A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems
A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems
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孤立波到 KdV 类特征值问题的 Hamiltonian-Krein（不稳定性）指数理论
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
A. Stefanov
中科院分区：
文献类型：
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作者：
T. Kapitula;A. Stefanov
哈密顿-克莱因(不稳定性)指标用于确定哈密顿本征值问题JLu=λu的正实部本征值的个数，其中J是斜对称的，而L是自伴的。如果J有一个有界逆，则该指标是良好建立的，它是由L约束作用于某个有限余维子空间的算子的负本征值的个数给出的。有一类重要的问题，即KdV型问题，J没有有界逆。在本文中，我们克服了这一困难，得到了KdV型特征值问题的指标。利用该指数讨论了KdV类问题和Benjamin-Bona-Mahony类问题同宿行波谱的稳定性。
The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem JLu=λu , where J is skew‐symmetric and L is self‐adjoint. If J has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator L constrained to act on some finite‐codimensional subspace. There is an important class of problems—namely, those of KdV‐type—for which J does not have a bounded inverse. In this paper, we overcome this difficulty and derive the index for eigenvalue problems of KdV‐type. We use the index to discuss the spectral stability of homoclinic traveling waves for KdV‐like problems and Benjamin—Bona—Mahony‐type problems.