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

这个项目处理某些类型的方程，在各种物理环境中模拟波的传播。其中包括：(1)超冷原子集合的结构；(2)在介质中移动的光波，其折射率取决于波的强度，允许波的自聚焦；(3)在浅通道中的水波；(4)在等离子体中的电子和离子中移动的波。在重要的数学问题中，从基本物理定律中严格推导出这样的模型，以及对特殊类型的相干波解（不同地称为孤波、扭结、漩涡、单极子和爆破剖面）的精确研究。在过去的十年中，来自几个数学分支的新技术，特别是谐波分析、谱理论和哈密顿力学，已经导致了惊人的发展，并解决了许多曾经被认为难以解决的问题。PI概述了他和他的学生和合作者将通过调整和扩展这些新方法来解决的几个猜想。同时，这些猜想的解决有望扩展这些方法的适用性，并连接其他数学分支。首先，对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）的推导提出了几个猜想。PI将继续研究通过施加强各向异性限制来降低维度的问题，并允许吸引相互作用导致聚焦NLS，这支持在最近的实验中观察到的明亮孤立波解。其次，提出了NLS奇点形成的几何结构问题。特别是，PI将考虑新的收缩球体爆破解的稳定性，椭圆上爆破的可能性，以及扎哈罗夫系统（等离子体中朗缪尔波坍缩的模型）中爆破的精确描述。第三，PI建议考虑暗孤子、涡旋和单极子的动力学，它们是波函数在空间边界上具有非平凡拓扑结构的解，在正弦戈登方程等方程中，受到半经典扰动。第四，研究完全可积模型中非摄动相互作用多孤子的动力学，特别是受半经典摄动影响的修正Korteweg-de Vries （mKdV）和NLS方程。

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.