论文标题
NLS类型方程的Whitham调制理论的严格理由
Rigorous justification of the Whitham modulation theory for equations of NLS type
论文作者
论文摘要
我们研究了周期性行驶波解决方案对非线性Schrödinger类型方程的模量稳定性。特别是，我们证明，由于缓慢的调制近似值所产生的准线性方程式的特性满足了相同的方程式，最多可以变化变化，因为线性化频谱的正常形式越过了来源。这种正常形式取自Arxiv：1910.05392，Leisman等人。通过对约旦链的分析来计算原点附近线性化操作员的光谱。我们使用Whitham的正式调制理论得出调制方程，尤其是应用于平均Lagrangian的变异原理。我们使用Leisman等人严格理论中假定的通用条件。指导调制方程的均匀化。由于特性方程与线性理论的正常形式之间的一致性，我们表明，Whitham系统的双曲度是基础波的调节稳定性的必要条件。
We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi-linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from arXiv:1910.05392, where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leisman et al. to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.
