Integrable systems in Geometry, Asymptotics and Inverse Problems

几何、渐近和反问题中的可积系统

• 批准号: RGPIN-2016-06660
• 负责人: Bertola, Marco
• 金额: $ 2.91万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=616442
• 关键词: Integrable; systems; Geometry; Asymptotics; Inverse

Integrable systems consist in a special class of overdetermined sets of partial differential (or difference) equations. They appear in several contexts in slightly different guises, including Random Matrix Theory, Moduli Spaces of Riemann surfaces and connections, Stochastic Processes and Inverse problems. A common thread to all these instances is the possibility of reformulation in terms of a particular boundary value problem for matrix-valued analytic functions, or what is now commonly referred to as a Riemann-Hilbert problem (RHP). The proposed research seeks to both advance the general understanding of RHPs as well as their application to several outstanding problems. Some of the specific goals include the study of the Poisson geometry underlying deformation theory of RHPs, extending also to RHPs on Riemann surfaces. In the context of intersection numbers on the moduli space of curves (Gromov-Witten invariants), the generating function can be associated also to a RHP and this has the benefit of leading to a rigorous asymptotic analysis. The generating function of intersection numbers between fundamental classes in the moduli space of curves is obtained from a matrix integral (Kontsevich); the integral is known to provide a formal solution of the KdV hierarchy. From works of G. Moore's, the formal connection with isomonodromic deformations (thus, indirectly, RHP) was observed. However an analytic approach is still missing. It is a goal to complete this description in terms of a suitable RHP for a matrix of fixed size of the Kontsevich integral and generalizations thereof. This in particular will shed light on non-formal properties of the generating function, such as the nonlinear Stokes' phenomenon (analytic resummation). Another application of RHPs is in solving inverse problems. Here the project focuses on the inverse (and forward) scattering theory of the nonlinear Schroedinger equation: in the semiclassical limit as Planck's constant is sent to zero is considered also in the study of elongated phases of Bose--Einstein condensates and in oceanography, where it has been proposed as describing the underlying mechanism of formation of the so-called rogue waves and for the ``three sister'' rogue waves. A second outstanding inverse problem originates in the area of medical imaging (tomography); in order to reduce irradiation of patient's tissue, the (ill-posed) problem of inversion with partial data must be addressed. Then the open question is the degree of instability of the reconstruction, which is translated into a question about asymptotic behavior of singular values and singular functions for a certain integral operator.

可积系统包含一类特殊的偏微分（或差分）方程超定集。它们出现在几个上下文中略有不同的伪装，包括随机矩阵理论，黎曼曲面和连接的模空间，随机过程和逆问题。所有这些例子的一个共同点是，对于矩阵值解析函数的特定边值问题，或者现在通常被称为黎曼-希尔伯特问题（RHP），可以重新表述。拟议中的研究旨在促进对RHP的普遍理解以及它们在几个突出问题中的应用。 一些具体的目标包括研究泊松几何基础变形理论的RHP，也延伸到RHP的黎曼曲面。 在曲线的模空间上的交集数（Gromov-Witten不变量）的上下文中，生成函数也可以与RHP相关联，这有利于导致严格的渐近分析。在曲线的模空间中的基本类之间的相交数的生成函数是从矩阵积分（Kontsevich）获得的;该积分已知提供KdV层次的形式解。从G.摩尔的，形式上的联系与isomonodromic变形（因此，间接，RHP）进行了观察。 然而，分析方法仍然缺失。目标是根据Kontsevich积分的固定大小的矩阵的合适RHP及其推广来完成此描述。这将特别阐明生成函数的非形式性质，如非线性斯托克斯现象（解析重传）。 RHP的另一个应用是求解逆问题。在这里，该项目侧重于非线性薛定谔方程的逆（和前向）散射理论：在普朗克常数被发送到零的半经典极限中，也被认为是玻色-爱因斯坦凝聚体和海洋学的拉长阶段的研究，在那里，它被提议描述所谓的流氓波和"三姐妹“流氓波形成的基本机制。 第二个突出的逆问题起源于医学成像（断层扫描）领域;为了减少对患者组织的照射，必须解决部分数据反演的（不适定）问题。然后，开放的问题是重建的不稳定程度，这是转化为一个问题的奇异值和奇异函数的渐近行为的某一积分算子。