SGER: Stimulated Raman Scattering in the Transient Limit

描述瞬态极限下受激拉曼散射的方程是非线性的、可积的。然而，该系统在长距离传播后的物理行为与其他可积系统（如光纤中的光脉冲）有很大不同。在光纤中，光脉冲分解成许多孤子和色散波贡献，而在受激拉曼散射的瞬态状态下，解倾向于依赖于（距离）x（时间）的相似解。这最后一个结果虽然可以从以前的数值结果中推断出来，但直到最近才得到证明，而且它的许多结果还有待阐明。第一个值得关注的问题是第二次斯托克斯生成和有限T2对自相似解的可能实验观测所施加的限制。我们将调查这一现象。我们还将探讨瞬态拉曼效应对限制与Ti：蓝宝石激光器结合使用的拉曼降频器转换效率的影响。我们还将探索类似技术在光纤光栅、双能级系统、光折变材料和其他系统中的应用程度，这些系统要么已经显示出自相似性，要么因为它们具有长期记忆而成为有希望的候选系统。最后，在数学方面，我们将与Tom Seidman合作研究这些解的稳定性，并与其他数学家合作获得频率失谐时自相似情况的渐近公式。

The equations which describe stimulated Raman scattering in the transient limit are nonlinear and integrable. However, the physical behavior of this system after propagating, long distances is quite different from other integrable systems such as light pulses in optical fibers. In optical fibers, the light pulses break up into a number of solitons and a dispersive wave contribution, while in the transient regime of stimulated Raman scattering, the solutions tend toward a similarity solution which depends on (distance) x (time). This last result, while inferrable from previous numerical results, was only recently proved, and many of its consequences remain to be elucidated. The first issue of concern is the limitation which second Stokes generation and finite T2 impose on possible experimental observation of the self- similar solution. We will investigate this phenomenon. We will also explore the impact of the transient Raman effect on limiting the conversion efficiency of Raman frequency downshifters which are used in conjunction with Ti:Sapphire lasers. We will also explore the extent to which similar techniques can be applied to fiber gratings, two-level systems, photorefractive materials, and other systems which have either already been shown to exhibit self- similarity or are promising candidates because they have long-term memory. Finally, on the mathematical front, we will collaborate with Tom Seidman in studying the stability of these solutions and with other mathematicians in obtaining asymptotic formulae for the self-similar case when the frequencies are detuned.