Nonlinera Dynamics of Complex Systems

1. 将孤子理论扩展到包括非线性系统中不稳定性的影响。特别是，振幅调制被考虑。在非线性薛定谔方程中，我们得到了时间t和空间x交换的非线性演化方程。我们称之为不稳定非线性薛定谔方程（UNLS）。UNLS方程是一个完全可积系统，适用于瑞利-泰勒问题和电子束等离子体等物理系统。对于小波数kappa， UNLS方程是不适定的，因为增长率是无界的。应用约化微扰理论，可以解决这一难题，得到新的振幅方程。新方程的优点是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。利用非均匀质量分布的一维非线性晶格，考虑两种情况；缓慢变化的波和载波调制。进一步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、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的能级密度是遍历的。研究了一个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。作为一个特例，我们得到了两个线极化波的光孤子。我们研究了一类曲线运动由曲率和扭转决定的模型。该几何方法等价于零特征值的AKNS形式。研究了作为聚合物模型的随机游走的拓扑性质。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