Linear and Nonlinear Waves

9706780在本项目中，我们研究了一类非齐次非线性波动方程的大时间行为。这门课包括与辐射相互作用的孤立波解。其目的是发展不可积多通道问题的散射理论。具有位势项和非线性的Klein-Gordon方程就是一种考虑的情况。另一个例子是具有一个以上束缚态的势的非线性薛定谔方程。该提案的其他部分涉及N体色散方程的谱理论，特别是证明了这种哈密顿量的局部衰变和不存在奇异连续谱。该项目研究辐射波和波在介质中的大时间行为。发展了寻找自发或受激发的分子辐射的解析方法。这种辐射的性质在各种应用中被用来确定源的结构和化学性质。该项目的第二部分是关于波在介质中传播的问题。这种波经常改变介质，从而影响波的移动方式。这种自影响导致了在真空中看不到的具有新的和复杂的行为的非线性波动方程。最引人注目的例子是在有限时间内形成的奇点。它们被用来解释各种现象，从黑洞的形成到空气和等离子体的湍流。另一种现象是在介质中以异常高的稳定性运动的波团的形成，就像光纤中的孤子一样。目前的项目致力于分析非线性波和孤子的大时间辐射和相互作用。这包括研究光纤中的缺陷和杂质对高速通信中使用的光孤子的运动和结构的影响。

9706780 Soffer In this project the analysis of large time behavior of a class of nonhomogeneous and nonlinear wave equations is pursued. This class include solitary wave solutions interacting with radiation. The aim is to develop the scattering theory for nonintegrable multichannel problems. The Klein-Gordon equation with potential term and nonlinearity is one case to be considered. The other example is the nonlinear Schrodinger equation with a potential holding more than one bound state. Other parts of the proposal deal with the spectral theory of N-body dispersive equations, in particular the proof of local decay and absence of singular continuous spectrum for such hamiltonians. The project deals with the large time behavior of radiation waves and waves in media. Analytic methods to find the radiation of molecules, spontaneous or stimulated, are developed. The nature of this radiation is used in various applications to determine the structure and chemical properties of the source. The second part of the project deals with waves travelling through a medium. Such waves often change the medium and thus affect the way the wave moves. Such self-influence lead to NONLINEAR wave equations with new and complicated behavior not seen in vacuum. The most striking examples are singularities that form in a finite time. These are used to explain diverse phenomena, from black hole formation to air and plasma turbulence. Another phenomenon is the formation of "clumps" of waves that move in the medium with unusually high stability, like solitons in optic fibers. The current project is devoted to the analysis of the large time radiation and interaction of nonlinear waves and solitons. This includes the study of the effect of defects and impurities in optic fibers on the motion and structure of optical solitons used in high speed communications.