Nonlinera Dynamics of Complex Systems

复杂系统的非线性动力学

基本信息

批准号：
03044040
负责人：
WADATI Miki
金额：
$ 3.52万
依托单位：
The Univ. of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for international Scientific Research
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1992
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-03044040/
关键词：
Soliton Nonlinear waves Random matrix Motion of curves Motion of surfaces Knot theory Inverse scattering method Optical soliton Illーposed problem Selberg積分逓減摂動法不安定非線形シコレディンガ-方程式

项目摘要

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

1. 孤子理论被扩展到包括非线性系统中的不稳定效应，我们将其称为不稳定非线性薛定谔方程（UNLS），UNLS方程是一个完全可积的系统，适用于诸如此类的物理系统。 Rayleigh-Taylor问题和电子束等离子体。2．UNLS方程由于生长而不适定。对于小波数kappa，速率没有限制。通过应用约简微扰理论，我们可以克服这个困难并得到一个新的振幅方程。新方程的优点是涵盖了混沌现象和孤子现象。 3．关于非线性波传播，我们使用具有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。二维晶格并得到具有非齐次效应的Kadomtsev-Petviash vili方程，我们预测了孤子变形、非线性反射和传输等有趣现象。4．在随机矩阵系综理论中，假设系综平均量。描述系综的几乎所有个体成员的属性（随机矩阵系综的遍历性）与经典正交相关的随机矩阵系综的水平密度。多项式被证明为ergodec。 5．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，得到了两个线偏振波的光学孤子。 6. 我们研究了一类模型，使得曲线的运动由曲率和扭转决定。几何方法被证明与零时的 AKNS 形式等效。 7. 研究了作为聚合物模型的随机游走的拓扑性质。