Reseach on a nonlinear evolution equation including chaos and soliton.

研究了一类具有不稳定性、耗散性和色散性的非线性发展方程的初值问题。分别在强色散和弱色散情况下，发现了由一系列具有相同振幅的孤子组成的相干（平衡）解或表现出局域脉冲状结构的不规则涨落的混沌解.结果表明，在无限维系统中，这些解的行为可以用局域类孤子脉冲的相互作用系统地描述，即，一个只有几个自由度的系统。主要研究结果如下：1.从相干到混沌阶段的时空演化的解决方案，可以定性地很好地描述了弱相互作用的脉冲，其中每个是稳定的解决方案，以原来的发展方程。弱色散情况下脉冲尾部的振荡结构是boun存在的原因 ...更多信息 d状态的脉冲间距离，从而解释了初值问题中脉冲间距离在一定区域内取一定定值的数值结果。在强色散的单调尾情形下，脉冲相互作用的影响变为排斥，这解释了数值模拟中脉冲渐近地趋向于周期性排列以适应周期边界条件的结果.利用非线性晶格模型研究了非对称类孤子脉冲序列的动力学行为。在三脉冲周期系统中，发现当相互作用的脉冲具有非对称的振荡结构时，系统会发生周期运动或混沌运动.通过与对称格点的比较，讨论了非对称格点的几个一般性质，并对五阶KdV孤子和Korteweg-de弗里斯孤子进行了讨论。结果表明，振荡结构引入混沌运动的可能性和晶格力的不对称性引入非保守性质，如不稳定性和耗散。少

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