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In this paper, we study the orbital stability for a four-parameter family ofperiodic stationary traveling wave solutions to the generalized Korteweg-deVries equation. In particular, we derive sufficient conditions for such asolution to be orbitally stable in terms of the Hessian of the classical actionof the corresponding traveling wave ordinary differential equation restrictedto the manifold of periodic traveling wave solution. We show this condition isequivalent to the solution being spectrally stable with respect toperturbations of the same period in the case of the Korteweg-de Vries equation,and in neighborhoods of the homoclinic and equilibrium solutions in the case ofa power-law nonlinearity.

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