Eur. Phys. J. Plus (2017) 132: 518
https://doi.org/10.1140/epjp/i2017-11755-6

Regular Article

Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its solitary-wave solutions via mathematical methods

¹ Mathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
² Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

^* e-mail: Aly742001@yahoo.com

Received: 21 June 2017
Accepted: 24 October 2017
Published online: 12 December 2017

Abstract

In this study, we presented the problem formulations of models for internal solitary waves in a stratified shear flow with a free surface. The nonlinear higher order of extended KdV equations for the free surface displacement is generated. We derived the coefficients of the nonlinear higher-order extended KdV equation in terms of integrals of the modal function for the linear long-wave theory. The wave amplitude potential and the fluid pressure of the extended KdV equation in the form of solitary-wave solutions are deduced. We discussed and analyzed the stability of the obtained solutions and the movement role of the waves by making graphs of the exact solutions.