New Applications of Soliton Theory in Nonlinear Optics

数学科学：孤子理论在非线性光学中的新应用[摘要]Kaup本项目的目的是研究用集体模描述某些非线性可积系统的方法。集体模式描述只使用几个参数来描述系统的重要特征，并且系统的演化可以通过这几个参数的演化来很好地理解。各种非线性光学系统的当前技术利益，如二次谐波产生，三波参数相互作用，受激拉曼散射和双光子传播，将被研究。我们试图确定可能对这些系统有用的集体模式的形式，然后确定如何从输入数据中收集这些集体模式的参数值。新型的集体模式，称为虚拟孤子，已经在二次谐波产生中被发现。这个项目的目的是开发新的方法来表达描述激光与物质相互作用的复杂方程的解。对于一些简单脉冲和这些脉冲的某些组合，给出这些方程精确解的简单数学表达式是已知的。其中一些脉冲与用于在光纤中传输信息的光脉冲类型相同。在这个项目中，我们将开发一种比目前可用的更显式的形式来表达方程的其他解的方法。在这个过程中，我们将发现有用的脉冲形式的新类别，并扩大易于理解的解决方案的类别。

NSF Award Abstract - DMS-0129714 Mathematical Sciences: New Applications of Soliton Theory in Nonlinear Optics Abstract DMS-0129714 Kaup The purpose of this project is to investigate methods for describing certain nonlinear integrable systems in terms of collective modes. A collective-mode description uses only a few parameters to describe the important features of a system, and the evolution of the system can be well understood in terms of the evolution of only these few parameters. Various nonlinear optical systems of current technological interest, such as second harmonic generation, three-wave parametric interactions, stimulated Raman scattering, and two-photon propagation, will be studied. We seek to determine forms of collective modes that may be useful for these systems, and then to determine how parameter values for these collective modes may be gleaned from input data. New types of collective modes, called virtual solitons, have already been discovered in second harmonic generation.The purpose of this project is to develop new ways to express solutions of the complicated equations describing interactions of laser light with matter. Simple mathematical expressions that give exact solutions of these equations are known for some simple pulses and certain combinations of such pulses. Some of these pulses are the same type as the optical pulses that are used to transmit information in optical fibers. In this project, we will develop ways to express other solutions of the equations in a more explicit form than is currently available. In the process, we will find new classes of useful pulse forms and broaden the class of well-understood solutions.