Solitary wave formation in the compressible Euler equations

TL;DR

This paper analyzes how perturbations evolve in a 1D compressible Euler gas, showing they form solitary waves and that shock formation depends on initial data amplitude, supported by asymptotic analysis and computational experiments.

Contribution

It introduces a dispersive nonlinear wave model for perturbations in the 1D Euler equations and validates the existence of large-time non-breaking solutions.

Findings

Solutions mainly consist of solitary waves

Shock formation occurs for larger initial data

Threshold for transition between behaviors is validated

Abstract

We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength perturbations approximately obey a system of dispersive nonlinear wave equations. Computational experiments demonstrate that solutions of the 1D Euler equations agree well with this dispersive model, with solutions consisting mainly of solitary waves. Shock formation seems to be avoided for moderate-amplitude initial data, while shock formation occurs for larger initial data. We investigate the threshold for transition between these behaviors, validating a previously-proposed criterion based on further computational experiments. These results support the existence of large-time non-breaking solutions to the 1D compressible Euler equations, as hypothesized in previous works.