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The modeling of unidirectional propagation of long water waves in dispersive media is presented. The Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are derived from water waves models. New traveling solutions of the KdV and BBM equations are obtained by implementing the extended direct algebraic and extended sech-tanh methods. The stability of the obtained traveling solutions is analyzed and discussed.

Many nonlinear evolution equations are playing important role in the analysis of some phenomena and including ion acoustic waves in plasmas, dust acoustic solitary structures in magnetized dusty plasmas, and electromagnetic waves in size-quantized films [

The Benjamin-Bona-Mahony (BBM) equation is well known in physical applications [

The BBM equation has been investigated as a regularized version of the KdV equation for shallow water waves [

The main mathematical difference between KdV and BBM models can be most readily appreciated by comparing the dispersion relation for the respective linearized equations. It can be easily seen that these relations are comparable only for small wave numbers and they generate drastically different responses to short waves. This is one of the reasons why, whereas existence and regularity theory for the KdV equation is difficult, the theory of the BBM equation is comparatively simple [

This paper is organized as follows: an introduction is in Section

In water wave equations, a two-dimensional inviscid, incompressible fluid with constant gravitational field is considered. The physical parameters are scaled into the definition of space,

The progressive wave solution of the first order system is

The derivation of the KdV equation in [

The following is given nonlinear partial differential equations (BBM and KdV equations) with two variables

We introduce an independent variable, where

We suppose that

Equating the highest-order nonlinear term and highest-order linear partial derivative in (

Setting the coefficients of (

Using Mathematica and Wu

We will employ the proposed methods to solve the KdV equation:

Figure

Travelling waves solutions (

Figure

Travelling waves solutions (

The stability of soliton solution is stable at

by comparing them with the coefficients of

Figure

Travelling waves solutions (

This soliton solution is stable if

Figure

Travelling waves solutions (

Figure

This soliton solution is stable if

The Benjamin-Bona-Mahony equation (

Figure

Travelling waves solutions (

Figure

Travelling waves solutions (

By implementing the extended direct algebraic and modified sech-tanh methods, we presented new traveling wave solutions of the KdV and BBM equations. We obtained the water wave velocity potential of KdV equation in periodic form and bright and dark solitary wave solutions by using the extended direct algebraic method. Using the modified sech-tanh method, the water wave velocity of KdV equation in form of bright and dark solitary wave solutions. The water wave velocity potentials of BBM equation are deduced in form of dark solitary wave solutions. The structures of the obtained solutions are distinct and stable.

The authors declare that there is no conflict of interests regarding the publication of this paper.