On the Instability of Solitary-wave Solutions for Fifth-order Water Wave Models
Angulo Pava, Jaime
Southwest Texas State University, Department of Mathematics
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).
Water wave model, Variational methods, Solitary waves, Instability
Angulo Pava, J. (2003). On the instability of solitary-wave solutions for fifth-order water wave models. <i>Electronic Journal of Differential Equations, 2003</i>(06), pp. 1-18.