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A Hamiltonian–Krein (Instability) Index Theory for Solitary Waves to KdV-Like Eigenvalue Problems



The Hamiltonian–Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem inline image, where inline image is skew-symmetric and inline image is self-adjoint. If inline image has a bounded inverse the index is well established, and it is given by the number of negative eigenvalues of the operator inline image constrained to act on some finite-codimensional subspace. There is an important class of problems—namely, those of KdV-type—for which inline image 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.

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