A Hamiltonian-Krein (instability) index theory for 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 $J L u=\lambda u$, 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.