Investigation of Chaotic Phenomena by a Soliton Lattice Model

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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