Linear and Nonlinear Waves

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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.