Modulation theory of soliton-mean flow in KdV equation with box type initial data

TL;DR

This paper develops a modulation theory for soliton-mean flow interactions in the KdV equation with box initial data, combining theoretical derivations with numerical verification to explain complex wave phenomena.

Contribution

It introduces a novel soliton modulation system derived from Whitham theory to analyze soliton interactions with large-scale flows in the KdV equation.

Findings

Explicit equations for soliton trajectory and amplitude changes.

Verification of theoretical predictions through numerical simulations.

Identification of exotic interaction phenomena with applications in fluid dynamics.

Abstract

For the KdV equation with box type initial data, the interaction between a trial soliton and large-scale dispersive mean flow is studied theoretically and numerically. The pure box initial value can cause rarefaction wave and dispersive shock wave, and can create an area of soliton train. The key to the interaction of soliton and mean flow is that the dynamic evolutions of the mean flow and the local soliton can be described by the same modulation system. The soliton modulation system is derived from the degenerations of the two-genus Whitham modulation system. Considering the influence of rarefaction wave, dispersive shock wave and soliton train on the trial soliton, in the framework of Whitham modulation theory, the equation describing the soliton trajectory and the changes in amplitude and phase shift are given explicitly. The predicted results are compared with the numerical simulations, which verifies the corrections of the theoretical analysis. The exotic interaction phenomena between soliton and mean flow found in this work have broad applications to shallow water soliton propagations and real soliton experiments in fluid dynamics.