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

近年来，对称性、守恒积分、哈密顿结构以及偏微分方程的相关方面引起了广泛的关注。与此同时，研究可积偏微分方程系统的几何方法激增。我的提案很大程度上建立在我当前资助的这些研究方向的基础上。许多部分非常适合理学硕士项目、博士论文和博士后工作，以及本科生的贡献。 双哈密尔顿可积系统和曲线/曲面流： 我最近使用几何框架方法导出了通用的多分量可积系统，该系统是非线性薛定谔（NLS）、修正的科特维格德弗里斯（mKdV）、具有双哈密尔顿结构的正弦戈登（SG）方程的群不变推广，由黎曼对称空间和半简单李群中的曲线流产生。 在与理学硕士学生的合作中，我们正在推导新型 NLS/mKdV 方程。来自埃尔米特和洛伦兹对称空间。我计划从四元数和格拉斯曼空间导出新的四元数和 eqns. 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导出群不变可积系统。 我打算扩展我的方法来导出 Lax 对。这将导致对 Drinfeld 和 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结果。我还打算将松弛对和双哈密顿结构从框架变量提升回曲线流。这将给出在 Lax 对和泊松括号上已知的抽象结果的明确表述。 概括我在 R^3 中所做的工作，我计划从对称空间/李群中的几何表面流导出 2 1 维可积系统。结果将产生巨大影响，因为目前已知的此类系统只有几种类型。 多分量孤子： 在与学生的合作中，我们正在研究 2 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“流氓波”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。在与一名博士生合作研究无粘可压缩流体方程时，我们推导了 R^n 中移动域的所有运动学和涡量积分。我最近将结果扩展到移动表面，得出了螺旋度和熵循环的有趣概括。 我计划为其他几个重要的流体方程获得移动域/表面上的所有此类守恒积分。在 R^n 和黎曼流形中。该结果将对 n 维的几何/拓扑研究具有基本意义。 Arnold&Khesin 阐述的流体动力学。 非线性偏微分方程的对称性、守恒定律和精确解： 出于物理和数学原因，对称不变解非常有用。我与合作者一起开发了一种基于几何群叶状结构的新颖对称方法，以找到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/色散相关的有趣的分析行为，包括临界功率的情况。我计划继续开发这种方法，并将其应用于其他物理和分析上有趣的偏微分方程。这将对对称方法和偏微分方程的分析产生很大的影响。 我还计划探索对称性和守恒定律之间的新联系，这源于我们在最近的书中总结的与布鲁曼的合作。

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.