Radiating solitary waves in coupled Boussinesq equations

TL;DR

This paper analyzes radiating solitary wave solutions in coupled Boussinesq equations, deriving asymptotic solutions with oscillatory tails and validating estimates through numerical simulations.

Contribution

It provides an analytical construction of radiating solitary waves with oscillatory tails in coupled Boussinesq equations, including explicit tail amplitude estimates.

Findings

Analytical solutions for radiating solitary waves with tails

Explicit estimates for tail amplitudes

Validation through numerical simulations

Abstract

In this paper we are consider radiating solitary wave solutions of coupled regularised Boussinesq equations. This type of solution consists of a leading solitary wave with a small-amplitude co-propagating oscillatory tail, and emerges from a pure solitary wave solution of a symmetric reduction of the full system. We construct an asymptotic solution, where the leading order approximation in both components is obtained as a particular solution of the regularised Boussinesq equations in the symmetric case. At the next order, the system uncouples into two linear non-homogeneous ordinary differential equations with variable coefficients, one correcting the localised part of the solution, which we find analytically, and the other describing the co-propagating oscillatory tail. This latter equation is a fourth-order ordinary differential equation and is solved approximately by two different methods, each exploiting the assumption that the leading solitary wave has a small amplitude, and thus enabling an explicit estimate for the amplitude of the oscillating tail. These estimates are compared with corresponding numerical simulations.