Relative Morse indices, Fredholm indices, and group velocities

We discuss Fredholm properties of the linearization of partial differential equations on cylindrical domains about travelling and modulated waves. We show that the Fredholm index of each such linearization is given by a relative Morse index which depends only on the asymptotic coefficients. Several strategies are outlined that help to compute relative Morse indices using crossing numbers of spatial eigenvalues, and the results are applied to prove the existence of small viscous shock waves in hyperbolic conservation laws upon adding small localized time-periodic source terms.