Symmetries, Conserved Integrals, Hamiltonian Flows, and Integrable Systems

近年来，偏微分方程的对称性、守恒积分、哈密顿结构和相关方面吸引了许多数学活动。与此同时，在应用数学和数学物理学中，几何方法被大量应用于可积系统的研究。我的建议在很大程度上建立在我目前资助的这些研究方向上。许多部分非常适合于硕士项目，博士论文和博士后工作，以及本科研究生的贡献。(1)可积系统和曲线/曲面流使用几何框架方法，我以前导出的多分量可积系统是群不变的推广的非线性薛定谔（NLS），修改Korteveg de弗里斯（mKdV），sine-Gordon（SG）方程的双Hamilton结构，产生于曲线流在黎曼对称空间和半单李群。这一工作将被推广并应用于获得新的包含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。 我也将把其他一些关于欧氏空间中的曲面流的工作推广到对称空间和李群，这将导致新的2+1维可积系统。 (2)峰是具有峰值指数轮廓的行波。它们出现在与水波理论相关的非线性色散波方程中。 我已经开始了对峰子方程的几个方面的研究，其中包括寻找一个与NLS方程相关的新的峰子系统。我还证明了一个非常一般的波动方程族具有多峰子解。我将继续研究这两个主题，以理解和研究峰子相互作用的有趣特性。 (3)流体流动中的守恒积分一个公开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，我解决了这个问题的两个重要类型的守恒积分的可压缩流体流动在n>1维。我还通过在平流表面上寻找新的守恒积分来扩展结果，这给出了螺旋度和环流的有趣概括。我计划获得进行类似的工作与自由边界的流体流动，可压缩磁流体力学。 (4)非线性偏微分方程的对称性与守恒律守恒律在非线性偏微分方程的研究中是非常重要的。对于有拉格朗日量的偏微分方程，所有的守恒律都可以通过诺特定理从对称性中得到，但是对于没有拉格朗日量的偏微分方程，这种联系失败了。在以前的工作发表在几篇论文和合著的书，我已经开发出一种算法的方法来找到任何给定的偏微分方程承认的守恒律是否有拉格朗日。 我计划继续发展和应用这种方法的偏微分方程的重要性，在应用数学和物理。

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.