Electronic Journal of Differential Equations (Jan 2003)
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 $$ u_t+u_{xxxxx}+bu_{xxx}=(G(u,u_x,u_{xx}))_x, $$ where $ G(q,r,s)=F_q(q,r)-rF_{qr}(q,r)-sF_{rr}(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 $H^2(mathbb{R})$ which is based in results obtained by Levandosky. The instability of this class of solitary-wave solutions is determined for $b eq0$, 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 Gonc calves Ribeiro. Moreover, our approach shows that the trajectories used to exhibit instability will be uniformly bounded in $H^2(mathbb{R})$.
