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Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations

Turaev, D and Zelik, S (2010) Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations (Unpublished)

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Abstract

We prove that the attractor of the 1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positive space-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillations in grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic proof for the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closed to a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring ones for all time. The temporal evolution of the well-separated multi-soliton structures is described by a weakly coupled lattice dynamical system (LDS) for the coordinates and phases of the solitons. We develop a version of normal hyperbolicity theory for the weakly coupled LDSs with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory may be of independent interest, the main difficulty addressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back to the initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatial or temporal modulation which could prevent the individual solitons in the grid from drifting towards each other and destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-time chaotic solutions for which the random soliton drift is arrested is large enough, so the corresponding space-time entropy is strictly positive.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Turaev, DUNSPECIFIEDUNSPECIFIED
Zelik, SUNSPECIFIEDUNSPECIFIED
Date : 17 February 2010
Related URLs :
Additional Information : This is an arXiv version of this paper. The published version can be found at http://dx.doi.org/10.3934/dcds.2010.28.1713 © 2010 Published by AIMS. All rights reserved.
Depositing User : Symplectic Elements
Date Deposited : 20 Jan 2015 12:32
Last Modified : 10 Apr 2015 13:33
URI: http://epubs.surrey.ac.uk/id/eprint/806858

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