Nonlinear Wave Motion

非线性波动表现在许多自然现象和工艺过程中。非线性效应在高功率信号和大震级波中最为突出：当一个波叠加在另一个波上时，这些波不像线性情况下那样只是相加，而是相互作用。这种波形成复杂的波型，有非常重要的应用，包括光纤和激光中的电磁波传播，空气动力学中的冲击波，以及海洋中的异常波。控制方程的非线性阻碍了对这些现象的数学分析。对于非线性方程，一般理论的表述往往是不可行的，求解需要逐个研究。这项研究工作的重点是解决一些开放的问题，这些问题将大大扩展数学家的能力，以发展更深层次的理解和解释在应用中广泛出现的大振幅波现象的行为。逆散射变换（IST）方法是为数不多的分析技术之一，它允许人们分析和获得数学物理中许多重要方程的解。IST将用于求解和理解具有PT（奇偶时间）对称性的新型非线性波动方程的性质。IST也可以用来描述某些类型的局部解，称为块状解，它们在所有方向上衰减。这些解的关键性质将被理解，它们将被连接到非平稳Schrödinger方程。研究工作还将针对一类称为弥散冲击波（DSWs）的冲击波现象进行调查。dsw是通过色散进行正则化的激波，与通过耗散进行正则化的标准激波不同。dsw在水波、玻色-爱因斯坦凝聚、非线性光学等领域有广泛的应用。DSWs的理论将得到扩展，以便在一维上获得对基础方程的改进近似，并将进行详细的多维分析。

Nonlinear wave motion is manifested in many natural phenomena and technological processes. Nonlinearity effects are most prominent for the high-power signals and large-magnitude waves: when one wave is superimposed on the other the waves are not just adding as would be true in the linear case but interact with each other. Such waves form complex wave patterns and have very important applications, including among others propagation of electromagnetic waves for fiber optics and lasers, shock waves in aerodynamics, and rogue waves in the ocean. Mathematical analysis of these phenomena is hampered by the nonlinearity of the governing equations. For nonlinear equations formulating a general theory is often unfeasible, and obtaining solutions requires a case-by-case study. This research effort focuses on solving a number of open problems that will substantially extend the ability of mathematicians to develop deeper understanding and explain the behavior of large-amplitude wave phenomena which arise widely in applications. The Inverse Scattering Transform (IST) method is one of the few analytical techniques that allows one to analyze and obtain solutions to a number of important equations in mathematical physics. IST will be used to obtain solutions and understand properties of new classes of nonlinear wave equations which exhibit PT (parity-time) symmetry properties.  IST can also be used to describe certain classes of localized solutions, termed lump solutions, which decay in all directions. The key properties of these solutions will be understood and they will be connected to the non-stationary Schrödinger equation. The research effort will also be directed to investigating a class of shock wave phenomena termed dispersive shock waves (DSWs). DSWs are shock waves which are regularized by dispersion, in contrast with standard shock waves which are regularized by dissipation. DSWs arise in many applications including water waves, Bose- Einstein condensates, nonlinear optics, etc. The theory of DSWs will be extended in order to obtain improved approximations to the underlying equations in one dimension, and a detailed multidimensional analysis will be developed.