Nonlinear Wave Motion

非线性波动涉及高功率信号如何传输以及如何在应用中广泛发生的研究。一些应用包括表面水波、海洋中的异常波、海啸、光波（例如波导和激光中出现的光波）、声波以及相关的冲击波。这项研究工作的重点是解决许多开放性问题，这些问题将大大扩展数学家解释应用中广泛出现的大功率波现象行为的能力，同时加深对这种行为的理解。通过积极参与这项研究，将培养一名博士后和本科生。非线性波运动的数学分析存在困难，因为基础方程是非线性的，与小功率线性波相比，其数学理解要少得多。本提案中讨论的可用于分析一类非线性波动方程的方法之一称为逆散射变换（IST）。 PI 具有使用此方法的经验，并发现了新类别的具有物理意义的方程，这些方程具有可以应用 IST 的有趣的基础对称性。该方法还适用于在所有方向上衰减的多维方程和相关解。后面这些解决方案的关键属性将被理解，并且它们可以与量子力学和光学中的基本方程联系起来。还将对一类称为色散冲击波（DSW）的波进行研究。 DSW 是本质上具有色散性的冲击波；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、等离子体物理学和玻色-爱因斯坦凝聚等。 DSW 的理论将得到扩展，以获得一维基础方程的改进近似，并将开发详细的多维分析。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.