Mathematical Sciences: Analysis of Solitary Waves and Multi-Solitons

这个项目将涉及分析无耗散的非线性色散系统理论中的孤立波、多孤子和相关现象。本课程将探讨孤立波分析中的几个问题，包括孤立波的存在性、唯一性和稳定性。数值研究将补充理论分析。此外，还将考虑各种近似方程的有效性和非线性偏微分方程组或积分-微分方程组的某些严格性质。非线性分析工具将得到一些数值模拟和逆散射理论的特殊方法的补充。孤立波是局部的、非线性的、前进的波，在非线性和色散效应之间取得平衡而生存。它们出现在许多物理环境中。严格的分析往往是相当困难的，因为问题从一开始就是非线性的。建立了预测孤立波的存在并捕捉其定性特征的非线性模型方程。要研究的系统出现在许多非线性科学领域，包括流体力学、非线性光学、固体物理学、海洋学和气象学。

This project will involve the analysis of solitary waves, multi- solitons, and related phenomena in the theory of nonlinear dispersive systems without dissipation. Several questions in the analysis of solitary waves will be examined, including their existence, uniqueness, and stability. Numerical investigations will supplement theoretical analysis. In addition, the validity of various approximate equations and certain rigorous properties of nonlinear dispersive systems of partial differential or integro-differential equations will be considered. The tools of nonlinear analysis will be supplemented by some numerical simulation and the special methods of inverse scattering theory. Solitary waves are localized, nonlinear, progressing waves that subsist on a balance struck between nonlinear and dispersive effects. They arise in many physical settings. Rigorous analysis is often quite difficult since the problem is nonlinear from the outset. Nonlinear model equations have been developed that predict the presence of solitary waves and capture their qualitative features. The systems to be studied arise in many areas of nonlinear science including fluid mechanics, nonlinear optics, solid state physics, oceanography, and meteorology.