Reseach on a nonlinear evolution equation including chaos and soliton.

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Initial value problems of a nonlinear evolution equation involving instability, dissipation and dispersion are investigated numerically and analytically. Coherent (equilibrium) solutions consisting of a sequence of solitons with the same amplitude or chaotic solutions showing irreqular fluctuations of localized pulse-like structures are found respectively for strongly or weakly dispersive cases. It is shown that the behaviours of these solutions in an infinite dimensional system can be described systematically by the interactions of localized soliton-like pulses, i.e., by a system with a few degrees of freedom. Main results obtained are as follows.1. The spatio-temporal evolutions of solutions ranging from coherent to chaotic stages can be qualitatively well described by weak interactions of pulses each of which is the steady solution to the original evolution equation. The oscillatory structure of a tail of the pulse for weakly dispersive cases is responsible for the existence of boun … More d state of pulses, which explains the numerical result that the inter-pulse distances in the initial value problem take certain fixed values in the definite regions. In cases of monotone tails for strongly dispersive cases, effects of pulse interactions become repulsive, which explains the result that the pulses asymptotically tend to be arranged periodically adjusting to the periodic boundary conditions in the numerical simulation.2. Dynamical behaviours of a sequence of asymmetric soliton-like pulses are investigated in terms of a nonlinear lattice model. In a three pulse periodic system, it is found that either periodic or chaotic motions occur when the individual interacting pulse has an asymmetric oscillatory tain structure.3. Several general properties of the asymmetric lattice are also discussed in comparison with the symmetric lattices for the Korteweg-de Vries and the fifth order KdV solitons. It is shown that oscillatory structures introduce a possibility of chaotic motions and asymmetry in the lattice forces introduces non-conservative properties like instability and dissipation. Less