Inverse Scattering and Partial Differential Equations

这位首席研究员将继续他在几何分析的三个领域的研究：(1)二维和一维空间中的完全可积的非线性色散方程，(2)混沌散射中的共振，(3)具有奇异势的线上薛定谔方程和狄拉克方程的逆散射。2+1维完全可积的色散方程描述了非线性表面波，表现出“块”和线-孤子行为，并提供了具有Kähler靶的薛定谔映射的例子。主要研究人员和合作者将利用泛函、调和和整体分析工具，将逆散射和规范变换技术发展成严格的分析方法，用于研究2+1维完全可积的色散方程。他们寻求这些动力系统的轨道结构和稳定性；孤子解的存在、分类和稳定性；以及流动的哈密顿结构的完整图景。最初，PI将集中于Davey-Stewartson方程、Novikov-Veselov方程、Kadomtsev-Petviashvili方程和Ishimori方程作为测试案例，其动力学行为可以得到深入研究。渐近双曲流形(AH)和复双曲流形(CH)是具有简单几何结构的变曲率流形，而混沌测地线流形的特征是紧凑的、分形的陷集。因此，它们是研究混沌散射的天然目标，用于研究测地线的陷阱集与共振分布之间的关系。开发近似(AH)和精确(CH)轨迹公式的工具，主要研究人员和合作者将获得关于动态数据方面的共振分布的估计。继续他关于线上奇异位势的逆散射的工作，首席调查者将研究薛定谔方程和狄拉克系统的波算子、m-函数和逆散射映射。这项工作的目标将是发展类似的西蒙的A-函数，并研究具有奇异初始数据的NLS和KdV方程的解的定性行为。完全可积和混沌动力系统是无限维动力系统的重要“极端”情况，它出现在应用科学的许多不同领域。一维完全可积方法(表示为由散射数据确定的Riemann-Hilbert问题的解)给出了可积偏微分方程组、随机矩阵系综、圆和直线上的正交多项式以及组合问题的解的非常精确的渐近性。PI试图为振动d-bar问题发展类似的渐近方法，以确定二维完全可积偏微分方程解、平面上正交多项式的渐近性和正态矩阵分布的渐近性。这种分析将需要新的技术和结果来进行谐波分析。在另一个极端，混沌动力系统的量子化是当前研究的热点：在拟议的研究中，我们将在一个几何环境中研究经典囚禁和量子混沌之间的关系，其中动力学和散射都可以进行详细的分析。

The Principle Investigator will continue his research in three areas of geometric analysis: (1) Completely integrable, nonlinear dispersive equations in two space and one time dimensions, (2) resonances in chaotic scattering, and (3) inverse scattering for Schrödinger and Dirac-type equations on the line with singular potentials. Completely integrable dispersive equations in 2 + 1 dimensions describe nonlinear surface waves, exhibit "lump" and line-soliton behavior, and provide examples of Schrödinger maps with Kähler targets. The Principal Investigator and collaborators will develop the techniques of inverse scattering and gauge transformations into rigorous analytical methods for the study of completely integrable dispersive equations in 2+1 dimensions, using the tools of functional, harmonic, and global analysis. They seek a complete picture of the orbit structure and stability of these dynamical systems; existence, classification, and stability of soliton solutions; and Hamiltonian structure of the flows. Initially the PI will concentrate on the Davey-Stewartson, Novikov-Veselov, Kadomtsev-Petviashvili and Ishimori equations as test cases whose dynamical behavior can be studied in depth. Asymptotically hyperbolic (AH) and complex hyperbolic (CH) manifolds are variable-curvature manifolds with simple geometry at infinity and chaotic geodesic flows characterized by a compact, fractal trapped set. As such they are natural targets for the investigation of chaotic scattering, to study the relationship between the trapped set of geodesics and the distribution of resonances. Developing the tools of approximate (AH) and exact (CH) trace formulae, the Principal Investigator and collaborators will obtain estimates on the distribution of resonances in terms of dynamical data. Continuing his work on inverse scattering for singular potentials on the line, the Principal Investigator will study wave operators, m-functions, and inverse scattering maps for Schrödinger equations and Dirac systems. The goal of this work will be to develop analogues of Simon's A-function and to study qualitative behavior of solutions to the NLS and KdV equations with singular initial data. Completely integrable and chaotic dynamical systems are important "extremal" cases of infinite-dimensional dynamical systems which occur in many different areas of applied science. The completely integrable method in one dimension (formulated as the solution of a Riemann-Hilbert problem determined by scattering data) gives remarkably precise asymptotics for solutions of integrable partial differential equations, random matrix ensembles, orthogonal polynomials on the circle and the line, and combinatorial problems. The PI seeks to develop analogous asymptotic methods for the oscillatory d-bar-problems that determine solutions of completely integrable partial differential equations in two dimensions, asymptotics of orthogonal polynomials in the plane, and asymptotics of normal matrix distributions. This analysis will require new techniques and results in harmonic analysis. At the other extreme, the quantization of chaotic dynamical systems is an area of intensive current research interest: in the proposed research we will study the relationship between classical trapping and quantum chaos in a geometrical setting where the dynamics and scattering are amenable to a detailed analysis.