Stabilization of soliton instabilities by higher order dispersion: KdV-type equations

The stability of the solitons of the fifth order KdV-type equations with arbitrary power nonlinearities is studied. We show that a sufficient condition of the soliton stability with respect to small perturbations is a minimum of the Hamiltonian, constrained by the constancy of the momentum. Some other forms of this condition and consequences from them are derived. The behavior of the Hamiltonian for a class of finite soliton perturbations is investigated. The obtained results demonstrate that higer order dispersion, under certain conditions, stabilizes soliton instabilities, which is in agreement with numerical experiments.