Investigation of Chaotic Phenomena by a Soliton Lattice Model

本研究的目的是通过空间局域结构的叠加来描述某些非线性演化方程中出现的各种“孤子”或“混沌”现象。提出了一种孤立子格子模型，该模型近似于具有不稳定性、耗散和色散的演化方程的解，用于研究混沌行为的性质。这个想法被广泛应用于几个耗散色散方程在一个或两个空间维度，也描述了强混沌。主要研究结果如下：1.讨论了对称和非对称振荡孤子格子模型下色散和耗散系统中的混沌行为。研究发现，晶格中的非对称性与非保守性有关，并且超过3个脉冲与振荡尾结构的相互作用会引入运动的不规则性.不稳定KdV-Burgers方程的解具有混沌行为 ...更多信息 对于强色散情况。定常激波结构的发展与该方程的色散关系相一致的强耗散的情况下。从理论上用精确包络孤子解的叠加描述了Ginzburg-Landau方程中的混沌谱，这对数值计算得到的混沌谱提供了满意的最低近似.数值求解二维耗散-色散方程，发现一维解一般是不稳定的，并产生二维局域结构。在弱色散情况下，整体行为是混沌的，而在强色散情况下，二维局域脉冲结构发展并松弛成二维空间中的“准晶格”排列.数值计算发现，圆柱对称孤立波成为Zakharov-Kuznetsov方程的基础。这种孤立波在孤立时稳定传播，但在碰撞情况下表现为“准孤立子”，因为在碰撞过程中发生振幅变化和涟漪的产生。少

The purpose of this research is to describe a variety of "soliton" or "chaos" phenomena arising in certain nonlinear evolution equations by means of a superposition of spatially localized structures. A soliton lattice model which approximates solutions of an evolution equation with instability, dissipation and dispersion is proposed to investigate the properties of chaotic behaviors. The idea is applied extensively to several dissipative-dispersive equations in one- or two-space dimension and also to descriptions of strong chaos. The obtained results are summarized as follows.1. Chaotic behaviors in dispersive or dissipative systems are discussed in relation to symmetric or asymmetric oscillatory soliton lattice model. The finding is that an asymmetry in lattice is associated with non-conservative property and interactions of more than 3 pulses with oscillatory tail structure introduce irregularities in the motion.2. Solutions of the unstable KdV-Burgers equation show chaotic behaviors … More for strongly dispersive case. Stationary shock structures develop for strongly dissipative case in consistent with the dispersion relation for this equation. Spectra of chaos in the Ginzburg-Landau equation are described theoretically by a superposition of exact envelope soliton solutions, which provides a satisfactory lowest approximation to numerically obtained spectra.3. Two-dimensional dissipative-dispersive equation is solved numerically to show that one-dimensional solutions are generally unstable and two-dimensionally localized structures are generated. For weakly dispersive case, overall behaviors are chaotic, while, for strongly dispersive case, two-dimensionally localized pulse structures develop and they relax into a "quasi-lattice" arrangement in two-dimensional space.4. It is found numerically that cylindrically symmetric solitary waves become fundamental to the Zakharov-Kuznetsov equation. This solitary wave travels stably when isolated, but behaves as "quasi-soliton" in case of collision, because changes in amplitude and generations of ripple take place during collisions. Less