On the orbital (in)stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations

In this paper we generalize previous work on the stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric 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 show that the linear 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. Different solutions of the integrable modified KdV equation are studied analytically in detail, while numerical computations come to our aid for the nonintegrable cases with a fifthand sixth-order nonlinearity. The stability question for KdV has been answered when the period of the perturbation is the same as that of the underlying cnoidal wave. However, up until now the question of the orbital stability of these waves with respect to periodic perturbations whose period is an integer multiple of the wave period was still open, as in this case there are eigenvalues with negative Krein signature, i.e., the wave is not a local minimizer of a constrained energy. By using the integrable structure associated with KdV we are able to show that these energetically unstable waves are indeed orbitally stable.