Self-similar wave breaking in dispersive Korteweg-de Vries hydrodynamics

TL;DR

This paper analyzes the self-similar formation of dispersive shock waves during wave breaking in Korteweg-de Vries hydrodynamics, providing explicit solutions for power-law initial profiles.

Contribution

It derives closed-form self-similar solutions of the Whitham modulation equations for wave breaking with power-law profiles in KdV dynamics.

Findings

Dispersive shock waves form at different edges depending on pulse sign.

Self-similar solutions are obtained for arbitrary power-law profiles.

The dynamics are governed by the Whitham modulation equations.

Abstract

We discuss the problem of breaking of a nonlinear wave in the process of its propagation into a medium at rest. It is supposed that the profile of the wave is described at the breaking moment by the function $(-x)^{1/n}$ ($x<0$, positive pulse) or $-x^{1/n}$ ($x>0$, negative pulse) of the coordinate $x$. Evolution of the wave is governed by the Korteweg-de Vries equation resulting in formation of a dispersive shock wave. In the positive pulse case, the dispersive shock wave forms at the leading edge of the wave structure, and in the negative pulse case at its rear edge. The dynamics of dispersive shock waves is described by the Whitham modulation equations. For power law initial profiles, this dynamics is self-similar and the solution of the Whitham equations is obtained in a closed form for arbitrary $n>1$.