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A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV‐Like Eigenvalue Problems
Author(s) -
Kapitula Todd,
Stefanov Atanas
Publication year - 2014
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12031
Subject(s) - mathematics , korteweg–de vries equation , eigenvalues and eigenvectors , bounded function , hamiltonian (control theory) , mathematical analysis , inverse , homoclinic orbit , hamiltonian system , instability , mathematical physics , pure mathematics , nonlinear system , physics , mathematical optimization , quantum mechanics , bifurcation , geometry
The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem J L u = λ u , 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.