Stability of Small Periodic Waves in Fractional KdV-Type Equations
Stability of Small Periodic Waves in Fractional KdV-Type Equations
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分数阶KdV型方程中小周期波的稳定性
DOI：
10.1137/120894397
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
Mathew A. Johnson
中科院分区：
文献类型：
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作者：
Mathew A. Johnson
考虑了色散和非线性对KdV型非线性偏微分方程周期行波解稳定性的影响，包括广义KdV方程和Benjamin-Ono方程.在这项调查中，我们考虑的谱稳定性的平衡态的小扰动所产生的这种解决方案。我们的分析的一个关键特征是一个非局部的Floquet理论，适合于分析相关的线性化运营商的$L^2（\mathbb{R}）$频谱的发展。然后，使用谱微扰理论，我们推导出的非线性的功率和分数色散算子的符号，确定的频谱稳定性和不稳定性的任意小的本地化扰动之间的关系。
We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDEs of KdV type, including generalized KdV and Benjamin--Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonlocal Floquet-like theory that is suitable to analyze the $L^2(\mathbb{R})$ spectrum of the associated linearized operators. Using spectral perturbation theory then, we derive a relationship between the power of the nonlinearity and the symbol of the fractional dispersive operator that determines the spectral stability and instability to arbitrary small localized perturbations.