Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation

In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg-de Vries (gKdV) equation ut = uxxx + f(u)x. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solution. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case when f(u) = u (the Korteweg-de Vries equation) and in a neighborhood of the solitary wave if f(u) = u for some integer p ≥ 1.