The Hamiltonian-Krein (instability) index is concerned with determining the number of eigenvalues with positive real part for the Hamiltonian eigenvalue problem JLu=λu, 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. © 2013 by the Massachusetts Institute of Technology.
