Symmetries, Conserved Integrals, Hamiltonian Flows, and Integrable Systems

对称性、守恒积分、哈密顿结构和偏微分方程的相关方面近年来吸引了大量的数学活动。与此同时，应用数学和数学物理中出现了大量研究可积系统的几何方法。我的提案在很大程度上建立在我目前资助的这些研究方向上。许多部分非常适合硕士项目、博士论文和博士后工作，以及本科生的研究贡献。(1)可积系统和曲线/曲面流利用几何框架方法，我已经推导了多分量可积系统，这些系统是非线性薛定谔（NLS）、修正Korteveg de Vries （mKdV）、具有双哈密顿结构的正弦戈登（SG）方程的群不变推广，产生于黎曼对称空间和半单李群中的曲线流。这项工作将被推广和应用于获得新的四元数和八元数变量的可积方程，这将是可积系统理论的一大进步。我还将把欧几里得空间表面流的一些早期工作推广到对称空间和李群，这将导致新的2+1维的可积系统。(2) Peakon方程Peakon是指具有峰值指数剖面的行波。它们出现在与水波理论相关的非线性色散波动方程中。我已经开始研究峰值方程的几个方面，其中包括寻找一个与NLS方程相关的新的峰值系统。我还证明了一类非常一般的波动方程具有多峰解。我将继续研究这两个主题，以理解和研究峰值交互的有趣特性。流体流动中的守恒积分一个开放问题是确定流体流动基本方程的所有守恒积分。在最近的工作中，我解决了两种重要类型的守恒积分的可压缩流体流动在n>1维的问题。我还通过在平流表面上发现新的守恒积分扩展了结果，它给出了螺旋和循环的有趣推广。我计划对具有自由边界的流体流动和可压缩磁流体力学进行类似的工作。(4)非线性偏微分方程的对称性和守恒律守恒律在非线性偏微分方程的研究中非常重要。对于有拉格朗日量的偏微分方程，所有的守恒定律都可以通过诺特定理从对称性中得到，但对于没有拉格朗日量的偏微分方程，这种联系就失效了。在之前发表的几篇论文和合著的一本书中，我开发了一种算法方法，可以找到任何给定偏微分方程所承认的守恒定律，无论它是否具有拉格朗日量。我计划继续发展并将这种方法应用于应用数学和物理中重要的偏微分方程。

Symmetries, conserved integrals, Hamiltonian structures, and related aspects of PDEs have attracted much mathematical activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable systems in applied mathematics and mathematical physics. My proposal builds significantly on these directions of research in my current grant. Many parts are well suited for MSc projects, PhD theses, and postdoctoral work, as well as contributions by undergrad research students. (1) Integrable systems and curve/surface flows Using geometric frame methods, I have previously derived multi-component integrable systems which are group-invariant generalizations of nonlinear Schrodinger (NLS), modified Korteveg de Vries (mKdV), sine-Gordon (SG) equations with a bi-Hamiltonian structure, arising from curve flows in Riemannian symmetric spaces and semisimple Lie groups. This work will be extended and applied to obtain new integrable equations involving quaternionic and octonionic variables, which will be a large advance in the theory of integrable systems. I also will generalize some other earlier work on surface flows in Euclidean space to symmetric spaces and Lie groups,, which will lead to new integrable systems in 2+1 dimensions. (2) Peakon equations Peakons are travelling waves that have a peaked exponential profile. They arise in nonlinear dispersive wave equations connected to water wave theory. I have begun to work on several aspects of peakon equations, which includes finding a new peakon system related to the NLS equation. I have also shown that a very general family of wave equations possesses multi peakon solutions. I will continue to work on these two topics to understand and look at interesting features of peakon interactions. (3) Conserved integrals in fluid flow An open problem is to determine all conserved integrals for the fundamental equations of fluid flow. In recent work, I settled this problem for two important types of conserved integrals for compressible fluid flow in n>1 dimensions. I also extended the results by finding new conserved integrals on advected surfaces, which gave interesting generalizations of helicity and circulation. I plan to obtain carry out similar work for fluid flow with free boundaries, and for compressible magnetohydrodynamics. (4) Symmetries and conservation laws of nonlinear PDEs Conservation laws are very important in the study of nonlinear PDEs. For PDEs that have a Lagrangian, all conservation laws can be obtained from symmetries through Noether's theorem, but this connection fails for PDEs without a Lagrangian. In previous work published in several papers and a co-authored book, I have developed an algorithmic method to find the conservation laws admitted by any given PDE whether or not it has a Lagrangian. I plan to continue to develop and apply this method to PDEs of importance in applied mathematics and physics.