A model for shock wave chaos

TL;DR

This paper introduces a new mathematical model for shock wave chaos that captures key features of detonations, including instability and chaotic behavior, aligning with complex reactive Euler equation solutions.

Contribution

The paper presents a simplified model equation that accurately reproduces chaotic shock wave phenomena observed in detonations, offering a new tool for analysis.

Findings

Model predicts chaotic shock waves similar to detonations

Reproduces bifurcation and chaos onset

Captures shock formation in reaction zones

Abstract

We propose the following model equation: \[u_{t}+1/2(u^{2}-uu_{s})_{x}=f(x,u_{s}), \] that predicts chaotic shock waves. It is given on the half-line $x<0$ and the shock is located at $x=0$ for any $t\ge0$. Here $u_{s}(t)$ is the shock state and the source term $f$ is assumed to satisfy certain integrability constraints as explained in the main text. We demonstrate that this simple equation reproduces many of the properties of detonations in gaseous mixtures, which one finds by solving the reactive Euler equations: existence of steady traveling-wave solutions and their instability, a cascade of period-doubling bifurcations, onset of chaos, and shock formation in the reaction zone.