Transverse Orbital Stability of Periodic Traveling Waves for Nonlinear Klein-Gordon Equations

In this paper, we establish the orbital stability of a class of spatially periodic wave train solutions to multidimensional nonlinear Klein-Gordon equations with periodic potential. We show that the orbit generated by the one-dimensional wave train is stable under the flow of the multidimensional equation under perturbations which are, on one hand, coperiodic with respect to the translation or Galilean variable of propagation, and, on the other hand, periodic (but not necessarily coperiodic) with respect to the transverse directions. That is, we show their transverse orbital stability. The class of periodic wave trains under consideration is the family of subluminal rotational waves, which are periodic in the momentum but unbounded in their position.