On the Instability of Solitary-wave Solutions for Fifth-order Water Wave Models

Abstract

This work presents new results about the instability of solitary-wave solutions to a generalized fifth-order Korteweg-deVries equation of the form uₜ + uₓₓₓₓₓ + buₓₓₓ = (G(u, uₓ>, uₓₓ))ₓ, where G(q, r, s) = Fq(q, r) - rFqr(q, r) - sFrr(q, r) for some F(q, r) which is homogeneous of degree p + 1 for some p > 1. This model arises, for example, in the mathematical description of phenomena in water waves and magneto-sound propagation in plasma. The existence of a class of solitary-wave solutions is obtained by solving a constrained minimization problem in H2(ℝ) which is based in results obtained by Levandosky. The instability of this class of solitary-wave solutions is determined for b ≠ 0, and it is obtained by making use of the variational characterization of the solitary waves and a modification of the theories of instability established by Shatah & Strauss, Bona & Souganidis & Strauss and Gonçalves Ribeiro. Moreover, our approach shows that trajectories used to exhibit instability will be uniformly bounded in H2(ℝ).