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

本课题的研究重点是波浪动力学分析。波的传播和其他特性是现代科学技术的支柱。量子波控制着纳米世界，电磁波表现为光、激光、热，并负责化学反应和电子设备。引力理论也可以用爱因斯坦波动方程来描述。通过这样的波动方程来分析复杂系统是极其困难的，即使是超高速的计算机也无法完成这项工作。因此，对方程的深入理解为解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。本文研究了数学物理基本方程的特解，即相干态的行为。特别地，重点放在这些解决方案在长时间内的扰动影响上，以努力控制这些解决方案的稳定性、寿命和演变。大多数情况下考虑的是被称为孤子的解，它是一些小范围内的自困波团。激光束通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起着关键作用。本文将研究非线性薛定谔方程的孤子动力学。虽然对所有初始数据的这些方程的解的完全理解似乎是遥远的，但提出了一个新的方向：通过确定在时间上绝热的过程，将分析具有大势项的孤子与辐射的相互作用以及彼此之间的相互作用。将使用PI先前开发的相关绝热色散理论和新的计划方法。它有望填补我们对非线性散射的一个基本方面的理解上的空白：具有大扰动的孤子的相互作用过程。由PI发起的对非齐次非线性的长距离型的研究，是许多非线性散射问题（如Kink散射）的基础，揭示了一种新的、微妙的共振现象：这种系统的一些不变Sobolev范数的无界增长。这项研究的目的是了解这些新过程的含义。线性方程的色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发展另一种抽象理论。通过避免研究线性方程的显式（本征）解或基本解的需要，可以将色散理论扩展到新的问题类别，包括流形上的动力学。

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.