Amplitude Decay of Solitary Waves - asymptotic and numerical results

TL;DR

This paper investigates the decay behavior of solitary waves in perturbed Korteweg-de Vries equations, combining asymptotic analysis with numerical simulations to understand tail formation and validate theoretical predictions.

Contribution

It provides a detailed asymptotic analysis of amplitude decay in perturbed solitary waves and confirms findings through numerical simulations, focusing on the Burgers-Korteweg-de Vries equation.

Findings

Slowly decaying tail behind soliton observed

Explicit core solution derived

Asymptotic results validated by numerical simulations

Abstract

The relevance of perturbed forms of the Korteweg-de Vries equation to a range of physical problems is discussed. Solutions which are perturbations of solitary travelling wave solutions are then considered, focussing predominantly on the Burgers-Korteweg-de Vries equation. Asymptotic analysis demonstrates the appearance of a slowly decaying tail behind a core soliton-like solution. The solution in the tail region is determined in the form of a convolution integral involving the Airy function, while the core solution is obtained explicitly. Asymptotic results are fully validated by comparison with numerical results, obtained using a pseudospectral scheme.