Research Article
Travelling Wave Solutions for the Coupled IBq Equations by
Using the tanh-coth Method
Omer Faruk Gozukizil
1
and Samil Akcagil
2
1
Department of Mathematics, Sakarya University, 9054187 Sakarya, Turkey
2
Vocational School of Pazaryeri, Bilecik S ¸eyh Edebali University, 9011800 Bilecik, Turkey
Correspondence should be addressed to Omer Faruk Gozukizil; farukg@sakarya.edu.tr
Received 4 May 2013; Accepted 9 December 2013; Published 19 January 2014
Academic Editor: Ch. Tsitouras
Copyright © 2014 O. F. Gozukizil and S. Akcagil. Tis 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.
Based on the availability of symbolic computation, the tanh-coth method is used to obtain a number of travelling wave solutions
for several coupled improved Boussinesq equations. Te abundant new solutions can be seen as improvement of the previously
known data. Te obtained results in this work also demonstrate the efciency of the method.
1. Introduction
We consider the following two coupled improved Boussinesq
(IMBq) equations of Sobolev type:
−
2
−
=(,)
, ∈ R,>0,
−
2
−
=(,)
, ∈ R,>0,
(1)
where and are given nonlinear functions, (,) and
(,) are unknown functions, and is a constant that has
been derived to describe bidirectional wave propagation in
several studies, for instance, in a Toda lattice model with a
transversal degree of freedom, in a two-layered lattice model,
and in a diatomic lattice. D´ e Godefroy has studied (1) as the
Cauchy problem under certain conditions and showed that
the solution for the Cauchy problem of this system blows up
in fnite time [1]. Wang and Li have considered the Cauchy
problem for (1), proved the existence and uniqueness of the
global solution, and given sufcient conditions of blow-up
of the solution in fnite time by convex methods [2]. Te
Cauchy problem for (1) has been studied and established the
conditions for the global existence and fnite-time blow-up of
solutions in Sobolev spaces
×
for >1/2 [3]. For more
information, we refer the reader to [3] and references therein.
Chen and Zhang have considered the initial boundary
value problem for the system of the generalized IMBq type
equations:
−
−
=()
, 0<<
1
,>0,
−
=()
, 0<<
1
,>0,
(2)
where (,) and (,) are unknown functions, ,>0
are constants, and and are the given nonlinear functions.
As a result, they have proved the existence and uniqueness
of the global generalized solution and the global classical
solution [4].
Rosenau is concerned with the problem of how to
describe the dynamics of a dense lattice via the system
=(+2)
+
1
12
,
=(2+
2
+
2
)
+
1
12
,
(3)
where ,>0 are constants, and pointed out that (3) was a
convenient vehicle to study the dynamics of a dense lattice [5].
In [6], a transversal degree of freedom was introduced
in the Toda lattice. For diferent order of magnitude of the
longitudinal and transversal strains, coupled and uncoupled
Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2014, Article ID 486269, 14 pages
http://dx.doi.org/10.1155/2014/486269