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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758908
• 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.

可积系统由一类特殊的超定偏微分（或差分）方程组组成。它们以略有不同的形式出现在多种环境中，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。 所有这些实例的共同点是根据矩阵值解析函数的特定边值问题重新表述的可能性，或者现在通常称为黎曼-希尔伯特问题（RHP）。拟议的研究旨在增进对 RHP 的一般理解及其在几个突出问题上的应用。一些具体目标包括研究 RHP 变形理论的泊松几何，也扩展到黎曼曲面上的 RHP。在曲线模空间上的交数（Gromov-Witten 不变量）的背景下，生成函数可以是也与 RHP 相关，这有利于进行严格的渐近分析。曲线模空间中基本类之间交集数的生成函数由矩阵积分 (Kontsevich) 获得；众所周知，积分提供了 KdV 层次结构的形式解。从 G. Moore 的作品中，观察到与等单向变形（因此，间接地，RHP）的形式联系。然而，仍然缺少分析方法。目标是根据 Kontsevich 积分的固定大小矩阵的合适 RHP 及其概括来完成此描述。这尤其将揭示生成函数的非形式特性，例如非线性斯托克斯现象（解析恢复）。 RHP 的另一个应用是解决反问题。在这里，该项目重点关注非线性薛定谔方程的逆（和前）散射理论：在半经典极限中，当普朗克常数被发送到零时，在玻色-爱因斯坦凝聚的拉长相和海洋学的研究中也考虑到了，其中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。那么，悬而未决的问题是重构的不稳定程度，它被转化为关于某个积分算子的奇异值和奇异函数的渐近行为的问题。