"Современные тенденции в развитии науки:
перспективы и практика"
106
SOLITON AND PERIODIC WAVE SOLUTIONS OF THE LOADED
NONLINEAR EVOLUTION EQUATIONS
Fakhriddin Abdikarimov
Phd student, Khorezm Mamun Academy
Sadokat Maylieva
Master degree student, Urgench State University
Abstract:
In this article, we establish new traveling wave solutions for the
loaded Benjamin-Bona-Mahony and the loaded modified Benjamin-Bona-Mahony
equation by the functional variable method. The performance of this method is
reliable and effective and gives the exact solitary wave solutions and periodic wave
solutions. All solutions of these equations have been examined and three-
dimensional graphics of the obtained solutions have been drawn by using the
MATLAB program. We get some traveling wave solutions, which are expressed by
the hyperbolic functions and trigonometric functions. This method is effective in
finding exact solutions of many other similar equations.
Key words:
loaded Benjamin-Bona-Mahony equation, loaded modified
Benjamin-Bona-Mahony equation, hyperbolic functions, trigonometric functions,
periodic wave solutions, solitary wave solutions, functional variable method.
Benjamin-Bona-Mahony (BBM) equation is well known in the analysis of
the surface waves of long wavelength in liquids, hydromagnetic waves in a cold
plasma, acoustic-gravity waves in compressible fluids, and acoustic waves in
harmonic crystals and it describes the model for propagation of long waves which
incorporates nonlinear and dissipative effects [1]. In the last two decades, various
versions of the BBM equation have been investigated in the literature [2].
In 1972, Benjamin, Bona, and Mahony formulated a model equation for the
unidirectional propagation of small-amplitude long waves on the surface of water
in a channel [3]. A general form of the BBM equation is
0,
x
t
x
txx
u
u
uu
u
+ −
−
=
where
( , )
u x t
is an unknown function,
x
R
,
0
t
,
is any constant.
The BBM equation has been investigated as a regularized version of the KdV
equation for shallow water waves [4]. In certain theoretical investigations the
equation is studied as a model for long waves and from the standpoint of existence
and stability, the equation offers considerable technical advantages over the KdV
equation [5]. In addition to shallow water waves, the equation applies to the study
of drift waves in plasma or the Rossby waves in rotating fluids. Under certain
«Современные тенденции в развитии науки:
перспективы и практика»
107
conditions, it also provides a model of one-dimensional transmitted waves.
The modified Benjamin-Bona-Mahony equation is a special type of the BBM
equation. By changing the nonlinear term of the form
(
2)
n
x
u u n
=
the new
modified form is obtained as follows:
2
0,
x
t
x
txx
u
u
u u
u
+ −
−
=
BBM equation can be solved by many methods. This equation is solved by
(
)
G / G
- the expansion method [6], the exp-function method [7, 8], the homotopy
perturbation method [9, 10], and the variation iteration method [11]. Zabusky
and Kruskal investigated the interaction of solitary waves and the recurrence of
initial states [12]. The Adomian decomposition method is another method to
design some of the exact solitary wave solutions of the generalized form of the
BBM equation [13]. Besides the analytical and exact solutions of the BBM
equation, many numerical techniques from different families are developed and
implemented for the numerical solutions to various evolution problems for the
BBM equation [14, 15].
In this article, we consider the following the loaded BBM equation and the
loaded modified BBM equation
1
( ) (0, )
0,
x
t
x
txx
x
u
u
uu
u
t u
t u
+ −
−
+
=
2
2
( ) (0, )
0,
x
t
x
txx
x
u
u
u u
u
t u
t u
+ −
−
+
=
where
( , )
u x t
is an unknown function,
x
R
,
0
t
,
and
are constants,
1
( )
t
and
2
( )
t
are the given real continuous functions.
We construct exact travelling wave solutions of the loaded BBM equation
and modified BBM equation by the functional variable method. All solutions of
these equations have been examined and three-dimensional graphics of the
obtained solutions have been drawn by using the MATLAB program. We get some
traveling wave solutions, which are expressed by the hyperbolic functions and
trigonometric functions. The functional variable method is flexible, reliable and
straightforward to find solutions of some nonlinear evolution equations arising in
engineering and science.
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Benjamin-Bona-Mahony equation, Communications in Nonlinear Science and
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"Современные тенденции в развитии науки:
перспективы и практика"
108
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