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December  2010, 28(4): 1713-1751. doi: 10.3934/dcds.2010.28.1713

Analytical proof of space-time chaos in Ginzburg-Landau equations

Dmitry Turaev 1, and  Sergey Zelik 2,

1. 

Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
2. 

Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  October 2009 Revised  February 2010 Published  June 2010

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 LDS's with continuous time and establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos in discrete-time LDS's. 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.
Keywords: extended systems, attractors of PDE's in unbounded domains, soliton interaction, normal hyperbolicity, center-manifold reduction, multi-pulse solutions, lattice dynamical systems..
Mathematics Subject Classification: 35Q30, 37L3.
Citation: Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
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