Spontaneous formation of singularities through critical collapse

该项目致力于研究在各种生物和物理系统中出现的崩溃现象的数学性质。例如，当强大的激光束进入透明介质（如普通的玻璃）时，可能会发生坍缩。介质与激光束的相互作用导致自聚焦，当强光在介质内部产生有效透镜并在传播过程中放大自己，直到达到光振幅的奇异点。在数学上，奇点意味着对应的数学解趋于无穷。在流体力学和细菌生长（如大肠杆菌）中，也会发生定性上类似的塌陷现象。在后一种情况下，细菌之间的交流（通过一种称为化学引诱剂的化学物质）导致细菌菌落自发聚集到一个非常高（几乎单一）的细菌密度。模拟崩塌现象的相应非线性偏微分方程组的解在有限时间（爆破）内自发形成奇点。膨胀通常伴随着溶液空间范围的急剧收缩，因此有“坍缩”一词。在奇异点附近，底层非线性现象发生质变；最初的数学模型失去了适用性，其他机制变得更加重要，例如光学介质中的光击穿和等离子体的形成，或者细菌聚集和细菌菌落形成多细胞生物。本文将重点研究非线性Schrödinger方程（NLSE）、Keller-Segel方程（KSE）和Davey-Stewartson方程（DSE）中的坍缩现象，它们是研究临界二维有限时间奇点的原型方程。这些方程是非线性科学中最普遍和最广泛的方程之一，在非线性光学、流体力学和生物学中有着广泛的应用。自60年代初激光问世以来，对坍缩的理解变得尤为迫切。50多年的研究为NLSE和KSE产生了完善的崩溃理论。然而，直到最近，为什么这些理论结果从未被直接模拟或实验证实，这仍然是一个谜。解释是，现有的理论需要不切实际的大振幅才能适用。本项目将在NLSE和KSE中为实际振幅发展详细的崩溃缩放和崩溃正则化理论。首席研究员还将开发多年来一直缺乏的DSE坍缩缩放理论。将努力使理论成为应用的实际工具。微扰和非微扰方法都将被使用，以及匹配的渐近技术和广泛的超级计算。NLSE、KSE和DSE的不同应用将促进非线性光学、流体力学、玻色-爱因斯坦凝聚和生物学等领域的思想交流。

This project is devoted to the study of mathematical nature of phenomena of collapse that arise in a variety of biological and physical systems. For example, collapse may occur when a powerful laser beam enters a transparent medium, such as usual glass. Interaction of the medium with the laser beam results in self-focusing when the intense light creates an effective lens inside the medium and amplifies itself during propagation until the singularity of the light amplitude is reached. Mathematically a singularity means that the corresponding mathematical solution approaches infinity. Qualitatively similar collapse phenomena occur in hydrodynamics as well as in the bacterial growth (for example, for E. Coli). In the latter case, communications between bacteria (through a chemical substance called chemoattractant) cause spontaneous aggregation of the bacterial colony to a very high (almost singular) bacterial density. Solutions of the corresponding nonlinear systems of partial differential equations that model phenomena with collapse experience spontaneous formation of singularities in finite time (blow-up). Blow-up is often accompanied by a dramatic contraction of the spatial extent of a solution, hence the term "collapse." Near the singularity point there is a qualitative change in underlying nonlinear phenomena; the initial mathematical models lose their applicability and other mechanisms become more important, such as optical breakdown and formation of plasma in optical media, or bacterial crowding and formation of multicellular organisms from the bacterial colony.This research will focus on phenomena of collapse in the Nonlinear Schrödinger equation (NLSE), Keller-Segel equation (KSE) and Davey-Stewartson equation (DSE) which are archetypal equations to study finite time singularities in the critical dimension two. These equations are among most universal and widespread equations in nonlinear science with numerous applications in nonlinear optics, hydrodynamics and biology. The need in understanding collapse became especially pressing since the advent of lasers in early 60s. More than 50 years of research produced well-established collapse theories for NLSE and KSE. However, until recently it remained a puzzle why these theoretical results were never confirmed by either direct simulations or in experiments. The explanation is that the existing theories required unrealistically large amplitudes for their applicability. This project will develop a detailed theory of collapse scaling and collapse regularization in NLSE and KSE for the realistic amplitudes. The principal investigator will also develop a DSE collapse scaling theory, which has been lacking for many years. The effort will be made to make the theory a practical tool for applications. Both perturbative and nonperturbative approaches will be used, as well as matched asymptotic techniques and extensive supercomputing. Diverse applications of NLSE, KSE and DSE will promote cross-fertilization of ideas across the fields of nonlinear optics, hydrodynamics, Bose-Einstein condensation and biology.