Solitons on the rarefactive wave background via the Darboux transformation

TL;DR

This paper investigates how solitons interact with rarefactive waves in the KdV equation, demonstrating that transmitted solitons can be derived via Darboux transformation while trapped solitons do not persist long-term.

Contribution

It introduces a method to obtain transmitted solitons using Darboux transformation and shows trapped solitons vanish over time through numerical simulations.

Findings

Transmitted solitons can be derived using Darboux transformation.

Trapped solitons do not persist in long-term dynamics.

Numerical simulations confirm the disappearance of trapped solitons.

Abstract

Rarefactive waves and dispersive shock waves are generated from the step-like initial data in many nonlinear evolution equations including the classical example of the Korteweg-de Vries (KdV) equation. When a solitary wave is injected on the step-like initial data, it is either transmitted over the background or trapped in the rarefactive wave. We show that the transmitted soliton can be obtained by using the Darboux transformation for the KdV equation. On the other hand, no trapped soliton can be obtained by using the Darboux transformation and we show with numerical simulations that the trapped soliton disappears in the long-time dynamics of the rarefactive wave.