Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation

We study the singularly perturbed (sixth-order) Boussinesq equation recentl y introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. This e quation describes the bi-directional propagation of small amplitude and lon g capillary-gravity waves on the surface of shallow water for bond number l ess than but very close to 1/3. On the basis of far-field analyses and heur istic arguments, we show that the traveling wave solutions of this equation are weakly non-local solitary waves characterized by small amplitude fast oscillations in the far-field. Using various analytical and numerical metho ds originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation, we obtai n weakly non-local solitary wave solutions of the singularly perturbed (six th-order) Boussinesq equation and provide estimates of the amplitude of osc illations which persist in the far-field. (C) 2001 IMACS. Published by Else vier Science B.V. All rights reserved.