Spontaneous formation of singularities through critical collapse

通过临界崩溃自发形成奇点

基本信息

批准号：
1412140
负责人：
Pavel Lushnikov
金额：
$ 24万
依托单位：
University of New Mexico
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-08-01 至 2017-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1412140&HistoricalAwards=false
关键词：
Spontaneous formation singularities through critical

项目摘要

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.

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（例如通常的玻璃）时，可能会崩溃。当强烈的光在传播过程中产生有效的镜头，直到达到光放大器的奇异性，培养基与激光束的相互作用会导致自我关注。从数学上讲，奇异性意味着相应的数学解决方案接近无穷大。定性相似的崩溃现象发生在流体动力学和细菌生长中（例如，对于大肠杆菌）。在后来的情况下，细菌（通过称为化学吸引剂的化学物质）之间的通信导致细菌菌落的赞助聚集至非常高（几乎是单数）的细菌密度。偏微分方程的相应非线性系统的解决方案，以有限时间（爆炸）在有限时间内对现象进行模拟现象的赞助。爆炸通常是通过解决方案的空间范围的巨大收缩来实现的，因此“崩溃”一词。在奇异点附近，基本的非线性现象存在质量变化。最初的数学模型失去了其适用性和其他机制变得更加重要，例如光学介质中的光学分解和血浆形成，或细菌拥挤和从细菌群体中形成多细胞生物的细菌，这将重点介绍非线性schrödinger方程（NLSE），KEELERETION（NLSE）的崩溃现象（KSE SSEGERITAND）（KSEGELSENTION）（KSEGELSEGERATION）（KSEGELSEN），（DSE）是研究临界维度二的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宽度的方程之一，在非线性光学，流体动力学和生物学中有许多应用。自60年代初激光前进以来，理解崩溃的需求变得尤为紧迫。超过50年的研究为NLSE和KSE产生了良好的崩溃理论。但是，直到最近，直接模拟或实验中都从未证实这些理论结果的原因。解释是，现有的理论需要不切实际的大放大器才能适用。该项目将开发出NLSE和KSE中崩溃缩放和崩溃调节的详细理论，用于现实的放大器。主要研究者还将开发出多年来缺乏的DSE崩溃理论。将努力使该理论成为应用程序的实用工具。将使用扰动和非扰动方法，以及匹配的不对称技术和广泛的超级计算。 NLSE，KSE和DSE的各种应用将促进非线性光学，流体动力学，Bose-Einstein凝结和生物学领域的思想的交叉侵入。