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

本项目的研究重点是波浪动力学分析。波的传播和其他性质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。引力理论也是由爱因斯坦波动方程描述的。通过这样的波动方程分析复杂系统是非常困难的，即使是超级快速的计算机也无法完成这项工作。因此，更深入地理解方程提供了新的工具，解决方案的相关功能。与此同时，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，重点是在很长一段时间内，在努力控制这种解决方案的稳定性，寿命和演变的干扰的影响。大多数情况下考虑的解决方案称为孤子-这是一群自陷波在一些小的领域。激光束通过光纤形成的孤子在快速通信和未来计划的其他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。虽然完全理解这些方程的所有初始数据的解决方案似乎遥远，一个新的方向提出：孤子与辐射的相互作用，与大的潜在条款，并与彼此，将通过识别过程进行分析是绝热的时间。将使用PI先前开发的相关绝热色散理论和新计划的方法。它有望填补我们在理解非线性散射的一个基本方面的空白：孤子与大扰动的相互作用过程。由PI发起的对长程非均匀非线性的研究揭示了一种新的微妙的共振现象：这种系统的某些不变Sobolev范数的无约束增长，这种非均匀非线性是许多非线性散射问题（例如扭结散射）的基础。研究的目的是了解这些新过程的影响。线性方程组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发展一种替代的抽象理论。通过避免研究线性方程的显式（特征）解或基本解的需要，人们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.