Nonlinera Dynamics of Complex Systems

1.孤子理论扩展到包括非线性系统的不稳定性的影响。特别是，幅度调制被认为是。在非线性薛定谔方程中，我们得到了一个时间t和空间x互换的非线性发展方程.我们称之为不稳定的非线性Schorodinger（UNLS）方程。UNLS方程是一个完全可积的系统，适用于Rayleigh-Taylor问题和电子束等离子体等物理系统. UNLS方程是不适定的，因为增长率对于小波数Kappa是没有界的。应用约化微扰理论，我们可以克服这个困难，得到一个新的振幅方程。新方程的优点在于它包含了混沌现象和孤子现象.我们研究了非均匀性对非线性波传播的影响。使用一维非线性晶格与非均匀质量分布，我们考虑两种情况下，慢变波和调制的载波。F ...更多信息 进一步将理论推广到二维格点，得到了具有非齐次效应的Kadomtsev-Petviashvili方程。我们预测了一些有趣的现象，如孤子的变形、非线性反射和透射.在随机矩阵系综理论中，假设系综平均量描述了系综中几乎所有个体成员的性质（随机矩阵系综的遍历性）。证明了与经典正交多项式相关的随机矩阵系综的能级密度是遍历的。研究了耦合非线性薛定谔方程。该方程描述了群速度几乎相等的两个单色波的非线性调制。作为特例，得到了两个线偏振波的光孤子.研究了一类曲线的运动由曲率和挠率决定的模型。几何方法被证明是等价的AKNS形式主义与零特征值。研究了作为聚合物模型的无规游动的拓扑性质。少

1. Soliton theory is extended to include effects of instability in nonlinear systems. In particular, amplitude modulations are considered. We obtain a nonlinear evolution equation where time t and space x are exchanged in the nonlinear Schrodinger equation. We call it unstable nonlinear Schorodinger (UNLS) equation. The UNLS equation is a completely integrable system and is applicable to physical systems such as Rayleigh-Taylor problem and electron-beam plasma.2. The UNLS equation is ill-posed since the growth rate is not bounded for small wavenumber kappa. By applying reductive perturbation theory, we can remedy this difficulty and obtain a new amplitude equation. The new equation has an advantage that it encompass chaos phenomena and soliton phenomena.3. We examine the effect of inhomogeneity on the nonlinear wave propagations. Using a one-dimensional nonlinear lattice with non-uniform mass distribution, we consider two cases ; slowly varying waves and modulations of carrier waves. F … More urther, we extend the theory into two-dimensional lattice and obtain the Kadomtsev-Petviash vili equation with inhomogeneous effects. We predict interesting phenomena such as deformation of solitons, and nonlinear reflections and transmissions.4. In the theory of random matrix ensembles, it is assumed that ensemble-averaged quantities describe the properties of almost all individual members of the ensembles (the ergodicity of random matrix ensembles). The level densities of random matrix ensembles related to classical orthogonal polynomials are proved to be ergodec.5. A coupled nonlinear schrodinger equation is studied. The equation describes non-linear modulations of two monochromatic waves whose group velocities are almost equal. As a special case, optical solitons for two linearly polarized waves are obtained.6. We study a class of models such that a motion of curves is determined by the curvature and torsion. The geometrical approach is shown to be equivalent to the AKNS formalism with zero eigenvalue.7. Topological properties of random walks as a model of polymers are studied. Less