On the spectra of periodic waves for infinite-dimensional Hamiltonian systems

We consider the problem of determining the spectrum for the linearization of an infinite-dimensional Hamiltonian system about a spatially periodic traveling wave. By using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators JγLγJγLγ, where JγJγ is skew-symmetric with bounded inverse and LγLγ is symmetric with compact inverse. Our main result relates the number of unstable eigenvalues of the operator JγLγJγLγ to the number of negative eigenvalues of the symmetric operator LγLγ. The compactness of the resolvent operators allows us to greatly simplify the proofs, as compared to those where similar results are obtained for linearizations about localized waves. The theoretical results are general, and apply to a larger class of problems than those considered herein. The theory is applied to a study of the spectra associated with periodic and quasi-periodic solutions to the nonlinear Schrödinger equation, as well as periodic solutions to the generalized Korteweg–de Vries equation with power nonlinearity.