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NEW EXACT SOLUTIONS FOR THE LOADED KORTEWEG-DE VRIES
AND THE LOADED MODIFIED KORTEWEG-DE VRIES BY THE
FUNCTIONAL VARIABLE METHOD
Bazar Babajanov
Doctor of Sciences in Physical and Mathematical Sciences, Department of Applied
Mathematics and Mathematical Physics, Urgench State University
Sadokat Maylieva
Master degree student, Urgench State University
Abstract:
In this paper, we construct exact traveling wave solutions of the
loaded Korteweg-de Vries and the loaded modified Korteweg-de Vries by the
functional variable method. The performance of this method is reliable and effective
and gives the exact solitary and periodic wave solutions. All solutions to these
equations have been examined and 3D graphics of the obtained solutions have been
drawn using the MATLAB program. We get some traveling wave solutions, which
are expressed by the hyperbolic functions and trigonometric functions. The
graphical representations of some obtained solutions are demonstrated to better
understand their physical features, including bell-shaped solitary wave solutions,
singular soliton solutions, and solitary wave solutions of kink type. Our results reveal
that the method is a very effective and straightforward way of formulating the exact
traveling wave solutions of non-linear wave equations arising in mathematical
physics and engineering.
Key words:
the loaded Korteweg-de Vries equation, the loaded modified
Korteweg-de Vries equation, periodic wave solutions, soliton wave solutions,
functional variable method.
The investigation of exact traveling wave solutions to non-linear evolution
equations plays an important role in the study of non-linear physical phenomena.
These equations arise in several fields of science, such as fluid dynamics, physics
of plasmas, biological models, non-linear optics, chemical kinetics, quantum
mechanics, ecological systems, electricity, ocean, and sea. One of the most
important non-linear evolution equations is the Korteweg De Vries (KdV)
equation.
The KdV equation was first observed by John Scott Russell in experiments,
and then Lord Rayleigh and Joseph Boussinesq studied it theoretically. Finally, in
1895, Korteweg and De Vries formulated a model equation to describe the
aforementioned water wave, which helped to prove the existence of solitary
waves. In the mid-1960s, Zabusky and Kruskal discovered the remarkably stable
particle-like behavior of solitary waves. The KdV equation is especially
"Современные тенденции в развитии науки:
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important due to the potential application of different properties of electrostatic
waves in the development of new theories of chemical physics, space
environments, plasma physics, fluid dynamics, astrophysics, optical physics,
nuclear physics, geophysics, dusty plasma, fluid mechanics, and different other
fields of applied physics [1, 2, 3, 4].
In recent years, studying electrostatic waves specifically to discuss different
properties of solitary waves in the field of soliton dynamics has played a
significant role for many researchers and has received considerable attention
from them. The ion acoustic solitary wave is one of the fundamental non-linear
wave phenomena appearing in plasma physics. In 1973, Hans Schamel studied a
modified Korteweg-de Vries equation for ion-acoustic waves. The modified KdV
equation has been applied widely in the molecular chain model, the generalized
elastic solid, and so on [5, 6, 7]. Non-linear interactions between low-hybrid
waves and plasmas can be described well by using the modified KdV equation [8].
In arterial mechanics, a model is widely used in which the artery is
considered as a thin-walled prestressed elastic tube with a variable radius (or
with stenosis), and blood as an ideal fluid [9]. The governing equation that models
weakly nonlinear waves in such fluid-filled elastic tubes is the modified KdV
equation
2
6
( )
0,
t
x
xxx
x
u
u u
u
h t u
−
+
−
=
where
t
- is a scaled coordinate along the axis of the vessel after static
deformation characterizing axisymmetric stenosis on the surface of the arterial
wall.
x
- is a variable that depends on time and coordinates along the axis of the
vessel.
( )
h t
- is a form of stenosis and characterizes the average axial velocity of
the fluid.
We suppose that a form of stenosis
( )
h t
proportional to
(0, )
u
t
and we
consider the loaded KdV and the loaded modified KdV equation
1
6
( ) (0, )
0,
t
x
xxx
x
u
uu
u
t u
t u
−
+
+
=
2
2
12
( ) (0, )
0,
t
x
xxx
x
u
u u
u
t u
t u
−
+
+
=
where
( , )
u x t
is an unknown function,
x
R
,
0
t
,
and
are any constants,
1
( )
t
and
2
( )
t
are the given real continuous functions.
We establish exact traveling wave solutions of the loaded KdV and the
loaded modified KdV 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. The traveling wave solutions obtained via this method
are expressed by hyperbolic functions and trigonometric functions. The graphical
representations of some obtained solutions are demonstrated to better
understand their physical features, including bell-shaped solitary wave solutions,
«Современные тенденции в развитии науки:
перспективы и практика»
105
singular soliton solutions, and solitary wave solutions of kink type. This method
presents wider applicability for handling non-linear wave equations.
References:
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