Soliton Dynamics and Scattering Theory

研究的重点是理解孤子动力学的数学方面和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。对于空间中局部的吸引非线性，可能存在解，然后解可以无色散地移动，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重要作用。方程是非线性的，需要新的技术来理解这类方程的解的大时间行为。 使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，其中孤子解是通过从势阱隧穿而产生的。这一孤子形成的预测和细节将进一步研究。特别地，该方法为证明非线性薛定谔方程解的非标准能量估计的先验估计提供了一种新的途径。第二部分的研究涉及详细的衰减时间的波动方程的解决方案Schwarzschild和克尔流形。特别是作为初始数据的角动量的函数的衰减率被追求。这将为广义相对论中普赖斯的一个经典猜想提供严格的证明。特别地，人们考虑光能位于适当活性电介质材料内部的势阱中的情况。然后，发展的局域化的能量是由一个新的数学形式，以适应这种情况。结果表明，孤立子波可以从井中产生，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期望轮廓的孤子，有时被称为孤子枪。这种方法已经在PI以前的工作基础上的一些实验中观察到。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。解决这个问题的方法导致了一个新的，更普遍的理论，解决积分方程的Voltera型，也导致了严格的数学验证的一些经典命题在物理文献中有关的行为辐射波了黑洞。

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.