On the instability of solitary-wave solutions for fifth-order water wave models

Angulo Pava, Jaime

Electronic Journal of Differential Equations, Tome 2003 (2003).

Résumé

Summary: 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 $p greater than 1$. 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 $b\neq0$, 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 Goncalves Ribeiro. Moreover, our approach shows that the trajectories used to exhibit instability will be uniformly bounded in $H^2(\mathbb{R})$.

Détail

Classification : 35B35, 35B40, 35Q51, 76B15, 76B25, 76B55, 76E25
Keywords: water wave model, variational methods, solitary waves, instability