  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.
