A numerical dressing method for the nonlinear superposition of solutions of the KdV equation

TL;DR

This paper unifies two numerical methods to accurately compute complex solutions of the KdV equation, including superpositions of solitons and periodic solutions, for all space and time values.

Contribution

It introduces a combined numerical approach to construct and compute nonlinear superpositions of KdV solutions, integrating initial-value and finite-genus methods.

Findings

Able to compute superposition solutions accurately

Handles asymptotically periodic and soliton solutions

Applicable for all x and t values

Abstract

In this paper we present the unification of two existing numerical methods for the construction of solutions of the Korteweg-de Vries (KdV) equation. The first method is used to solve the Cauchy initial-value problem on the line for rapidly decaying initial data. The second method is used to compute finite-genus solutions of the KdV equation. The combination of these numerical methods allows for the computation of exact solutions that are asymptotically (quasi-)periodic finite-gap solutions and are a nonlinear superposition of dispersive, soliton and (quasi-)periodic solutions in the finite (x,t)-plane. Such solutions are referred to as superposition solutions. We compute these solutions accurately for all values of x and t.