Resonant One Dimensional Nonlinear Geometric Optics
Resonant One Dimensional Nonlinear Geometric Optics
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DOI：
10.1006/jfan.1993.1065
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发表时间：
1993-05
影响因子：
1.7
通讯作者：
J. Joly;G. Métivier;J. Rauch
中科院分区：
文献类型：
--
作者：
J. Joly;G. Métivier;J. Rauch
摘要 本文研究了一维空间中振荡波列的存在性及其共振相互作用，严格证明了弱非线性光学相应展开式的有效性。我们考虑半线性和拟线性系统，后者在激波形成之前。该研究的一些重要特点如下。 (1) 我们证明了具有弱非线性光学控制的渐近展开式的精确解族的存在性。允许使用具有可变系数、非恒定背景场和非线性相位的方程。我们的弱横向假设使我们能够在甚至以前不存在正式理论的情况下证明扩展的合理性。 (2)我们对共振进行了详细的研究。与这种共振相关的几何形状与平面网理论有关。 (3)我们研究轮廓的光滑度。它们的规律性由类似于描述一维奇点传播的总和定律决定。 (4) 扩展被证明是合理的，直至轮廓的分解，该分解与精确解的适当定义的分解一致。
Abstract In this paper, we study the existence and resonant interaction of oscillatory wave trains in one space dimension, giving a rigorous proof of the validity of the corresponding expansions of weakly nonlinear optics. We consider both semilinear and quasilinear systems, the latter before shock formation. Some important features of the study are the following. (1) We prove the existence of families of exact solutions which have asymptotic expansions governed by weakly nonlinear optics. Equations with variable coefficients, nonconstant background fields and nonlinear phases are permitted. Our weak transversality hypotheses allow us to justify expansions where even a formal theory did not exist before. (2) We make a detailed study of resonances. The geometry associated with such resonances is related to the theory of planar webs. (3) We study the smoothness of the profiles. Their regularity is ruled by a sum law analogous to that describing the propagation of singularities in one dimension. (4) The expansions are justified up to the breakdown of the profiles which coincides with a suitably defined breakdown for exact solutions.