Solitary Wave Interactions in the Cubic Whitham Equation

Abstract

The vortical Whitham equation is modeled with quadratic and cubic nonlinearity, satisfying the unidirectional dispersion relation used to describe the propagation of nonlinear waves in the presence of a vertically sheared current of constant vorticity. In this article, we neglect the quadratic nonlinearity to numerically investigate solitary wave interactions. We show that the geometric Lax categorization is satisfied; however, an algebraic categorization based on the ratio of the initial solitary wave amplitudes is not possible. Specifically, our numerical simulations indicate that for solitary waves with large amplitudes, the interactions maintain two well-separated crests. Additionally, for solitary waves of different polarities, we find that wave-breaking may occur.