On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations

In this paper we generalize previous work on the spectral and orbital stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skewsymmetric operator J is singular. We assume that J restricted to the orthogonal complement of its kernel has a bounded inverse. With this assumption and some further genericity conditions we (a) derive an unstable eigenvalue count for the appropriate linearized operator, and (b) show that the spectral stability of the wave implies its orbital (nonlinear) stability, provided there are no purely imaginary eigenvalues with negative Krein signature. We use our theory to investigate the (in)stability of spatially periodic waves to the generalized KdV equation for various power nonlinearities when the perturbation has the same period as that of the wave. Solutions of the integrable modified KdV equation are studied analytically in detail, as well as solutions with small amplitudes for higher-order pure power nonlinearities. We conclude by studying the transverse stability of these solutions when they are considered as planar solutions of the generalized KP-I equation.