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

[403596 . Forest/Overman]主要项目涉及控制光纤脉冲传输的非线性薛定谔型方程。标量NLS是单模光纤的模型方程，而NLS方程的耦合系统描述双折射光纤、开关器件、光纤-光纤链路等。虽然标量NLS的可积理论和近可积微扰方法已经得到了很好的发展，但这些基本结果对于耦合NLS方程仍然是开放的。我们建议在本提案中发展几个相互关联的项目：耦合NLS脉冲串解的频谱变换构建；可积不稳定性的显式表征数值编码来实现光谱变换，并结合完整的pde编码来实现耦合NLS pde的精确光学摄动；用近可积摄动方法来理解观测到的模拟。最终，我们的目标是预测和理解当前光纤技术的鲁棒性，最佳传输和交换限制，与行业合作者合作并推动。光纤技术是现代通信和电子高速公路的基础。这种技术的优势可以通过精确的模型和沿着这些不同光纤传输的仿真来清楚地实现。虽然长距离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更复杂的模型方程和需要理解的完全不同的问题。光纤的基本控制方程由一个非线性薛定谔方程组成，而新的应用是由耦合的薛定谔方程控制的。NLS孤子的理论和非线性诊断工具尚不清楚耦合的NLS pdes-这些有价值的方法正是我们提出的目标。我们打算与学术界和工业界的光学专家合作，在具有实际意义的方程上开发这些方法，并重点关注技术问题。

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.