Mathematical Sciences: Nearly Integrable Nonlinear Wave Phenomena:Theory and Applications"

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9403596 Forest/Overman The primary project relates to the equations of nonlinear Schrodinger type which govern pulse transmission in fiber optics. Scalar NLS is the model equation for monomode fibers, while coupled systems of NLS equations describe birefringent fibers, switching devices, fiber-fiber links, etc. While the integrable theory and near-integrable perturbation methods are well-developed for scalar NLS, these fundamental results are still open for coupled NLS equations. We propose to develop several interconnected projects in this proposal: the spectral transform construction of pulse train solutions for coupled NLS; explicit characterization of integrable instabilites; numerical codes to implement the spectral transform in conjunction with full pde codes for the precise optical perturbations of coupled NLS pdes; nearly integrable perturbation methods to understand the observed simulations. Ultimately, our goal is to predict and understand the robust, optimal transmission and switching limits of present fiber optic technology, in cooperation with and driven by industry collaborators. Fiber optics technology is fundamental to modern communications and the electronic superhighway. Advantages in this technology can clearly be achieved by accurate models and simulation of the transmission along these various fibers. While long-distance communication lines are now on sound mathematical and engineering grounds, various new fibers and applications lead to more complicated model equations and quite different issues need to be understood. The fundamental governing equations for fiber optics consist of a single nonlinear Schrodinger (NLS) equation, while the newer applications are governed by coupled NLS equations. The theory and nonlinear diagnostic tools for the solitons of NLS are NOT yet known for the coupled NLS pdes--these valuable methods are precisely the goals of our proposal. We intend to develop these methods on the equations of practical interes t, with the collaboration of optics experts in academics and industry, and focused on the problems of technological concern.