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Analytical proof of space-time chaos in Ginzburg-Landau equations. (English) Zbl 1213.35376
The authors consider the one-dimensional quintic complex Ginzburg–Landau equation with a broken phase symmetry, \[ u_t = (1 + i \beta) u_{xx} - (1 + i \delta) u + (i + \rho) |u|^2 u - (\epsilon_1 + i \epsilon_2) |u|^4 u + \mu, \] where \(\beta\), \(\delta\), \(\rho\), \(\epsilon_1\), \(\epsilon_2\), and \(\mu\) are some real parameters and \(\mu\) is small.
The authors prove that the attractor of this evolutionary equation 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 equations.
The analytic proof is developed for the existence of two-soliton configurations with chaotic temporal behavior. The constructed solutions are closed to a grid of such chaotic soliton pairs. Every pair in the grid is well spatially separated from the neighboring pairs for all times. The temporal evolution of the multi-pulse structures is described by a weakly coupled lattice dynamical system for the coordinates and phases of the solitons.

MSC:
35Q56 Ginzburg-Landau equations
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B41 Attractors
35Q51 Soliton equations
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