色散介质中的短脉冲近似
Short Pulses Approximations in Dispersive Media
摘要：
本文导出了具有快速振荡初值的非线性双曲型方程组解的各种近似。我们首先提供误差估计所谓的缓慢变化的包络线，全色散，薛定谔近似的维纳代数，这个功能框架使我们能够精确的条件，这些模型的有效性，我们特别给出了一个严格的证明“实用规则”，作为一个标准的使用缓慢变化的包络线近似（SVEA）。我们还讨论了这些模型的扩展到短脉冲和更一般的大频谱波，如啁啾脉冲。然后，我们推导出并严格证明了一个改进的频率色散的修改后的薛定谔方程。数值计算，然后提出，证实了理论预测。
英文摘要：
We derive various approximations for the solutions of nonlinear hyperbolic systems with fastly oscillating initial data. We first provide error estimates for the so-called slowly varying envelope, full dispersion, and Schrödinger approximations in a Wiener algebra; this functional framework allows us to give precise conditions on the validity of these models; we give in particular a rigorous proof of the ``practical rule'' which serves as a criterion for the use of the slowly varying envelope approximation (SVEA). We also discuss the extension of these models to short pulses and more generally to large spectrum waves, such as chirped pulses. We then derive and justify rigorously a modified Schrödinger equation with improved frequency dispersion. Numerical computations are then presented, which confirm the theoretical predictions.
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