Soliton Dynamics and Scattering Theory

这项研究的重点是了解孤子动力学的数学方面，以及波传播和散射中的相关主题。PI考虑一维和三维上的非线性薛定谔方程。对于具有吸引力的、位于空间中的非线性，可能存在解，这些解可以在没有色散的情况下移动，直到它们被扰动。这类方程在描述光学器件、玻色-爱因斯坦凝聚流体以及流体动力学和等离子体物理中的其他非线性色散问题中起着至关重要的作用。这些方程是非线性的，需要新的技术来理解这类方程解的大时间行为。利用薛定谔方程的流体力学公式，可以详细地分析一种全新的情况，即通过势井隧穿产生孤子解。这种孤子形成的预测和细节将被进一步研究。特别地，该方法为证明非标准能量估计的非线性薛定谔方程解的先验估计提供了一种新的途径。第二部分研究了Schwarzschild流形和Kerr流形上波动方程解的时间衰减性。特别地，求出了作为初始数据角动量函数的衰减率。这将为广义相对论中经典的普赖斯猜想提供严格的证明。详细研究涉及光学器件的新过程。特别是，人们考虑了这样一种情况，其中光能位于适当活跃的介电材料内的势井中。然后，通过一种适应于这种情况的新的数学形式，导出了局域能量的发展。结果表明，孤波可以从井中产生，并且该理论能够确定孤波的大小和速度。因此，它允许通过调节势垒的形状和输入能量来产生具有所需轮廓的孤子，有时被称为孤子枪。在PI以前工作的基础上，这种方法已经在一些实验中被观察到。研究的第二部分是黑洞产生的流形上波的传播和衰减的数学分析。这个问题的解决方法导致了一种新的、更一般的Voltera型积分方程组的求解理论，也导致了对物理文献中关于黑洞辐射波行为的一些经典猜想的严格的数学验证。

The research is focused on understanding the mathematical aspects of soliton dynamics and related topics in wave propagation and scattering. The PI considers the nonlinear Schroedinger equation on one and three dimensions. For attractive nonlinearities localized in space solutions may exist, which can then move without dispersion, until they are perturbed. These class of equations play now a a critical role in describing optical devices, Bose-Einstein condensate fluids and other nonlinear dispersive problems in fluid dynamics and plasma physics. The equations being nonlinear, requires new techniques to understand the large time behavior of the solutions of this class of equations. Using the hydrodynamic formulation of Schroedinger equation it was possible to analyze in detail a completely new type of situations, in which a soliton solution is created through tunneling from a potential well. The prediction and the details of this soliton formation will be further studied. In particular, the method offers a new way of proving a-priori estimates on the solutions of nonlinear Schrodinger equation which are not of the standard energy estimates. A second part of the research involves the detailed decay in time of solutions of the wave equation on Schwarzschild and Kerr manifolds. In particular the decay rates as a function of the angular momentum of the initial data is pursued. This will provide a rigorous proof of a classical conjecture of Price in the theory of General Relativity.New processes involving optical devices are studied in detail. In particular one considers a situation in which light energy is located in a potential well, inside a properly active dielectric material. Then, the development of the localized energy is derived by a new mathematical formalism adapted to this situation. It is shown that soliton waves can emerge from the well, and the theory is capable of determining their size and speed. As such, it allows, by tuning the shape of the potential well and the incoming energy to produce solitons with desired profile, sometimes referred to as soliton guns. This approach has already been observed in some experiments based on previous works by the PI. A second part of the research is the mathematical analysis of wave propagation and decay on manifolds generated by black holes. The approach to this problem led a to a new, more general theory of solving integral equations of the Voltera type, and also led to a rigorous mathematical verification of some classical conjectures in the physics literature concerning the behavior of radiated waves off a black hole.