Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system --

TL;DR

This paper rigorously validates the use of Whitham's modulation equations for dissipative systems with conservation laws, specifically in the context of a generalized Ginzburg-Landau system, through detailed error estimates.

Contribution

It provides the first rigorous error estimates confirming the validity of Whitham's modulation equations for a dissipative Ginzburg-Landau system with conservation law.

Findings

Error estimates show close approximation of true solutions by modulation equations

Analytic smoothing and Fourier analysis techniques are effective in the proof

Validation extends to systems with dissipation and conservation laws

Abstract

It is well-established that Whitham's modulation equations approximate the dynamics of slowly varying periodic wave trains in dispersive systems. We are interested in its validity in dissipative systems with a conservation law. The prototype example for such a system is the generalized Ginzburg-Landau system that arises as a universal amplitude system for the description of a Turing-Hopf bifurcation in spatially extended pattern-forming systems with neutrally stable long modes. In this paper we prove rigorous error estimates between the approximation obtained through Whitham's modulation equations and true solutions to this Ginzburg-Landau system. Our proof relies on analytic smoothing, Cauchy-Kovalevskaya theory, energy estimates in Gevrey spaces, and a local decomposition in Fourier space, which separates center from stable modes and uncovers a (semi)derivative in front of the relevant nonlinear terms.