# Nonlinear Instability of Periodic Traveling Waves

## TL;DR

This paper investigates the nonlinear instability of zero solutions in reaction-diffusion systems with periodic coefficients, using spectral analysis and the Bloch transform to identify conditions leading to instability.

## Contribution

It characterizes nonlinear instability for periodic reaction-diffusion systems with essential spectrum instability, extending results to dissipative conservation law systems.

## Key findings

- Identifies conditions for nonlinear instability based on spectral properties.
- Uses Bloch transform to analyze the growth of perturbations.
- Extends instability analysis to dissipative conservation laws.

## Abstract

We study the local dynamics of $L^{2}\left(\mathbb{R}\right)$-perturbations to the zero solution of spatially $2\pi$-periodic coefficient reaction-diffusion systems. In this case the spectrum of the linearization about the zero solution is purely essential and may be described via the point spectrum of a one-parameter family of Bloch operators. When this essential spectrum is unstable, we characterize a large class of initial perturbations which lead to nonlinear instability of the trivial solution. This is accomplished by using the Bloch transform to construct an appropriate projection to capture the maximum amount of linear exponential growth associated to the initial perturbation arising from the unstable eigenvalues of the Bloch operators. This result is also extended to dissipative systems of conservation laws.