Nonlinear Wave Motion

非线性波动与高功率信号如何传输的研究有关，在应用中广泛存在。一些应用包括表面水波、海洋中的异常波、海啸、光波（诸如在波导和激光器中发生的光波）、声波以及相关联的冲击波。这项研究工作的重点是解决一些开放的问题，这将大大扩展数学家的能力，以解释在应用中广泛出现的大功率波现象的行为，同时发展对这种行为的更深入的理解。通过积极参与本研究，将培养一名博士后和一名本科生。非线性波浪运动的数学分析存在困难，因为基本方程是非线性的，其数学理解远不如小功率线性波浪。逆散射变换（IST）是一种分析非线性波动方程的方法。PI有这种方法的经验，并发现了新的物理意义的方程与有趣的基本对称性，IST可以应用。该方法也适用于多维方程和相关的解决方案，在所有方向上衰减。这些后者解决方案的关键属性将被理解，它们可以连接到量子力学和光学的基本方程。还将对一类称为分散冲击波（DSW）的波进行研究。DSW是本质上是色散的冲击波;基本方程是色散的而不是耗散的。因此，DSW不像大气冲击波那样受到耗散的强烈影响。DSW出现在许多应用中，包括水波、非线性光学、等离子体物理和玻色-爱因斯坦凝聚等。DSW的理论将被扩展，以获得改进的近似的基本方程在一个维度和一个详细的多维分析将被开发。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Nonlinear wave motion relates to the study of how high-power signals are transmitted and occur widely in applications. Some of the applications include surface water waves, rogue waves in the ocean, tsunamis, optical waves such as those that occur in waveguides and lasers, acoustic waves, and associated shock waves. This research effort focuses on solving a number of open problems that will substantially extend the ability of mathematicians to explain the behavior of large power wave phenomena that arise widely in applications while developing a deeper understanding of this behavior. A postdoctoral associate and undergraduate students will be trained through active participation in this research.The mathematical analysis of nonlinear wave motion presents difficulties because the underlying equations are nonlinear for which there is far less mathematical understanding than small power linear waves. One of the methods that can be used to analyze a class of nonlinear wave equations discussed in this proposal is termed the inverse scattering transform (IST). The PI has experience with this method and has found new classes of physically significant equations with interesting underlying symmetries to which IST can be applied. The method also applies to multidimensional equations and associated solutions which decay in all directions. Key properties of these latter solutions will be understood, and they can be connected to fundamental equations in quantum mechanics and optics. Research will also be conducted on a class of waves called dispersive shock waves (DSWs). DSWs are shock waves that are dispersive in nature; the underlying equations are dispersive not dissipative. Consequently, DSWs are not like atmospheric shock waves which are are strongly affected by dissipation. DSWs arise in many applications including water waves, nonlinear optics, plasma physics, and Bose-Einstein condensation amongst many others. 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.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.