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

This work presents new results about the instability of solitarywave 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 magnetosound 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 6= 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 the trajectories used to exhibit instability will be uniformly bounded in H2(R).