Soliton Dynamics and Scattering Theory

孤子动力学和散射理论

• 批准号: 0903651
• 负责人: Avraham Soffer
• 金额: $ 20.48万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0903651&HistoricalAwards=false
• 关键词: Soliton; Dynamics; Scattering; Theory

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.

该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的相关主题。 PI 考虑一维和三维的非线性薛定谔方程。因为空间解中可能存在局部有吸引力的非线性，然后解可以在没有色散的情况下移动，直到它们受到扰动。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学中的其他非线性色散问题中发挥着关键作用。这些方程是非线性的，需要新技术来理解此类方程解的大时间行为。 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，其中通过势井的隧道效应创建孤子解。这种孤子形成的预测和细节将得到进一步研究。特别是，该方法提供了一种证明非线性薛定谔方程解的先验估计的新方法，该方程不是标准能量估计。研究的第二部分涉及史瓦西流形和克尔流形上波动方程解的时间详细衰减。特别是追踪作为初始数据角动量函数的衰减率。这将为广义相对论中普莱斯的经典猜想提供严格的证明。详细研究了涉及光学设备的新过程。特别地，考虑这样一种情况，其中光能位于适当活性介电材料内部的势阱中。然后，局域能量的发展是通过适应这种情况的新数学形式导出的。结果表明，孤子波可以从井中出现，并且该理论能够确定其大小和速度。因此，它允许通过调整势阱的形状和输入能量来产生具有所需轮廓的孤子，有时称为孤子枪。这种方法已经在 PI 之前的工作基础上的一些实验中被观察到。研究的第二部分是对黑洞产生的流形上的波传播和衰变进行数学分析。解决这个问题的方法导致了一种新的、更通用的解决 Voltera 型积分方程的理论，并且还导致了对物理文献中有关黑洞辐射波行为的一些经典猜想的严格数学验证。