Mathematics and Computers in Simulation 55 (2001) 393–405 Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation Prabir Daripa ∗ , Ranjan K. Dash Department of Mathematics, Texas A&M University, College Station, TX 77843, USA Received 1 October 2000; accepted 31 October 2000 Abstract We study the singularly perturbed (sixth-order) Boussinesq equation recently introduced by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159]. This equation describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number less than but very close to 1/3. On the basis of far-field analyses and heuristic 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 methods originally devised to obtain this type of weakly non-local solitary wave solutions of the singularly perturbed (fifth-order) KdV equation, we obtain weakly non-local solitary wave solutions of the singularly perturbed (sixth-order) Boussinesq equation and provide estimates of the amplitude of oscillations which persist in the far-field. © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Capillary-gravity waves; Singularly perturbed Boussinesq equation; Weakly non-local solitary waves; Asymptotics beyond all orders; Pseudospectral method 1. Introduction In this paper we study the singularly perturbed (sixth-order) Boussinesq equation η tt = η xx + (η 2 ) xx + η xxxx + ǫ 2 η xxxxxx (1) where ǫ is a small parameter. This equation was originally introduced by Daripa and Hua [1] as a regularization of the ill-posed classical (fourth-order) Boussinesq equation which corresponds to ǫ = 0 in Eq. (1). It is well-known that the fourth-order Boussinesq equation possesses the traveling-solitary-wave solutions (see [2,3]). The physical relevance of Eq. (1) in the context of water waves was recently addressed by Dash and Daripa [4]. It was shown that this equation actually describes the bi-directional propagation of small ∗ Corresponding author. E-mail address: prabir.daripa@math.tamu.edu (P. Daripa). 0378-4754/01/$20.00 © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0378-4754(00)00288-3