Electron. J. Diff. Eqns., Vol. 2003(2003), No. 06, pp. 1-18.

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

Jaime Angulo Pava

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 $p greater than 1$. 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 $H^2(\mathbb{R})$.

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.

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Jaime Angulo Pava
Department of Mathematics, IMECC-UNICAMP
C.P. 6065, CEP 13083-970-Campinas
Sao Paulo, Brazil
email: angulo@ime.unicamp.br

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