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.
