Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 06, pp. 1-18.

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

Author: Jaime Angulo Pava (UNICAMP, Campinas, Brazil)
         
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 $b\neq0$,
 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 Gon\c calves Ribeiro. Moreover, 
 our approach shows that the trajectories used to exhibit instability
 will be uniformly bounded in $H^2(\mathbb{R})$.
 
   
Submitted August 13, 2002. Published January 10, 2003.
Math Subject Classifications: 35B35, 35B40, 35Q51, 76B15, 76B25, 76B55, 76E25.
Key Words: Water wave model; variational methods; solitary waves; instability.