Dynamics of Dispersive PDE

本项目研究色散偏微分方程，这是描述物理系统中波动行为的方程。 因此，色散偏微分方程可以描述光学，流体或等离子体中的现象，仅举几个应用领域。 为了确定特定方程作为给定现象的模型的有效性，确定该方程是否有解以及如果有解，该解如何取决于物理参数可能是有帮助的。 对于某些非线性薛定谔方程，以及某些可用作水中波浪模型的方程，主要研究人员将研究这些方程何时缺乏解，或具有不连续依赖于数据的解。 这将表明，这些特定的模型只有有限的有效性的应用程序。 作为该项目的另一部分，对于其他一些离散偏微分方程族，主要研究者将确定这些方程何时具有或不具有某些特殊形式的解。 了解这些解，例如永久形式的波或时间周期波，可以更深入地了解相应的物理现象在长时间间隔内的行为。首席研究员将研究拟线性薛定谔方程和截断级数水波模型的适定性和不适定性。 对于拟线性薛定谔方程，在一定的非退化条件下，适定性定理已经得到了证明;在本文中，将研究当非退化条件被破坏时的不适定性. 对于水波的截断级数模型，将使用复变量方法和仿微分工具研究适定性/不适定性与色散强度之间的关系。 对于非线性薛定谔方程和非线性色散方程的一般家庭，主要研究者将使用小除数估计，规范形式和色散平滑估计，以证明在某些参数制度的小振幅空间周期，时间周期波的不存在。 此外，存在的时间周期，空间周期波的色散方程与高导数非线性（特别是与毛细波在流体中的系统）将被研究，使用小因子方法和技术的傅立叶分析。

This project studies dispersive partial differential equations, which are equations describing wave-like behavior in physical systems. As such, dispersive partial differential equations can describe phenomena in optics, fluids, or plasmas, to name just a few areas of application. To determine the validity of a particular equation as a model for a given phenomenon, it can be helpful to determine whether the equation has a solution, and if it does, how this solution depends upon the physical parameters. For certain nonlinear Schrodinger equations, and for certain equations which can be used as models for waves in water, the principal investigator will study when these equations either lack solutions, or have solutions which depend discontinuously upon the data. This will indicate that these particular models have only limited validity for applications. As another part of the project, for some other families of dispersive partial differential equations, the principal investigator will determine when the equations either do or do not possess solutions with certain special forms. Understanding such solutions, such as waves of permanent form or time-periodic waves, can lead to a deeper understanding of the behavior of the corresponding physical phenomena over long intervals of time.The principal investigator will study well-posedness and ill-posedness for quasilinear Schrodinger equations and for truncated series water wave models. For quasilinear Schrodinger equations, well-posedness theorems have been proved under certain non-degeneracy hypotheses; in this work, ill-posedness will be studied when the non-degeneracy condition is violated. For truncated series models of water waves, the relationship between well-posedness/ill-posedness and the strength of dispersion will be investigated using complex variable methods and tools from paradifferential calculus. For nonlinear Schrodinger equations and for general families of nonlinear dispersive equations, the principal investigator will use small divisor estimates, normal forms, and dispersive smoothing estimates to demonstrate the non-existence of small-amplitude spatially periodic, time-periodic waves in certain parameter regimes. Also, existence of time-periodic, spatially-periodic waves for dispersive equations with high-derivative nonlinearities (especially for systems related to capillary waves in fluids) will be studied, using small divisor methods and techniques of Fourier analysis.