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«Zamonaviy dunyoda amaliy fanlar: muammolar va
yechimlar» nomli ilmiy, masofaviy, onlayn konferensiya
INTEGRATION OF THE NONLINEAR SCHRÖDINGER EQUATION
WITH SELF-CONSISTENT SOURCE VIA INVERSE SCATTERING
METHOD
Bekdurdiyev Zufarbek Shodlik ugli
Master student of Urgench State University
https://doi.org/10.5281/zenodo.6629053
Annotation.
In this work is shown that the “finite density” solution of the
nonlinear Schrodinger equation with self-consistent source, can be found by the
inverse scattering problem for the Dirac’s type operator.
Introduction.
The nonlinear Schrödinger equation (NSE)
0
2
2
xx
t
u
u
u
iu
,
const
is used in many areas of physics. This equation belongs to the class of equations
that can be solvable using the inverse scattering method for a Dirac-type
operator. This was shown in the works of V.E. Zakharov and A.B. Shabat [1], L.A.
Takhtadjan and L.D. Fadeev [2], M. Ablowitz, D. Kaup, A. Newell and H. Segur [3].
The sign of the constant
corresponds to the attraction (
<0, focusing case)
and repulsion (
>0, defocusing case) of particles. In the focusing case the
problem of a finite number of particles and their bound states has a physical
meaning. In the classical limit, this is modeled by vanishing boundary value
problems. In the defocusing case, interest is the problem corresponding to a gas
of particles with a finite density.
In [4], V.K. Melnikov obtained evolutions of scattering data with respect to
t for a self-adjoint Dirac operator with a potential that is an NSE solution with a
self-consistent source of integral type. However, we note that in the above works
NSE was considered in the class of "rapidly decreasing" functions, i.e. conditions
that vanish in a certain way as the coordinate tends to infinity.
In connection with the application to specific physical problems, it became
necessary to consider NLS not only in the class of rapidly decreasing functions,
but also in classes of functions of a special form. First, in the work of
V.E.Zakharov and A.B. Shabat [5], NSE was integrated in the class of “finit dense”
functions, i.e., functions for which
2
( , )
i
it
u x t
e
,
( , )
0
x
u x t
as
x
. The n-
soliton solution of the NSE in the case of a finite density was found in [6].
Formulation of the problem.
We consider the integration of the following
system of equations
2
*
*
1,
2,
2,
1,
1
2
2
(
)
N
t
xx
n
n
n
n
n
iu
u u
u
i
,
(1)
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
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,
,...,
2
,
1
,
0
,
2
,
1
,
2
,
1
,
2
*
,
1
N
n
i
u
x
i
u
x
n
n
n
n
n
n
n
n
(2)
N
n
i
u
x
i
u
x
n
n
n
n
n
n
n
n
,...,
2
,
1
,
0
,
2
,
1
*
,
2
,
1
,
2
,
1
, (3)
with initial value
0
( ,0)
( )
u x
u x
,
(4)
where the bar means complex conjugation and
j
,
1, 2, ...,
j
N
are the
eigenvalues and function
0
( )
u x
satisfy the following properties:
1.
0
0
(1
) ( , )
(1
) ( , )
i
i
x u x t
e
dx
x u x t
e
dx
,
2. The equation
0
1
1
2
2
0
( )
(0)
( )
d
u x
y
y
dx
L
y
i
y
y
d
u x
dx
,
x
R
can have
N
number of eigenvalues. Here, the function
0
( )
u
x
is a complex
conjugation of
0
( )
u x
.
We also assume that the eigenfunctions
T
n
n
n
)
,
(
,
2
,
1
(
T
n
n
n
)
,
(
,
2
,
1
)
corresponding to
( )
n
t
this eigenvalues satisfy the following normalizing
conditions
det{
( , ),
( , )}
( )
T
k
k
k
s t
s t
t
,
N
k
,...,
2
,
1
,
(5)
Here
( )
k
t
,
1, 2,...,
j
N
are given and the continuous functions of
t
.
The main goal of this work is to study the integration of the nonlinear
Schrodinger equation via inverse scattering problem in the class of
( , )
u x t
function, which is sufficiently smooth and tends to its limits rapidly enough
when
x
and satisfies the condition
0
0
2
2
2
2
)
,
(
)
1
(
)
,
(
)
1
(
dx
e
t
x
u
x
dx
e
t
x
u
x
t
i
i
t
i
i
0
,
)
,
(
2
1
dx
x
t
x
u
k
k
k
.
(6)
Let the function
)
,
(
t
x
u
be a solution of equation (1), from the class of functions
(5). Consider an operator with a potential
)
,
(
t
x
u
that is a solution to the problem
under consideration and find the evolution from
t
the scattering data.
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
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Necessary information from scattering theory.
Consider the system of linear
equations on the real line (
)
x
(
)
0
L
I f
,
(7)
where
( , )
f
f x
is vector-column function and
( , )
( )
( , )
u x t
x
L t
i
u x t
x
,
0
t
.
There we present some necessary facts for our further exposition from the
theory of the direct and inverse scattering problem for the system of equations
(7).
We define the Jost solutions of the system (7) with the following asymptotic
values
2
2
2
2
1
~
, as
,
(
)
(
)
~
, as
,
1
ipx
i
i
t
i
i
t
ipx
e
x
i
p
e
i
p
e
e
x
(8)
2
2
2
2
(
)
~
, as
,
1
1
~
, as
,
(
)
i
i
t
ipx
ipx
i
i
t
i
p
e
e
x
e
x
i
p
e
where
2
2
)
(
p
,
(9)
here and below we will use the standard Pauli matrices
0
1
1
0
1
,
0
0
2
i
i
,
1
0
0
1
3
.
For real
(
2
2
), path of square root is fixed by the condition
sign
p
sign
)
(
. The Riemann surface
of a function
)
(
p
consists of two
instances
and
a complex plane
C
with cuts along the real axis from
to
and from
to
with properly identified cut edges (see [5]). The function
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
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)
(
p
is introduced on
the formula (8), where
0
Im
p
on the sheets
. In
what follows, for convenience, we will often omit the dependence of the function
)
(
p
on
. Thus, in formulas where and is involved, it is always assumed that
)
(
p
is a function of
.
It can be shown that
det( , )
0
d
dx
and
det( , )
0
d
dx
.
(10)
From (8) and (10) it follows that
2
2 (
)
det( , )
d
p
p
dx
,
2
2 (
)
det( , )
d
p
p
dx
.
(11)
For real
p
and
pairs of vector functions
{ , }
and
{ , }
form a fundamental
system of solutions to (7), so, there is a functions
( , ), ( , )
a t
b t
that for solutions
{ , }
and
{ , }
( , , )
( , ) ( , , )
( , ) ( , , )
x t
a t
x t
b t
x t
, as
из
]
,
[
\
1
R
.
(12)
The coefficients
)
,
(
t
a
and
)
,
(
t
b
are called transition coefficients. From
relations (10) and (11) we obtain
1
)
,
(
)
,
(
2
2
t
b
t
a
,
(13)
where the functions
( )
a
and
( )
b
are independent of
x
and
2
( , )
det( ( , , ), ( , , ))
2 (
)
a
t
x
t
x
t
p
p
,
(14)
2
( , )
det( ( , , ), ( , , ))
2 (
)
b
t
x
t
x
t
p
p
.
The function
)
,
(
t
a
admit an analytic continuation in
into the plane
. The
function
)
,
(
t
a
has the asymptotics
1
1
)
,
(
O
t
a
, as
0
Im
(15)
and
1
)
,
(
)
(
O
e
t
a
i
, as
0
Im
.
(16)
Besides, in the plane
the function
)
,
(
t
a
has a finite number of zeros at the
points
k
(
1, 2, ...,
)
k
N
, and these points are the eigenvalues of the operator
L
.
It follows from representation (14) that if
0
)
,
(
t
a
n
, then the columns
( , , )
x
t
and
( , , )
x
t
are linearly dependent at
n
, i.e.,
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
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( ,
, )
( ) ( ,
, )
n
n
n
x
t
c t
x
t
,
N
n
,...,
2
,
1
.
(17)
Note that the vector-functions
(
)
( , )
,
1, 2,...,
( , )
n
n
n
n
d
c
d
h x t
n
N
a
t
.
The following integral representations hold for the Jost solutions
2
2
2
2
(
)
(
)
( , , )
( , , )
,
1
1
i
i
t
i
i
t
x
ipx
i
p
i
p
e
e
x
t
e
K x y t
dy
(18)
where
)
,
,
(
)
,
,
(
)
,
,
(
)
,
,
(
)
,
,
(
22
21
12
11
t
y
x
K
t
y
x
K
t
y
x
K
t
y
x
K
t
y
x
K
.
In representation (18) the kernel
( , , )
K x y t
does not dependent on
and the
related to the potential
( , )
u x t
as the following:
2
2
21
2
( , , )
( , )
i
i
t
K
x x t
e
u x t
,
(19)
It is well known that the components of the kernel
)
,
,
(
t
y
x
K
for
y
x
are
solutions of the system of Gelfand-Levitan-Marchenko integral equations:
0
)
(
)
,
(
)
(
)
,
(
x
ds
y
s
F
s
x
K
y
x
F
y
x
K
,
x
y
(20)
where
)
(
)
(
)
(
)
(
)
(
1
2
*
2
1
x
F
x
F
x
F
x
F
x
F
,
x
z
z
i
e
z
x
e
2
2
)
,
(
2
2
2
2
1
1
2
( )
1
1
( )
( , ) ( , )
( ,
)
( )
4
2
( , )
1
i
i
t
i
i
t
N
n
n
n
n
n
n
i e
F x
c t
e
r z t e x z
dz
e x z
z
F x
a z t z
iz
.
Definition.
The set of the quantities
{ ( , ), ( , ),
( ), ( ),
1,2,..., }
n
n
a
t b
t
t c t n
N
is
called the scattering data for equation (7).
Evolution of scattering data.
If the potential
)
,
(
t
x
u
in the system of
equations (7) depends on
t
, then its solution
f
must also depend on
t
. Let this
time dependence have the form
( , , )
f
A x t
f
t
,
(21)
where
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
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2
2
2
*
*
2
2
2
2
2
x
x
i
i u
u
iu
A
u
iu
i
i u
.
The compatibility condition for linear systems (7) and (21) is
[ , ]
L
L A
G
t
,
(22)
where
0
0
g
G
g
.
Let
( , , )
x
t
be a Jost solution of the equation
( )
L t
. By differentiating
this relation with respect to
t
, we obtain the equation
L
L
t
t
t
.
(23)
By substituting
L
t
(22) into (23), we obtain the equation
(
)(
)
L
A
iG
t
,
(24)
whose solution we seek in the form
( , )
( , )
A
x t
x t
t
.
(25)
For the functions
( , )
x t
and
( , )
x t
, we obtain the equation
3
3
G
x
x
.
(26)
By multiplying Eq. (26) by
1
and
1
, we obtain
2
2
1
1
,
2 (
)
2 (
)
i
G
i
G
x
p
p
a
x
p
p
a
.
(27)
Relation (8) implies that
2
(2
)
A
i p
i
t
as
x
therefore, from
(21) we have
2
( , )
0,
( , )
2
a x t
x t
i p
i
as
x
. By solving (27), we
obtain
2
1
1
1
1
( , )
,
( , )
2
x
x
x t
G ds
x t
G ds
i p
i
a
a
.
Therefore, relation (13) can be represented in the form
2
1
1
1
1
2
x
x
A
G ds
G ds
i p
i
t
a
a
(28)
By using (25) and by passing to the limit as
x
in (28), we obtain
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«Zamonaviy dunyoda amaliy fanlar: muammolar va
yechimlar» nomli ilmiy, masofaviy, onlayn konferensiya
2
1
,
2 (
)
i
a
G ds
p
p
2
2
2
1
1
(2
)
2 (
)
2 (
)
i
i
b
b
i p
i
b
G ds
G ds
p
p a
p
p a
.
As in the continuous spectrum, one can show that
2
2
1
(4
2
)
2
(
)
n
n
n
n
n
n
n
n
n
i
с
i p
i
c
h R ds
p
p
,
1
1
2
2
n
n
n
G dx
d
dt
dx
,
1,2,3,
,
n
N
.
Theorem 1.
If the function
( , )
u x t
is a solution of the equation (1) in the class of
functions (3), then the scattering data of the system (7) with the function
( , )
u x t
depend on
t
as follows:
2
1
,
2 (
)
i
a
G ds
p
p
2
2
2
1
1
(2
)
2 (
)
2 (
)
i
i
b
i p
i
b
G ds
G ds
p
p a
p
p a
,
2
2
1
(4
2
)
2
(
)
n
n
n
n
n
n
n
n
n
i
с
i p
i
c
h R ds
p
p
,
1
1
2
2
n
n
n
n
n
G
dx
d
dt
dx
,
1,2,3,
,
n
N
.
The obtained relations determine completely the evolution of the scattering data
for the system (7), which allow as to find the solution of the problem (1)-(3) by
using the inverse scattering problem method.
Corollary.
If we get
*
*
1,
2,
2,
1,
1
( , )
2
(
)
N
n
n
n
n
n
g x t
i
then
1
1
0,
0,
G dx
G dx
1
1
( ),
( )
n
n
n
n
n
n
n
h G dx
i
t
G dx
t
.
30
«Zamonaviy dunyoda amaliy fanlar: muammolar va
yechimlar» nomli ilmiy, masofaviy, onlayn konferensiya
In this case
( )
n
n
d
t
dt
,
2
(4
2
( ))
n
n
n
n
n
n
с
i p
i
i
t c
.
Example.
Let
1
1
2
1
1
2
0
ip x
ip x
i
i
t
ip x
ip x
e e
ce e
u
e
e
ce
.
Where
1
, , ,
p
and
c
are positive numbers. In this case, the scattering data
system of equations (7) with potential
0
u
has
1
1
1
1
( , )
p
p
a t
p
p
,
( , )
0
b t
,
2
2
1
1
1
(
)
i
i
t
i
p
c
c
e
,
1
1
,
p
ip
Using results theorem 1, we can find
1
( )
n
d
t
dt
,
2
1
0
( )
exp (4
2
( ))
t
n
n
n
n
с t
c
i p
i
i
d
.
Solving the inverse problem we get
2
1
1 1
1 1
0
1
2
2
1
1 1
1 1
0
1
(4
2
( ))
2
(4
2
( ))
( , )
t
t
ip x
i p
i
i
d
ip x
i
i
t
ip x
i p
i
i
d
ip x
e e
ce e
u x t
e
e
ce
,
1
1
1
1
1
ip x
ip x
c
e
ce
,
2
1
1
2
1
1
2
1
(
)
1
i
i
t
ip x
ip x
i
p
e
c
e
ce
,
2
1
1
2
1
1
1
2
(
)
1
i
i
t
ip x
ip x
i
p
e
e
ce
1
1
1
1
2
2
2
'
2
1
1
1
1
1
1
1
1
1
2
(
)
,
2
(
)
ip x
ip x
i
i
ip x
ip x
c
ix
c
a e
c e
e
e
ce
p
p
p
p c
1
1
2
2
1
ip x
ip x
e
ce
2
1
1
1
1
2
2
2
'
2
1
1
1
1
1
1
2
(
) ,
2
ip x
ip x
i
i
t
i
i
ip x
ip x
i
c
e
ix
c
a e
e
c e
p
e
ce
p
p
1
1
2
.
References
:
1.
Захаров В. Е., Шабат А. Б. Точная теория двумерной
самофокусировки и одномерной автомодуляции волн и нелинейной среде.
// ЖЭТФ, 1971, Т61, №1, с. 118-134.
31
«Zamonaviy dunyoda amaliy fanlar: muammolar va
yechimlar» nomli ilmiy, masofaviy, onlayn konferensiya
2.
Тахтаджян Л.А., Фаддеев Л.Д. Гамильтонов подход в теории
солитонов. // М.Наука. 1986 г.
3.
Ablowitz М., Каир D.. Newell A., Segur Н. The Inverse Scattering
Transform-Fourier Analysis for Nonlinear Problems // Stud. Appl. Math. - USA,
1974. - LIII, №. - pp.249- 315.
4.
Melnikov V.K. Integration of the nonlinear Schrodinger equation
with a source. // Inverse Problem, 1992, V.8, pp. 133-147.
5.
Захаров B.E., Шабат А.Б. О взаимодействии солитонов в
устойчивой среде. // ЖЭТФ, 1973, Т.64, №5, стр. 1627-1639
6.
Yan-Chow Ma. The perturbed plane-wave solutions of the Cubic
Schrodinger Equation. // Studies in Applied Mathematics, 1979, №60, pp.43-58.
7.
Уразбоев Г.У., Мамедов К.А. О модифицированном уравнении
КдФ с самосогласованным источником в случае движущихся собственных
значений. // Вестник ЕГУ им. И.А.Бунина, выпуск 8, 2005, стр.84-94, серия
"Математика. Компьютерная математика", №1
8.
Карпман В.И., Маслов Е.М. // ЖЭТФ. 1977. Т.73. вып.2(8). С.537-
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