Long-time evolution of pulses in the Korteweg-de Vries equation in the absence of solitons revisited: Whitham method

TL;DR

This paper uses Whitham modulation theory to analytically describe the long-time evolution of pulses in the Korteweg-de Vries equation without solitons, focusing on dispersive shock and rarefaction waves.

Contribution

It provides a novel analytical approach to describe the structure and self-similar behavior of dispersive shock waves in the KdV equation.

Findings

Analytic description of dispersive shock wave structure

Comparison with numerical simulations confirms the theory

Identification of self-similar behavior near the shock edge

Abstract

We consider the long-time evolution of pulses in the Korteweg-de Vries equation theory for initial distributions which produce no soliton, but instead lead to the formation of a dispersive shock wave and of a rarefaction wave. An approach based on Whitham modulation theory makes it possible to obtain an analytic description of the structure and to describe its self-similar behavior near the soliton edge of the shock. The results are compared with numerical simulations.