Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results

TL;DR

This paper numerically investigates a class of KdV solutions that combine localized solitons with (quasi)periodic backgrounds, revealing complex dispersive and soliton interactions relevant for physical applications.

Contribution

It provides the first computational analysis of finite-genus solutions with soliton perturbations, bridging the gap between analytical theory and numerical simulation.

Findings

Demonstrates coexistence of solitons and (quasi)periodic backgrounds in KdV solutions.

Shows dispersive tail behaviors and soliton interactions through numerical simulations.

Highlights relevance to physical phenomena like localized sea swell perturbations.

Abstract

All solutions of the Korteweg -- de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi)periodic. In this paper, we discuss a class of solutions that is a nonlinear superposition of these two cases: their asymptotic state for large $|x|$ is (quasi)periodic, but they may contain solitons, with or without dispersive tails. Such scenarios might occur in the case of localized perturbations of previously present sea swell, for instance. Such solutions have been discussed from an analytical point of view only recently. We numerically demonstrate different features of these solutions.