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.