Ginzburg-Landau方程时空混沌的解析证明
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
摘要：
证明了具有破缺相对称性的一维五次复Ginzburg-Landau方程的吸引子对参数值的开集具有严格正的时空熵.通过研究一类Ginzburg-Landau型方程弱相互作用孤子网格中的混沌振荡，得到了该结果。我们提供了一个解析证明的存在性的两个孤子配置的混沌时间行为，并构造的解决方案，这是关闭到一个网格的这种混沌孤子对，在网格中的每一对空间上以及从相邻的所有时间分离。用弱耦合格子动力系统（LDS）描述了孤子坐标和相位的时间演化。我们开发了一个版本的正常双曲性理论的弱耦合LDSs连续时间和建立他们的存在类似的Sinai-Bunimovich混沌离散时间LDSs的时空混沌模式。虽然LDS理论的一部分可能是独立的利益，在该文件中解决的主要困难与解除LDS的时空混沌解决方案回到初始PDE。我们在这里考虑的方程是时空自治的，即我们没有施加空间或时间调制，这可能会阻止网格中的单个孤子在有限时间内相互漂移并破坏分离良好的网格结构。然而，我们设法表明，一组时空混沌解的随机孤子漂移被逮捕是足够大的，所以相应的时空熵是严格正的。
英文摘要：
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.
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