Eur. Phys. J. Plus (2017) 132: 410
https://doi.org/10.1140/epjp/i2017-11709-0

Regular Article

The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom

¹ Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon
² Centre d’Excellence Africain en Technologies de l’Information et de la Communication (CETIC), University of Yaounde I, P.O. Box 812, Yaoundé, Cameroon
³ Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Maroua, P. O. Box 814, Maroua, Cameroon
⁴ Department of Mathematics, California State University-Northridge, 91330-8313, Northridge, CA, USA

^* e-mail: fokostinos@yahoo.fr

Received: 15 June 2017
Accepted: 16 August 2017
Published online: 3 October 2017

Abstract

The goal of this work is to investigate, analytically and numerically, the dynamics of gravity water waves with the effects of the small surface tension and the bottom topography taken into account. Using a third-order perturbative approach of the Boussinesq equation, we obtain a new third-order perturbed Korteweg-de Vries (KdV) equation which includes nonlinear, dispersive, nonlocal and mixed nonlinear-dispersive terms, describing shallow water waves with a non-flat bottom and the surface tension. We show by numerical simulations, for various bottom shapes, that this new third-order perturbed KdV equation can support the propagation of solitary waves, whose profiles strongly depend on the surface tension. In particular, we show that the instability observed in the numerical simulation can be suppressed by the inclusion of small surface tension.