Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators
Counting Unstable Eigenvalues in Hamiltonian Spectral Problems via Commuting Operators
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通过通勤算子计算哈密顿谱问题中的不稳定特征值
DOI：
10.1007/s00220-017-2898-6
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
D. Pelinovsky
中科院分区：
文献类型：
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作者：
M. Haragus;Jin Li;D. Pelinovsky
我们给出了JLin形式的线性算子的不稳定本征值的一般计数结果，其中JLin和L1分别是斜算子和自伴算子。假设存在一个自伴算子K使得该算子J在K处可交换，我们证明了J的不稳定本征值的个数有界于K的非正本征值的个数。作为应用，我们讨论了经典KP-II(Kadomtsev-Petviashvili)方程中一维周期行波的横向稳定性。我们证明了这些一维周期波对于一般的二维有界扰动是横向谱稳定的，其中包括纵向和横向的周期扰动和局域扰动，它们对于双周期扰动是横向线性稳定的。
We present a general counting result for the unstable eigenvalues of linear operators of the formJ Lin whichJandLare skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operatorKsuch that the operatorsJ LandJ Kcommute, we prove that the number of unstable eigenvalues ofJ Lis bounded by the number of nonpositive eigenvalues ofK. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev–Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.