Spontaneous formation of singularities through critical collapse

这个项目致力于研究在各种生物和物理系统中出现的崩塌现象的数学性质。例如，当强大的激光束进入透明介质时，可能会发生坍塌，例如通常的玻璃。当强光在介质内部产生一个有效的透镜并在传播过程中放大自己，直到达到光幅的奇点时，介质与激光的相互作用会导致自聚焦。从数学上讲，奇点意味着相应的数学解接近无穷大。在流体力学和细菌生长过程中也会出现类似的塌陷现象(例如，E.Coli)。在后一种情况下，细菌之间的交流(通过一种名为趋化剂的化学物质)导致细菌菌落自发聚集到非常高的(几乎是单一的)细菌密度。模拟崩溃现象的相应的非线性偏微分方程组的解在有限时间内经历自发形成奇点(爆破)。爆炸通常伴随着解决方案的空间范围的戏剧性收缩，因此有了术语“崩溃”。在奇点附近，潜在的非线性现象发生了质的变化；最初的数学模型失去了它们的适用性，其他机制变得更加重要，例如光学介质中的光学击穿和等离子体的形成，或者细菌拥挤和细菌菌落形成多细胞生物体。本文将重点研究非线性薛定谔方程(NLSE)、Keller-Segel方程(KSE)和Davey-Stewartson方程(DSE)中的崩溃现象，它们是研究临界二维有限时间奇性的原型方程。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。自从60年代初激光问世以来，对崩塌的了解变得尤为迫切。50多年的研究为NLSE和KSE提出了成熟的崩溃理论。然而，直到最近，为什么这些理论结果从未得到直接模拟或实验的证实，仍然是一个谜。解释是，现有的理论需要不切实际的大幅度才能适用。该项目将在NLSE和KSE中为真实的振幅开发一个详细的塌陷比例和塌陷正则化理论。首席研究员还将发展多年来一直缺乏的DSE坍塌比例理论。将努力使该理论成为实际的应用工具。将使用微扰和非微扰方法，以及匹配的渐近技术和广泛的超级计算。NLSE、KSE和DSE的不同应用将促进在非线性光学、流体力学、玻色-爱因斯坦凝聚和生物学领域的思想交流。

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.