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

We prove that the attractor of the1D quintic complex Ginzburg-Landau equation with a broken phase symmetry has strictly positivespace-time entropy for an open set of parameter values. The result is obtained by studying chaotic oscillationsin grids of weakly interacting solitons in a class of Ginzburg-Landau type equations. We provide an analytic prooffor the existence of two-soliton configurations with chaotic temporal behavior, and construct solutions which are closedto a grid of such chaotic soliton pairs, with every pair in the grid well spatially separated from the neighboring onesfor all time. The temporal evolution of the well-separated multi-soliton structures is describedby 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 timeand establish for them the existence of space-time chaotic patterns similar to the Sinai-Bunimovich chaos indiscrete-time LDS's. While the LDS part of the theory may be of independent interest, the main difficultyaddressed in the paper concerns with lifting the space-time chaotic solutions of the LDS back tothe initial PDE. The equations we consider here are space-time autonomous, i.e. we impose no spatialor temporal modulation which could prevent the individual solitons in the grid from drifting towards each otherand destroying the well-separated grid structure in a finite time. We however manage to show that the set of space-timechaotic solutions for which the random soliton drift is arrested is large enough, so the correspondingspace-time entropy is strictly positive.