Note on the single-shock solutions of the Korteweg-de Vries-Burgers equation

TL;DR

This paper revisits the classic shock solutions of the Korteweg-de Vries-Burgers equation, emphasizing their monotonic and unique nature in plasma physics, and contrasts them with solitary wave solutions.

Contribution

It clarifies the properties of shock solutions, highlighting their monotonicity and uniqueness, which are often overlooked in recent literature.

Findings

Shock solutions are monotonic and unique for a given plasma configuration.

These shocks cannot exhibit oscillatory or bell-shaped features.

Contrast with solitary wave solutions of the parent equations.

Abstract

The well-known shock solutions of the Korteweg-de Vries-Burgers equation are revisited, together with their limitations in the context of plasma (astro)physical applications. Although available in the literature for a long time, it seems to have been forgotten in recent papers that such shocks are monotonic and unique, for a given plasma configuration, and cannot show oscillatory or bell-shaped features. This uniqueness is contrasted to solitary wave solutions of the two parent equations (Korteweg-de Vries and Burgers), which form a family of curves parameterized by the excess velocity over the linear phase speed.