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

Tobias Haas, Björn de Rijk, Guido Schneider

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.

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

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.

Paper Structure

This paper contains 16 sections, 6 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Pattern forming systems with a conservation law
  3. Slow modulations of periodic wave trains
  4. Formal derivation of the WMEs
  5. The functional analytic set-up
  6. Main results and some remarks
  7. Construction of approximate solutions
  8. The structure of the problem
  9. Cauchy-Kovalevskaya theory in Gevrey spaces
  10. Approximate solutions for the perturbed problem
  11. Exact solutions and error estimates
  12. The linear problem
  13. Decomposition in center and stable modes
  14. The equations for the error
  15. The energy inequalities
  16. ...and 1 more sections

Key Result

Theorem 1.2

For $m > 3/2$, $\sigma_{0}, T_0 > 0$ there exists a constant $C_{wh} > 0$ such that the following holds. Let be a solution to the WMEs whi1b with and let $\check{r}(X,T)$ be the corresponding solution to the algebraic equation wyk2. Then, there exist $C,T_1, {\varepsilon}_0 > 0$ such that for all ${\varepsilon} \in (0,{\varepsilon}_0)$, there exists a solution to nea1 with where $V_{app}^{*,\v

Figures (1)

  • Figure 1: Solutions of the linearization about the trivial ground state of translationally invariant pattern-forming systems are proportional to $e^{ikx+\lambda_j(k) t}$. The left panel shows the relevant spectral curves $k \mapsto \lambda_j(k)$ in the stable situation for systems with a conservation law, whereas the right panel depicts the unstable situation.

Theorems & Definitions (15)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Remark 3.1
  • Remark 3.2
  • ...and 5 more