A Hamiltonian-Krein (instability) index theory for KdV-like eigenvalue problems

Abstract

The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem $JLu=\lambda u$, 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 BBM-type problems.

submitted

References in corpus (2)

• Uniqueness and Nondegeneracy of Ground States for $(-\Delta {)}^{s}Q+Q-Q^{\alpha +1}=0$ in $\mathbb{R}$
• An index theorem for the stability of periodic traveling waves of KdV type

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