Linear and Nonlinear Waves

9706780 Soffer 本文研究了一类非齐次非线性波动方程的大时间行为。 这一类包括孤立波解与辐射相互作用。 目的是发展不可积的散射理论 多渠道的问题。带势项和非线性项的Klein-Gordon方程是一种需要考虑的情形。另一个例子是非线性薛定谔方程的潜在持有一个以上的束缚态。 提案的其他部分涉及N体色散方程的谱理论，特别是局部衰变的证明， 缺乏奇异连续谱的这种汉密尔顿。 该项目涉及辐射波和介质中波的大时间行为。用分析方法找到辐射 自发的或受刺激的分子的形成。这种辐射的性质在各种应用中用于确定源的结构和化学性质。 该项目的第二部分涉及通过介质传播的波。这种波经常改变介质，从而影响波的运动方式。这种自影响导致非线性波动方程具有真空中所没有的新的和复杂的行为。最突出的例子是在有限时间内形成的奇点。这些是 用于解释各种现象，从黑洞形成到空气和等离子体湍流。另一个现象是 在介质中以异常高的稳定性运动的波的“团块”，就像光纤中的孤子。本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用于高速通信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.