Rigorous approaches to universality results in random matrix theory, integrable systems and nonlinear integrable wave equations

随机矩阵理论、可积系统和非线性可积波动方程中普遍性的严格方法

• 批准号: 261229-2011
• 负责人: Bertola, Marco
• 金额: $ 2.67万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570977
• 关键词: Rigorous; approaches; universality; results; random

The proposed project focuses on the study of universal behaviors in critical phenomena in the areas of 1) random matrix theory; 2) random point processes; 3) nonlinear wave equations (Korteweg de Vries, Nonlinear Schroedinger, etc.) These areas are tied with the theory of Painleve' equations, isomonodromic systems and Riemann-Hilbert problems (i.e. non-Abelian boundary value problems). The critical phenomena we address occur in the study of asymptotic properties with respect to a small (or large) parameter; for example in the area of integrable wave equations like the NonLinear Schroedinger equation (NLS) the small parameter is Planck's constant. The equation is advocated in the study of special elongated phases of Bose-Einstein condensates and the study of rogue waves in oceanography. The behavior of the generic solution exhibits radically different behaviors in well defined regions of spacetime which we can think of as different phases; the notion of universality refers to the behavior in a scaling region around points on the boundary separating different phases, e.g. the points of gradient catastrophe. Such a behavior is independent of the initial data and (conjecturally) stable under certain perturbations of the dynamics as well. Depending on the nature of the transition, the phenomena display different scales w.r.t. the small parameter. In the case of Random Matrices, the universality typically refers to the asymptotic behavior of the statistics of eigenvalues at edges or in the bulk (e.g. Tracy-Widom distribution, Sine-kernel, etc); these behaviors depend only on symmetry properties of the model. Along the same lines the transition of certain random point processes displays similar characteristics (for example the transition from the Pearcey to the Airy point process in self-avoiding random walks). Recent result include a (partial) proof of B. Dubrovin conjecture on the behavior of small--dispersion NLS in the neighborhood of the point of gradient catastrophe and its relation to the theory of Painleve' equations: a novel link with rogue waves (Peregrine solutions of NLS) has been made in the oscillatory regions.

拟议的项目着重于研究在1）随机矩阵理论领域中对批判现象中普遍行为的研究； 2）随机点过程； 3）非线性波方程（Korteweg de Vries，非线性Schroedinger等） 这些领域与Painleve方程，异词质体系统和Riemann-Hilbert问题（即非亚洲边界价值问题）的理论相关。我们解决的关键现象发生在渐近特性相对于小（或大）参数的研究中。例如，在非线性Schroedinger方程（NLS）之类的集成波方程的区域中，小参数是普朗克的常数。该方程在研究Bose-Einstein冷凝物的特殊伸长阶段和海洋学中流氓波的研究中提倡该方程式。通用解决方案的行为在时空定义明确的区域中表现出根本不同的行为，我们可以将其视为不同的阶段。普遍性的概念是指边界围绕不同阶段（例如梯度灾难的点。这种行为与初始数据无关，并且在动力学的某些扰动下（猜想）稳定。根据过渡的性质，现象显示了不同的量表W.R.T.小参数。在随机矩阵的情况下，普遍性通常是指在边缘或散装中特征值统计的渐近行为（例如，Tracy-Widom分布，正弦角，等）；这些行为仅取决于模型的对称特性。沿着相同的行，某些随机点过程的过渡显示出相似的特征（例如，从珍珠到自避免随机步行中的通风点过程的过渡）。 最近的结果包括杜布罗罗文氏芽孢杆菌猜想的（部分）证明对梯度灾难点附近的小分散NL的行为及其与潘leve方程理论的关系：与Rogue Wave（NLS的peregrine solutionse of Painleve'方程）的关系。