Ta'lim innovatsiyasi va integratsiyasi
44-son_2-to’plam_May-2025
ISSN: 3030-3621
16
A GENERALIZED DIRECT METHODS FOR THE LOADED NONLINEAR
DEGASPERIS-PROCESI EQUATION
M.M.Xasanov*, Sh.Sh.Omonov**
*Urgench State University, Urgench, Uzbekistan
**Tashkent State University of Economic, Tashkent, Uzbekistan
e-mail:
Abstract:
This paper is studied to finding the traveling wave solutions of the
loaded nonlinear Degasperis-Procesi equation.
By using the dynamical system theory
the nonlinear Degasperis-Processi equation are studied .The bounded travelling wave
solutions such as peakons are analytically described. The loaded Degasperis-Procesi
equation is converted to the ordinary differential equation which are solved for all
possible soliton solutions of Degasperis-Procesi equation. We construct exact
travelling wave solution for loaded nonlinear Degasperis-Procesi equation,and the
obtained solution agrees well with the previously known result.
Keywords:
Loaded nonlinear Degasperis-Procesi equation,Travelling wave
solution. AMS Subject Classification 2010:
1.
Introduction.
Degasperis and Procesi [1]showed, by the use of the method of asymptotic
integrability, that the PDE
1
t
xxxt
x
x
xx
xxx
u
u
b
uu
bu u
uu
(1)
cannot be completely integrable unless
2
b
or
3
b
.The case
2
b
is the following
Camassa-Holm (CH) shallow water equation (see [2])
3
2
t
xxt
x
x
xx
x
u
u
uu
u u
uu
(2)
which is well known to be integrable and topossess multi-peakon solutions. The
case
3
b
is the following Degasperis-Procesi (DP) shallow water equation
4
3
t
xxt
x
x
xx
xxx
u
u
uu
u u
uu
(3)
Although, the DP equation (3) has a similar form to the Ch equation (2), two
equations are pretty different. For two equations, the different isospectral problem and
the fact that there is no simple transformation of equation (3) into equation (2) imply
that equation (3) is different from equation (2) in the integrable structures and the form
of the conservation laws. The DP equation (3) is very interesting as it is an integrable
shallow water equation and presents a quite rich structure. Degasperis, Holm and Hone
[3-5]proved that equation (3) is integrable by constructing its lax pair, and admits
multi-peakon solutions, and explained connection with a negative flow in the Kaup-
kupershmidt hierarchy via a neciprocal transformation. Landmark and Szmigielski [6]
Ta'lim innovatsiyasi va integratsiyasi
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ISSN: 3030-3621
17
presented an inverse scattering approach for computing n-peakon solutions of the
equation (3). The blow-upphenomenon of equation (3) was discussed and the global
existence of the solution was proved in [7]. In [8,9] the Cauchy problem for equation
(3) was demonstrated. Much work on the DP equation (3) has been done [10-11].
It is well known that nonlinear phenomena exist everywhere. For example, they
exist influid physics, condensed matter physics ,biophysics, plasma physics, quantum
field theory ,particle physics and nonlinear optics etc. they also connect with our
everyday’s life. Generally speaking, nonlinear phenomena can be described by
nonlinear partial differential equations .
It is known that loaded differential equations have great practical applications.
In the literature [12-16], loaded differential equations are typically called equations
containing in the coefficients or in the right-hand side any functionals of the solution,
in particular, the values of the solution or its derivatives on many folds of lower
dimension. These types of equations were explored in the works of N.N. Nazarov and
N.N. Kochin. However, they did not use the term “loaded equation”. At first, the term
has been used in the works of A.M. Nakhushev, which the most general definition of a
loaded equation is given and various loaded equations are classified in details[16]. For
instance, loaded differential, integral, integro-differential, functional equations etc.,
and numerous applications are described.
Alternatively, the
(
/
)
G G
- expansion method [17-28] is also effective in finding
traveling wave solutions of nonlinear evolution equations. Integration of the loaded
modified Korteweg-de Vries (mKdV) equation in the class of periodic functions is
studied in [29].
(
/
)
G G
-expansion method wasused for the integrations of loaded
Korteweg-de Vries (KdV) equation and the loaded modified Korteweg-de Vries
(mKdV) equation in [30,31].
In this paper, the solution of the loaded nonlinear Degasperis-Procesi equation
is studied by usage of direct method.
Let us consider the following loaded nonlinear Degasperis-Procesi equation
4
3
( ) (0, )(4
)
t
xxt
x
x
xx
xxx
x
x
u
u
uu
u u
uu
t u
t
uu
сu
(4)
where
( , )
u x t
is an unknown function,
x
R
,
0
t
,
( )
t
-
is the given real continuous
function.
2.The loaded nonlinear Degasperis-Procesi Equation
Assume the solution of
loaded nonlinear Degasperis - Procesi equation as
(
),
u
x
ct
,
x ct
( )
u
Now, we determine partial derivatives
(
),
u
x
ct
derivative with respect to
variables
x
and
t
.
'
,
,
t
u x t
c
'
,
,
x
u
x t
''
,
,
xx
u
x t
'''
,
,
xxx
u
x t
'''
,
,
xxt
u
x t
c
(5)
Ta'lim innovatsiyasi va integratsiyasi
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ISSN: 3030-3621
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Then, by using above equation (4) is transformed into the following ordinary
differential equation
'
'''
'
'
''
'''
'
'
4
3
( ) (0, )(4
)
0.
c
c
t u
t
c
(6)
By one time integrating with respect to ξ, equation (6) becomes
''
2
'2
''
2
2
( ) (0, )(2
)
0,
c
c
t u
t
c
''
'2
2
(
)
(2
)
( ) (0, )(2
).
c
c
t u
t
c
(7)
Then we get the following substitutions
'
,
p
''
'
,
pp
( ),
p
p
equation (7) using substitutions made following visible
'
2
2
(
)
(2
)
( ) (0, )(2
),
c
pp
p
c
t u
t
c
and also by using following substitution
2
,
p
z
'
'
,
2
z
pp
we take the first order linear differential equation
'
2
2
(
)
(2
)
( ) (0, )(2
),
2
z
c
z
c
t u
t
c
2
'
2
(2
)
( ) (0, )(2
)
2
2
,
(
)
(
)
(
)
c
t u
t
c
z
z
c
c
c
(8)
and to find its solution, we perform the following calculations:
2
'
2
( ( ) (0, ) 1)(2
)
2
(
)
(
)
t u
t
c
z
z
c
c
2
2
1
2( ( ) (0, ) 1) (
)(2
)
(
)
z
k
t u
t
c
c
d
c
2
2
3
2
1
2( ( ) (0, ) 1) (3
2
)
(
)
z
k
t u
t
c
c
d
c
3
2
2
4
2
1
3
2
2( ( ) (0, ) 1)(
)
(
)
3
2
4
c
c
z
k
t u
t
c
if
0
k
,
z
function becomes like this.
2
2
3
4
2
( ( ) (0, ) 1)(
2
)
,
(
)
t u
t
c
c
z
c
2
2
2
2
2
(1
( ) (0, ))(
2
)
,
(
2
)
t u
t
c
c
z
c
c
2
(1
( ) (0, )),
z
t u
t
2
2
(1
( ) (0, )),
p
t u
t
1
( ) (0, ),
p
t u
t
Ta'lim innovatsiyasi va integratsiyasi
44-son_2-to’plam_May-2025
ISSN: 3030-3621
19
1
( ) (0, ),
d
t u
t
d
1
( ) (0, )
1
( ) (0, ) .
d
t u
t dx
c
t u
t dt
Now we find
by integrating both sides of the equation:
0
1
( ) (0, )
1
( ) (0, )
,
t
d
t u
t x
c
s u
s ds
0
ln
1
( ) (0, )
1
( ) (0, )
,
t
t u
t x
c
s u
s ds
C
0
1
( ) (0, )
1
( ) (0, )
.
t
t u
t x c
s u
s ds
Ce
Substituting equation
into
( , )
u x t
,we obtain the solution of
,
u x t
0
1
( ) (0, )
1
( ) (0, )
( , )
.
t
t u
t x c
s u
s ds
u x t
e
(9)
Result
.1) from
( , )
u x t
on
0
x
ni (9) we find the function
(0, )
u
t
.
2) Substituting the last function
(0, )
u
t
into formula (9), we get the solution of
loaded nonlinear Degasperis-Procesi equation.
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