Linear and Nonlinear Dispersive Waves: Solitons, Nonlinear Resonances and Spectral Theory

线性和非线性色散波：孤子、非线性共振和谱理论

• 批准号: 1600749
• 负责人: Avraham Soffer
• 金额: $ 15万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600749&HistoricalAwards=false
• 关键词: Linear; Nonlinear; Dispersive; Waves; Solitons

The research in this project is focused on the analysis of wave dynamics. Wave propagation and other properties are the backbone of modern Science and Technology. Quantum waves control the nano world, electromagnetic waves manifest as light, lasers, heat, and are responsible for chemical reactions and electronic devices. Gravity theory is described by Einstein wave equation as well. The analysis of complex systems via such wave equations is extremely difficult, and even super fast computers cannot do the job. Therefore, deeper understanding of the equations provides new tools for the relevant features of the solutions. At the same time, the acquired knowledge opens new directions of research in mathematics. In this proposal the behavior of special solutions, called coherent states, of fundamental equations of mathematical physics are studied. In particular, focus is given to the effect of disturbances of such solutions over long periods of time, in an effort to control the stability, life time and evolution of such solutions. The cases mostly considered are solutions called solitons- which are clumps of self trapped waves in some small domain. The solitons formed by laser beams going through an optical fiber play a key role in fast communication and other future planned optical devices.Soliton dynamics of the nonlinear Schroedinger equation will be studied. While the complete understanding of the solutions of such equations for all initial data seems remote, a novel direction is proposed: The interaction of solitons with radiation, with large potential terms, and with each other, will be analyzed by identifying processes which are adiabatic in time. Relevant adiabatic dispersive theory will be used, previously developed by the PI and new planned methods. It is expected to complete a gap in our understanding of a fundamental aspect of nonlinear scattering: interaction process of a soliton with large perturbation. The study of nonhomogeneous nonlinearities of the long range type, initiated by the PI, which are fundamental to many nonlinear scattering problems (e.g. Kink scattering), uncovered a new, subtle resonance phenomena: the unbound growth of some Invariant Sobolev norm of such systems. It is the aim of the research to understand the implications of these new processes. Dispersive estimates for linear equations play a central role in spectral and scattering theory. The PI's recent and future work is to develop an alternative, abstract theory. By avoiding the need to study the explicit (eigen) solutions or fundamental solutions of the linear equation, one can expand the dispersive theory to new classes of problems, including dynamics on manifolds.

该项目的研究重点是对波动力学的分析。波浪传播和其他特性是现代科学技术的骨干。量子波控制纳米世界，电磁波表现为光，激光，热，并负责化学反应和电子设备。爱因斯坦波方程也描述了重力理论。通过此类波方程对复杂系统的分析非常困难，甚至超级快速计算机也无法完成这项工作。因此，对方程式的更深入了解为解决方案的相关特征提供了新的工具。同时，获得的知识开辟了数学研究的新方向。在此提案中，研究了数学物理基本方程的特殊解决方案的行为，称为相干状态。特别是，将重点放在长时间内这种解决方案的扰动的影响，以控制这种解决方案的稳定性，寿命和演变。这些案例主要考虑的是称为孤子的解决方案，它们是在某些小域中的自被困波团的团块。通过光纤通过激光束形成的孤子在快速通信和其他未来计划的光学设备中起关键作用。将研究非线性Schroedinger方程的Soliton动力学。尽管对所有初始数据的方程解决方案的完全理解似乎是遥远的，但提出了一个新的方向：孤子与辐射的相互作用，具有较大的潜在项，彼此之间的相互作用，将通过识别时间上绝热的过程来分析。先前由PI和新计划的方法开发的相关绝热分散理论。有望在我们对非线性散射的基本方面的理解中填补差距：具有巨大扰动的孤子的相互作用过程。 PI启动的远距离类型的非官方非线性的研究是对许多非线性散射问题（例如扭结散射）的基础，发现了一种新的，微妙的共振现象：这种系统的某些不变的Sobolev Norm的不合可及的生长。研究的目的是了解这些新过程的含义。线性方程的分散估计在光谱和散射理论中起着核心作用。 PI最近和未来的工作是开发一种替代性的抽象理论。通过避免需要研究线性方程的显式（EIGEN）解决方案或基本解决方案，可以将分散理论扩展到新的问题，包括流形动态。