Symmetries, conserved integrals, Hamiltonian flows, and integrable systems.

对称性、守恒积分、哈密顿结构以及与偏微分方程相关的方面近年来引起了人们的广泛关注。与此同时，研究可积偏微分方程系统的几何方法激增。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。双哈密顿可积系统和曲线/曲面流动：本文利用几何框架方法，从黎曼对称空间和半单李群的曲线流动中，导出了具有双哈密顿结构的非线性薛定谔（NLS）、修正Korteveg de Vries （mKdV）、正弦戈登（SG）方程的群不变推广的通用多分量可积系统。在与硕士生的合作中，我们正在开发新型的NLS/mKdV方程。从厄米和洛伦兹对称空间。我计划从四元数和格拉斯曼空间中推导出新的四元数和矩阵类型。我还计划将类似的方法应用于一个重要的开放问题，即如何从齐次空间中的曲线流中导出群不变可积系统。我打算扩展我的方法来推导Lax对。这将导致对Drinfeld&Sokolov在仿射李代数中Lax对的普遍构造的几何理解，这是可积系统理论的核心成果。我还打算将Lax对和双哈密顿结构从坐标系变量中提回到曲线流中。这将给出已知的关于Lax对和泊松括号的抽象结果的显式表述。推广我在R^3中所做的工作，我计划从对称空间/李群的几何表面流中导出2+1维可积系统。研究结果将产生巨大影响，因为目前已知的这类系统只有少数几种。多分量孤子：在与学生的合作中，我们正在研究双分量mKdV孤子的相互作用。与单分量情况相比，这些解决方案表现出有趣的新特征，包括“异常波”的形成，这是应用数学中一个非常活跃的话题。我们将继续讨论n分量孤子。守恒积分：一个开放的问题是确定流体流动基本方程的所有守恒积分。和一个博士生一起研究无粘可压缩流体方程。，我们推导了R^n中运动域上的所有运动学积分和涡量积分。我最近将结果扩展到移动的表面，得到了螺旋和熵循环的有趣推广。我计划对其他几个重要的流体方程在运动域/表面上获得所有这些守恒积分。在R^n和黎曼流形中。这些结果将对n-dim的几何/拓扑研究具有重要意义。由Arnold&Khesin阐述的流体力学。对称性、守恒定律和非线性偏微分方程的精确解：对称不变解在物理和数学上非常有用。与合作者一起，我开发了一种基于几何群叶化的新型对称方法，用于寻找具有幂非线性的偏微分方程的显式不变和非不变解。对于我们所考虑的非线性波/热方程，这些解具有与爆炸、吸引子、衰减/色散相关的有趣的解析行为，包括临界幂的情况。我计划继续开发这种方法，并将其应用于其他物理上和分析上有趣的偏微分方程。这将对偏微分方程的对称方法和分析产生重大影响。我还计划探索对称性和守恒定律之间的新联系，这源于我们在新书中总结的与布鲁曼的工作。

Symmetries, conserved integrals, Hamiltonian structures, and related aspects of PDEs have attracted much activity in recent years. At the same time, there has been a surge of geometric approaches to the study of integrable PDE systems. 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.Bi-Hamiltonian integrable systems and curve/surface flows:Using geometric frame methods, I recently derived universal 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.In work underway with MSc students, we are deriving new types of NLS/mKdV eqns. from Hermitian and Lorentzian symmetric spaces. I plan to derive new quaternion and matrix types of eqns.from quaternion and Grassmannian spaces. I also plan to apply similar methods to an important open problem of how to derive group-invariant integrable systems from curve flows in homogeneous spaces. I intend to extend my method to derive Lax pairs. This will lead to a geometric understanding of Drinfeld&Sokolov's universal construction of Lax pairs in affine Lie algebras, which is a central result in the theory of integrable systems. I also intend to lift the Lax pairs and bi-Hamiltonian structures from the frame variables back to the curve flow. This will give an explicit formulation of abstract results known on Lax pairs and Poisson brackets.Generalizing work I have done in R^3, I plan to derive 2+1 dimensional integrable systems from geometric surface flows in symmetric spaces/Lie groups. The results will have a large impact because only a few types of such systems are presently known.Multi-component solitons:In work with students, we are studying interaction of 2-component mKdV solitons. Compared to the 1-component case, these solutions exhibit interesting, new features, including formation of "rogue waves", which is a very active topic in applied math. We will go on to n-component solitons. Conserved integrals:An open problem is to determine all conserved integrals for fundamental equations of fluid flow. In work with a PhD student on inviscid compressible fluid eqns., we derived all kinematic and vorticity integrals on moving domains in R^n. I recently extended the results to moving surfaces, yielding interesting generalizations of helicity and entropy circulation. I plan to obtain all such conserved integrals on moving domains/surfaces for several other important fluid eqns. in R^n and in Riemannian manifolds. The results will be of basic interest in the geometrical/topological study of n-dim. hydrodynamics elaborated by Arnold&Khesin. Symmetries, conservation laws, and exact solutions of nonlinear PDEs:Symmetry-invariant solutions are very useful for physical and mathematical reasons. With collaborators, I developed a novel symmetry method based on geometrical group foliations to find explicit invariant and non-invariant solutions to PDEs with power nonlinearities. For the nonlinear wave/heat equations we considered, these solutions have interesting analytical behavior related to blow-up, attractors, decay/dispersion, including cases of critical powers. I plan to continue developing this method and apply it to other physically and analytically interesting PDEs. This will have a large impact in symmetry methods and analysis of PDEs.I also plan to explore new connections between symmetries and conservation laws, stemming from work with Bluman summarized in our recent book.