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.

可集成的系统包括一类特殊的部分差分（或差异）方程组。它们出现在几种情况下以略有不同的形式出现，包括随机矩阵理论，Riemann表面和连接的模量空间，随机过程和反问题。所有这些实例的一个共同点是根据矩阵值分析函数的特定边界价值问题进行重新重新制定的可能性，或者现在通常称为Riemann-Hilbert问题（RHP）。拟议的研究旨在促进对RHP的一般理解，以及它们在几个杰出问题上的应用。 一些具体目标包括研究RHP的基础变形理论的泊松几何形状，并将其扩展到Riemann表面上的RHP。 在曲线模量空间（Gromov-witten不变性）上的相交数字的背景下，生成函数也可以与RHP相关联，这具有导致严格的渐近分析的好处。曲线模量空间中基本类之间的相交数的生成函数是从矩阵积分（Kontsevich）获得的；已知该积分提供了KDV层次结构的形式解决方案。从G. Moore's的作品中，观察到与等词异构变形的形式联系（因此，间接，RHP）。 但是，分析方法仍然缺少。它的目标是根据合适的RHP来完成此描述，以适用于Kontsevich积分和概括的固定尺寸的矩阵。这特别会阐明生成函数的非正式特性，例如非线性Stokes的现象（分析重新召集）。 RHP的另一个应用是解决反问题。 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. 第二个杰出的逆问题起源于医学成像领域（断层扫描）；为了减少患者组织的照射，必须解决（不适合）与部分数据的反转问题。然后，一个开放的问题是重建的不稳定性，这被转化为一个关于某个积分操作员的奇异值的渐近行为和奇异函数的问题。