Linear relaxation to planar Travelling Waves in Inertial Confinement Fusion

TL;DR

This paper analyzes the linear stability of planar travelling waves in a reaction-diffusion model derived from Inertial Confinement Fusion, showing exponential convergence to planar solutions and recovering a dispersion relation in certain limits.

Contribution

It provides a rigorous mathematical analysis of wave stability and asymptotic behavior in a model relevant to Inertial Confinement Fusion, connecting heuristic and numerical results.

Findings

Solutions become exponentially planar over time.

Recovered dispersion relation in the limit of small temperature ratio and large wavenumber.

Validated heuristic and numerical dispersion relations through rigorous analysis.

Abstract

We study linear stability of planar travelling waves for a scalar reaction-diffusion equation with non-linear anisotropic diffusion. The mathematical model is derived from the full thermo-hydrodynamical model describing the process of Inertial Confinement Fusion. We show that solutions of the Cauchy problem with physically relevant initial data become planar exponentially fast with rate $s(\eps',k)>0$, where $\eps'=\frac{T_{min}}{T_{max}}\ll 1$ is a small temperature ratio and $k\gg 1$ the transversal wrinkling wavenumber of perturbations. We rigorously recover in some particular limit $(\eps',k)\rightarrow (0,+\infty)$ a dispersion relation $s(\eps',k)\sim \gamma_0 k^{\alpha}$ previously computed heuristically and numerically in some physical models of Inertial Confinement Fusion.