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

该项目主要研究1）随机矩阵理论; 2）随机点过程; 3）非线性波动方程（Korteweg de弗里斯，Nonlinear Schroedinger等）。 这些领域与Painleve方程理论、isomonodromic系统和Riemann-Hilbert问题（即非阿贝尔边值问题）有关。我们解决的关键现象发生在研究渐近性质的一个小（或大）参数;例如，在该地区的可积波动方程，如非线性薛定谔方程（NLS）的小参数是普朗克常数。该方程在研究玻色-爱因斯坦凝聚体的特殊伸长相和海洋学中的异常波中得到了广泛的应用。一般解的行为在定义明确的时空区域中表现出完全不同的行为，我们可以将这些区域视为不同的相;普适性的概念是指在分隔不同相的边界上的点（例如梯度突变点）周围的标度区域中的行为。这样的行为是独立的初始数据和稳定的某些扰动下的动力学。根据过渡的性质，现象显示不同的尺度w.r.t.小参数。在随机矩阵的情况下，普适性通常是指边缘或块中特征值统计的渐近行为（例如Tracy-Widom分布，正弦核等）;这些行为仅取决于模型的对称性。沿着相同的线，某些随机点过程的过渡显示出类似的特征（例如，自避免随机游动中从Pearcey到Airy点过程的过渡）。 最近的结果包括B的一个（部分）证明。关于小色散NLS在梯度突变点附近的行为及其与Painleve方程理论的关系的Dubrovin猜想：在振荡区建立了与流氓波（NLS的Peregrine解）的新联系。

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.