On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations
On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations
复制标题
广义Korteweg-de Vries方程空间周期平稳解的谱和轨道稳定性
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
B. Deconinck
中科院分区：
文献类型：
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作者：
T. Kapitula;B. Deconinck
在本文中，我们推广了以前的工作的频谱和轨道稳定性的波的无限维哈密顿系统，包括那些情况下，反对称运营商\（\mathcal{J}\）是奇异的。我们假设\（\mathcal{J}\）限制到其核的正交补有一个有界逆。有了这个假设和一些进一步的一般性条件，我们（a）得出一个不稳定的本征值计数适当的线性化运营商，和（B）表明，频谱的稳定性波暗示其轨道（非线性）的稳定性，提供了没有纯粹的虚数特征值与负Krein签名。我们用我们的理论研究了当扰动与波的周期相同时，空间周期波对各种幂次非线性的广义KdV方程的（不）稳定性.详细分析了可积修正KdV方程的解，以及高阶纯幂非线性的小振幅解。最后，我们研究了这些解作为广义KP-I方程的平面解时的横向稳定性。
In this paper we generalize previous work on the spectral and orbital stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric operator \(\mathcal{J}\) is singular. We assume that \(\mathcal{J}\) restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higher-order pure power nonlinearities. We conclude by studying the transverse stability of these solutions when they are considered as planar solutions of the generalized KP-I equation.