Derivation of mean-field equations and dynamics of coherent structures in nonlinear dispersive PDE

非线性色散偏微分方程中平均场方程的推导和相干结构动力学

• 批准号: 1500106
• 负责人: Walter Strauss
• 金额: $ 17.95万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500106&HistoricalAwards=false
• 关键词: Derivation; mean; field; equations; dynamics

This project deals with certain types of equations modeling wave propagation in various physical contexts. Among these are (1) the structure of a collection of ultra-cold atoms (2) light waves moving through a medium whose index of refraction depends upon the strength of the wave allowing for the self-focusing of the wave (3) water waves in a shallow channel, and (4) waves that move through the electrons and ions in a plasma. Among the important mathematical issues are the rigorous derivation of such models from fundamental physical laws, and the precise study of special types of coherent wave solutions (variously referred to as solitary waves, kinks, vortices, and monopoles, and blow-up profiles). In the last decade, new techniques from several branches of mathematics, specifically harmonic analysis, spectral theory, and Hamiltonian mechanics, have led to striking developments and the resolution of many problems once deemed intractable. The PI has outlined several conjectures that he and his students and collaborators will address by adapting and extending these new methods. At the same time, the resolution of these conjectures is expected to extend the applicability of these methods and interface other branches of mathematics.First, several conjectures on the derivation of the nonlinear Schrodinger equation (NLS) as a limit of the fundamental quantum mechanical model for a many-body system of bosons are outlined. The PI will continue to investigate problems pertaining to dimensional reduction by the imposition of strong anisotropic confining, and allowing attractive interactions leading to the focusing NLS, which supports bright solitary wave solutions, as observed in recent experiments. Second, problems pertaining to the geometrical structure of singularity formation for NLS are proposed. In particular, the PI will consider the stability of new contracting sphere blow-up solutions, the possibility for blow-up on an ellipse, and a precise description of blow-up in the Zakharov system, a model for collapse of Langmuir waves in a plasma. Third, the PI proposes to consider the dynamics of dark solitons, vortices, and monopoles, which are solutions for which the wave function has nontrivial topological structure on the spatial boundary, in equations such as the sine Gordon equation, subjected to semiclassical perturbations. Fourth, the dynamics of non-perturbative interacting multi-solitons in completely integrable models, specifically the modified Korteweg-de Vries (mKdV) and NLS equations, subjected to semiclassical perturbation, will also be pursued.

这个项目涉及某些类型的方程在各种物理环境中模拟波的传播。 其中包括：（1）超冷原子的集合结构;（2）光波穿过介质，介质的折射率取决于波的强度，从而允许波的自聚焦;（3）浅通道中的水波;以及（4）穿过等离子体中的电子和离子的波。 重要的数学问题包括从基本物理定律中严格推导出这些模型，以及精确研究特殊类型的相干波解（称为孤立波、扭结、涡旋、单极和爆破剖面）。 在过去的十年中，来自数学几个分支的新技术，特别是调和分析，谱理论和哈密顿力学，已经导致了惊人的发展和许多问题的解决一度被认为是棘手的。 PI已经概述了他和他的学生和合作者将通过适应和扩展这些新方法来解决的几个问题。 同时，这些问题的解决有望扩展这些方法的适用性，并与其他数学分支相联系。首先，概述了作为玻色子多体系统基本量子力学模型极限的非线性薛定谔方程（NLS）的推导过程。 PI将继续调查有关的问题，通过实施强各向异性限制降维，并允许有吸引力的相互作用，导致聚焦NLS，支持明亮的孤波解决方案，在最近的实验中观察到的。 其次，提出了非线性最小二乘奇异性形成的几何结构问题。 特别是，PI将考虑新的收缩球爆破解决方案的稳定性，在椭圆上爆破的可能性，并在扎哈罗夫系统，朗缪尔波在等离子体中的崩溃模型爆破的精确描述。 第三，PI建议考虑暗孤子，涡旋和单极子的动力学，这些解的波函数在空间边界上具有非平凡的拓扑结构，在正弦戈登方程等方程中，受到半经典扰动。 第四，非微扰相互作用多孤子在完全可积模型，特别是修改Korteweg-de弗里斯（mKdV）和NLS方程，受到半经典扰动的动力学，也将被追求。