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波浪动力学分析。波传播和其他特性是现代科学技术的支柱。量子波控制纳米世界，电磁波表现为光、激光、热，负责化学反应和电子设备。引力理论也是由爱因斯坦波动方程描述的。通过这种波动方程来分析复杂系统是极其困难的，即使是超高速计算机也无法完成这项工作。因此，对方程的更深入理解为解决方案的相关特征提供了新的工具。同时，所获得的知识开辟了数学研究的新方向。在该提案中，研究了数学物理基本方程的特殊解（称为相干态）的行为。特别是，重点关注此类解决方案在较长时间内的扰动影响，以努力控制此类解决方案的稳定性、寿命和演化。最常考虑的情况是称为孤子的解决方案，它是某些小域中的自陷波团。激光束穿过光纤形成的孤子在快速通信和其他未来计划的光学器件中发挥着关键作用。将研究非线性薛定谔方程的孤子动力学。虽然完全理解所有初始数据的此类方程的解似乎很遥远，但提出了一个新的方向：孤子与辐射、大势项以及彼此之间的相互作用将通过识别绝热过程来分析。时间。将使用 PI 先前开发的相关绝热色散理论和新计划的方法。它有望填补我们对非线性散射基本方面理解的空白：孤子与大扰动的相互作用过程。由 PI 发起的长程非齐次非线性研究是许多非线性散射问题（例如扭结散射）的基础，揭示了一种新的、微妙的共振现象：此类系统的某些不变索博列夫范数的无约束增长。研究的目的是了解这些新过程的影响。线性方程的色散估计在光谱和散射理论中发挥着核心作用。 PI 最近和未来的工作是开发一种替代的抽象理论。通过避免研究线性方程的显式（特征）解或基本解，我们可以将色散理论扩展到新的一类问题，包括流形动力学。