Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2013, Article ID 468206, 8 pages
http://dx.doi.org/10.1155/2013/468206
Research Article
Solitary Wave Solutions of the Boussinesq Equation and
Its Improved Form
Reza Abazari
1
and Adem KJlJçman
2
1
Young Researchers and Elite Club, Ardabil Branch, Islamic Azad University, P.O. Box 5616954184, Ardabil, Iran
2
Department of Mathematics, Institute of Mathematical Research, University Putra Malaysia (UPM), 43400 Serdang, Malaysia
Correspondence should be addressed to Reza Abazari; abazari-r@uma.ac.ir
Received 17 July 2013; Revised 5 September 2013; Accepted 6 September 2013
Academic Editor: Guo-Cheng Wu
Copyright © 2013 R. Abazari and A. Kılıc¸man. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
is paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011) and (Kılıcman
and Abazari, 2012), that focuses on the application of (
/)-expansion method with the aid of Maple to construct more general
exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general
exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientist Joseph Valentin Boussinesq
(1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude.
Our work is motivated by the fact that the (
/)-expansion method provides not only more general forms of solutions but also
periodic, solitary waves and rational solutions. e method appears to be easier and faster by means of a symbolic computation.
1. Introduction
In the recent five decades, a new direction related to the
investigation of nonlinear evolution equations (NLEEs) and
processes has been actively developing in various areas of
sciences. Nonlinear evolution equations have been the impor-
tant subject of study in various branches of mathematical-
physical sciences such as physics, fluid mechanics, and chem-
istry. e analytical solutions of NLEEs are of fundamental
importance, since many of mathematical-physical models are
described by NLEEs. Among the possible solutions to NLEEs,
certain special form solutions may depend only on a single
combination of variables such as solitons. In mathematics and
physics, a soliton is a self reinforcing solitary wave, a wave
packet or pulse, that maintains its shape while it travels at con-
stant speed. Solitons are caused by a cancelation of nonlinear
and dispersive effects in the medium. e term “dispersive
effects” refers to a property of certain systems where the speed
of the waves varies according to frequency. Solitons arise
as the solutions of a widespread class of weakly nonlinear
dispersive partial differential equations describing physical
systems. e soliton phenomenon was first described by John
Scott Russell (1808–1882) who observed a solitary wave in the
Union Canal in Scotland. He reproduced the phenomenon
in a wave tank and named it the “wave of translation” (also
known as solitary wave or soliton) [1]. e soliton solutions
are typically obtained by means of the inverse scattering
transform [2] and be in dept their stability to the integrability
of the field equations.
In fluid mechanics, the Boussinesq approximation for
water waves is an approximation valid for weakly nonlinear
and fairly long waves. e approximation is named aſter
Joseph Valentin Boussinesq (1842–1929), who first derived
them in response to the observation by John Scott Russell of
the wave of translation [3, 4]. According to the 1872 paper of
Boussinesq, for water waves on an incompressible fluid and
irrotational flow in the (, ) plane, the boundary conditions
at the free surface elevation = (, ) are
+ k
− w = 0,
+
1
2
(k
2
+ w
2
) + = 0,
(1)
where k is the horizontal flow velocity component, k =
/, w is the vertical flow velocity component, w = /,