Grant-in-Aid for Overseas Scientific Survey.
|Research Institution||The Univ. of Tokyo|
WADATI Miki Faculty of Science, Univ. of Tokyo, Professor, 理学部, 教授 (60015831)
HARVEY Segur コロラド大学, 応用数学, 教授
MARK Ablowit コロラド大学, 応用数学, 教授
出口 哲生 東京大学, 理学部, 助手 (70227544)
矢嶋 徹 東京大学, 工学部, 助手 (40230198)
HARVEY SEGUR コロラド大学, 応用数学, 教授
MARK ABLOWIT コロラド大学, 応用数学, 教授
ABLOWITZ Mark J. Program in Applied Mathematics, Univ. of Colorado, Professor
SEGUR Harvey Program in Applied Mathematics, Univ. of Colorado, Professor
DEGUCHI Tetsuo Faculty of Science, Univ. of Tokyo, Research Associate
YAJIMA Tetsu Faculty of Engineering, Univ. of Tokyo, Research Associate
|Project Fiscal Year
1991 – 1992
Completed(Fiscal Year 1992)
|Budget Amount *help
¥5,500,000 (Direct Cost : ¥5,500,000)
Fiscal Year 1992 : ¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1991 : ¥2,500,000 (Direct Cost : ¥2,500,000)
|Keywords||Soliton / Nonlinear waves / Random matrix / Motion of curves / Motion of surfaces / Knot theory / Inverse scattering method / Optical soliton / ソリトン / 非線形波動 / ランダム行列 / ひもの運動 / 面の運動 / 結び目理論 / 逆散乱法 / 光ソリトン / 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|
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
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