分裂步傅里叶方法在非线性薛定谔方程中的平面波稳定性
Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation
AI总结：
本文研究分裂步傅里叶方法在非线性薛定谔方程中对平面波解的长期稳定性，通过线性稳定性条件和非共振条件证明其在高阶Sobolev范数下的稳定性。
摘要：
立方非线性薛定谔方程在 тор 上的平面波解近期已被证明在轨道上是稳定的。在初始数据的小高阶Sobolev范数下，平面波在长时间内保持稳定，这种时间可以延伸到小量参数的任意负幂。本文研究是否分裂步傅里叶方法的数值离散化能够继承这种普遍的长时间稳定性性质。这确实可以在线性稳定性条件和非共振条件下证明。如果时间步长受到CFL条件的限制，则可以验证这两种条件。证明首先使用了哈密顿约简和变换，然后在时间上使用调制傅里叶展开。它为数值解的结构提供了详细的洞察。
英文摘要：
Plane wave solutions to the cubic nonlinear Schrödinger equation on a torus have recently been shown to behave orbitally stable. Under generic perturbations of the initial data that are small in a high-order Sobolev norm, plane waves are stable over long times that extend to arbitrary negative powers of the smallness parameter. The present paper studies the question as to whether numerical discretizations by the split-step Fourier method inherit such a generic long-time stability property. This can indeed be shown under a condition of linear stability and a non-resonance condition. They can both be verified if the time step-size is restricted by a CFL condition in the case of a constant plane wave. The proof first uses a Hamiltonian reduction and transformation and then modulated Fourier expansions in time. It provides detailed insight into the structure of the numerical solution.
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