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

9403596森林/欧弗曼主要项目涉及控制光纤中脉冲传输的非线性薛定谔方程。标量NLS是单模光纤的模型方程，而耦合的NLS方程组描述了双折射光纤、开关器件、光纤-光纤链路等。尽管标量NLS的可积理论和近可积微扰方法已经发展得很好，但这些基本结果对于耦合NLS方程仍然是开放的。在这个方案中，我们提出了几个相互关联的项目：耦合NLS脉冲串解的谱变换构造；可积不稳定性的显式表征；用于实现谱变换的数值代码与用于耦合NLS偏微分方程组的精确光学微扰的完整PDE代码；理解观测模拟的几乎可积微扰方法。最终，我们的目标是通过与行业合作伙伴的合作和推动，预测和了解当前光纤技术的稳健、最佳传输和交换极限。光纤技术是现代通信和电子高速公路的基础。这项技术的优势显然可以通过对沿这些不同光纤的传输进行准确的建模和模拟来实现。虽然长途通信线路现在有良好的数学和工程基础，但各种新的光纤和应用导致了更复杂的模型方程，需要理解完全不同的问题。光纤的基本控制方程由单一的非线性薛定谔(NLS)方程组成，而新的应用是由耦合的NLS方程控制的。对于耦合的NLS偏微分方程组，目前还不知道NLS孤子的理论和非线性诊断工具--这些有价值的方法正是我们建议的目标。我们打算在实际兴趣方程的基础上，与学术界和工业界的光学专家合作，发展这些方法，并专注于技术关注的问题。

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.