for the Korteweg–de Vries equation. However, this algebraic categorisation does not apply for the second system. Additionally, qualitative and numerical analyses of solitary waves governed by each Whitham–Boussinesq system, including their amplitude-velocity relations, are presented and compared using two independent approaches.