## Abstract

We prove that the attractor of the ID quintic complex GinzburgLandau 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 GinzburgLandau type equations. We provide an analytic proof for the existence of twosoliton 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.

Original language | English |
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Pages (from-to) | 1713-1751 |

Number of pages | 39 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2010 |

Externally published | Yes |

## Keywords

- Attractors of pde's in unbounded domains
- Center-manifold reduction
- Extended systems
- Lattice dynamical systems
- Multipulse solutions
- Normal hyperbolicity
- Soliton interaction

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics