Symmetries, Conserved Integrals, Hamiltonian Flows, and Integrable Systems

近年来，偏微分方程组的对称性、守恒积分、哈密顿结构及其相关方面引起了人们的广泛关注。与此同时，在应用数学和数学物理中，出现了研究可积系统的几何方法的浪潮。我的建议在很大程度上建立在我目前拨款中的这些研究方向上。许多部分非常适合于硕士项目、博士论文和博士后工作，以及本科生的贡献。*(1)可积系统和曲线/曲面流*我以前用几何框架方法导出了多分量可积系统，它们是由黎曼对称空间和半单李群中的曲线流产生的具有双哈密顿结构的非线性薛定谔(NLS)、修正的Korteveg de Vries(MKdV)和Sine-Gordon(SG)方程的群不变推广。这项工作将被推广并应用于得到包含四元数和八元数变量的新的可积方程，这将是可积系统理论的一大进步。*我还将把欧氏空间中其他一些关于表面流的工作推广到对称空间和李群，这将导致2+1维的新的可积系统。*(2)Peakon方程*Peakons是具有峰值指数分布的行波。它们出现在与水波理论有关的非线性色散波动方程中。*我已经开始研究Peakon方程的几个方面，包括寻找与NLS方程相关的新的Peakon系统。我还证明了一个非常一般的波动方程族有多个峰解。我将继续研究这两个主题，以了解和研究Peakon交互的有趣特性。(3)流体流动中的守恒积分*一个悬而未决的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n&1维可压缩流体流动的两类重要的守恒积分问题。我还通过在平流表面上找到新的守恒积分来扩展结果，这给出了螺旋度和环流的有趣推广。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)非线性偏微分方程组的对称性和守恒律*守恒律在非线性偏微分方程组的研究中是非常重要的。对于有拉格朗日量的偏微分方程组，所有的守恒律都可以通过Noether定理从对称性得到，但对于没有拉格朗日量的偏微分方程组，这种联系是不成立的。在之前发表在几篇论文和一本合著的书中，我已经开发了一种算法方法来寻找任何给定的偏微分方程所允许的守恒律，无论它是否具有拉格朗日。*我计划继续开发这种方法，并将其应用于应用数学和物理中重要的PDE。

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.