Nonlinera Dynamics of Complex Systems

1.将孤子理论推广到非线性系统中的不稳定性效应。特别是，幅度调制被考虑在内。我们得到了一个非线性发展方程，其中时间t和空间x在非线性薛定谔方程中交换。我们称之为不稳定的非线性薛定谔(UNLS)方程。UNLS方程是一个完全可积系统，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，因为对于小波数kappa，增长率是不有界的。应用约化微扰理论，我们可以克服这一困难，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。我们研究了非均匀对非线性波传播的影响。利用具有非均匀质量分布的一维非线性晶格，我们考虑了两种情况：慢变波和载波调制。F…进一步，我们将该理论推广到二维晶格中，得到了具有非齐次效应的Kadomtsev-Petviash Vili方程。我们预测了一些有趣的现象，如孤子的变形，以及非线性反射和传输。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水平密度是遍历的。研究了一个耦合的非线性薛定谔方程。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。我们研究了一类由曲率和挠率决定曲线运动的模型。证明了几何方法等价于特征值为零的AKNS形式。研究了随机游动作为聚合物模型的拓扑性质。较少

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