Soliton Dynamics and Scattering Theory

研究的重点是理解孤子动力学的数学方面以及波传播和散射的相关主题。PI考虑一维和三维的非线性薛定谔方程。对于空间中的吸引非线性，可能存在解，它们可以无色散地运动，直到它们被扰动。这类方程现在在描述光学器件、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关键作用。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动力学公式，可以详细分析一种全新类型的情况，在这种情况下，孤子解是通过势阱的隧穿产生的。该孤子形成的预测和细节将进一步研究。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方法。研究的第二部分涉及Schwarzschild流形和Kerr流形上波动方程解的详细时间衰减。特别地，衰减率作为初始数据的角动量的函数被追求。这将为广义相对论中的Price经典猜想提供一个严格的证明。详细研究了涉及光学器件的新工艺。特别地，我们考虑这样一种情况，其中光能位于电势阱中，在适当活性的介电材料内。然后，用一种新的适合这种情况的数学形式推导了局域能量的发展。结果表明，孤子波可以从井中出现，并且该理论能够确定它们的大小和速度。因此，它允许通过调整势阱的形状和入射能量来产生具有所需轮廓的孤子，有时被称为孤子炮。这种方法已经在一些基于PI先前工作的实验中被观察到。研究的第二部分是对黑洞产生的流形上的波传播和衰减的数学分析。这个问题的解决方法导致了一个新的，更一般的理论来解决Voltera型的积分方程，也导致了对物理学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.