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Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto-Sivashinsky Phase Dynamics

Baalen, Guillaume van

In: Communications in Mathematical Physics, 2004, vol. 247, no. 3, p. 613-654

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    Title
    • Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto-Sivashinsky Phase Dynamics
    Author
    • Baalen, Guillaume van. Département de Physique Théorique, Université de Genève, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Communications in Mathematical Physics, 2004, vol. 247, no. 3, p. 613-654. Springer-Verlag
    Other Version
    Classification
    • Mathematics
    OAI-PMH Identifier
    • oai:doc.rero.ch:318023
    Summary
    Abstract:: We study the Complex Ginzburg-Landau initial value problem for a complex field u ∈ C, with α,β∈R. We consider the Benjamin-Feir linear instability region We show that for all and for all initial data u 0 sufficiently close to 1 (up to a global phase factor e iφ 0 ,φ0∈R) in the appropriate space, there exists a unique (spatially) periodic solution of space period L 0 . These solutions are small even perturbations of the traveling wave solution, and s,η have bounded norms in various L p and Sobolev spaces. We prove that apart from corrections whenever the initial data satisfy this condition, and that in the linear instability range the dynamics is essentially determined by the motion of the phase alone, and so exhibits ‘phase turbulence'. Indeed, we prove that the phase η satisfies the Kuramoto-Sivashinsky equation for times while the amplitude 1+α2 s is essentially constant