Linear and Nonlinear Multichannel Scattering

在这一建议中，主要研究两个问题，第一，非线性薛定谔波的长时间行为和散射理论，从一个支持许多束缚态的势项出发。这导致了对哈密顿色散方程的多通道非线性散射和亚稳态的分析。第二个主题是研究波动方程亚稳态多呼吸子解的辐射衰减。分析了多呼吸子解的线性化波动方程的大时间行为，发展了一种估计扰动多呼吸子解寿命的理论，并将其应用于非线性光学，特别是具有非均匀衍射轮廓的光导。在今天的通信系统中，光学器件和光纤的使用使这一提议中所研究的一些问题变得更为活跃。特别讨论了光纤中的缺陷和其它不规则性对其传输特性的影响问题。我们还分析了通过对介质进行修改以实现对噪声和通道间干扰的更好过滤而制成的一些新型光学器件。

There are two main topics of research in this proposal, first, thelarge time behaviour and scattering theory of nonlinear Schrodingerwaves off a potential term which supports many bound states. This leadsto the analysis of multichannel nonlinear scattering and metastabilityof hamiltonian dispersive equations. The second topic involves the study ofradiation damping for metastable mutltibreather solutions of the waveequation. The large time behaviour of the linearized wave equation aroundmultibreather solutions is analyzed; a theory to estimate the lifetimesof perturbed multibreather solutions is developed and applicationsto nonlinear optics, in particular optical guides with non-uniform diffractionprofiles.The use of optical devices and fibers in today's communication systemsmotivates some of the problems studied in this proposal. In particularwe concentrate on the problem of effects of defects and other irregularitiesin optical fibers on their transport properties. We also analyze somenovel optical devices made by modifying the medium in a way to achievebetter filtering of noise and interchannel interference.