Self-similar solutions of multi-dimensional nonlinear Schrödinger equations
The method of a choice of self-similar variables for the description of multi-dimension wave collapse (WC) evolution is offered. It is based on the requirement that self-similar variables should provide preservation of an average square of radius of a wave package. Proposed self-similar substitution...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2008 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations / S.F. Skoromnaya, V.I. Tkachenko // Вопросы атомной науки и техники. — 2008. — № 4. — С. 237-241. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860185314233942016 |
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| author | Skoromnaya, S.F. Tkachenko, V.I. |
| author_facet | Skoromnaya, S.F. Tkachenko, V.I. |
| citation_txt | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations / S.F. Skoromnaya, V.I. Tkachenko // Вопросы атомной науки и техники. — 2008. — № 4. — С. 237-241. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The method of a choice of self-similar variables for the description of multi-dimension wave collapse (WC) evolution is offered. It is based on the requirement that self-similar variables should provide preservation of an average square of radius of a wave package. Proposed self-similar substitutions do not break convertibility of the initial equation and allow us to present an average square of width of a wave package in a universal kind. In offered self-similar variables a spherical-symmetric stretched in the one of directions WC is investigated.
Запропоновано метод вибору автомодельних змінних для опису розвитку багатовимірного хвильового колапсу (ХК). Він полягає в тім, що шукані автомодельні змінні повинні забезпечувати збереження середнього квадрата радіуса хвильового пакета. Запропоновані автомодельні підстановки не порушують оборотності вихідного рівняння й дозволяють представити середній квадрат радіуса хвильового пакета в універсальному виді. У запропонованих автомодельних змінних розглянута часова динаміка сферично-симетричного, розтягнутого уздовж одного з напрямків ХК.
Предложен метод выбора автомодельных переменных для описания развития многомерного волнового коллапса (ВК). Он заключается в том, что искомые автомодельные переменные должны обеспечивать сохранение среднего квадрата радиуса волнового пакета. Предложенные автомодельные подстановки не нарушают обратимости исходного уравнения и позволяют представить средний квадрат радиуса волнового пакета в универсальном виде. В предложенных автомодельных переменных исследована временная динамика сферически-симметричного, растянутого вдоль одного из направлений ВК.
|
| first_indexed | 2025-12-07T18:04:24Z |
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| fulltext |
SELF-SIMILAR SOLUTIONS OF MULTI-DIMENSIONAL NONLINEAR
SCHRÖDINGER EQUATIONS
S.F. Skoromnaya, V.I. Tkachenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: tkachenko@kipt.kharkov.ua
The method of a choice of self-similar variables for the description of multi-dimension wave collapse (WC) evo-
lution is offered. It is based on the requirement that self-similar variables should provide preservation of an average
square of radius of a wave package. Proposed self-similar substitutions do not break convertibility of the initial
equation and allow us to present an average square of width of a wave package in a universal kind. In offered self-
similar variables a spherical-symmetric stretched in the one of directions WC is investigated.
PACS: 52.35.-g, 52.35.Fp
The wave collapse (WC) is a disastrous in time (ex-
plosive) increase of an energy density in a diminishing
volume. It is one of the fundamental phenomenons in
wave propagation in dispersive media.
There are many examples of WC formation. It is,
first of all, collapse of Langmuir waves [1], which ap-
pears under intensive plasma heating by electromagnetic
radiation or electron bunches. It is considered by many
researchers as one of the main mechanism of an energy
dissipation of Langmuir waves on accelerated electrons.
Further, it is worth to mention the phenomenon of the
light self-focusing in nonlinear environments, theoreti-
cally predicted by G.A. Askarjan [2] and experimentally
confirmed by N.F. Pilipetskiy and A.F. Rustamov [3],
self-focusing weakly nonlinear hydrodynamic perturba-
tions in a static space dusty background [4]. This is not
a full list of scenarios of WC formation. All these proc-
esses are described by one of fundamental equations of
a nonlinear physics − a nonlinear Schrödinger equation
(NSE) and are of considerable scientific and practical
interest.
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2008. № 4.
Серия: Плазменная электроника и новые методы ускорения (6), с.237-241. 237
In the multidimensional geometry (two-dimensional
and three-dimensional) WC, as phenomenon of a forma-
tion of a singularity for a finite time, plays the same
fundamental role, as a soliton in the one-dimensional
geometry. In the last case, their formation is conditioned
by the balance of two opposite processes − dispersive
diffusion of wave packet and its nonlinear focusing.
Moreover, this balance is stable.
The balance is broken under conditions of the multi-
dimensional WC (multidimensional NSE). The system
becomes nonstable, and nonlinear processes define the
dynamics of WC. Study of WC (conditions of appearing
and a temporal dynamics of an "explosive" increasing
the wave amplitude) is important since these issues are
crucial for estimation of the collapse efficiency, as non-
linear mechanism of wave energy transformations.
It should be noted that the development of WC in
different media is the subject of numerous researches.
For instance, in [1, 5, 6] conditions of the WC origin
have been discussed in details and criteria of choose of a
type of self-similar solutions for the numerical study of
NSE have been received. As a rule, the numerical calcu-
lations are required a lot of computing resources (CPU
time and an operative memory) and also are connected
with the development of complex multidimensional
difference algorithms, which are unstable (do not pro-
vide the conservation of NSE integrals). Last fact
proves to be true, for example, results of the work [6],
where there is a big discrepancy of the specified value
of initial amplitude (Fig.1,a) and received from Fig.2,a
arithmetically. This discrepancy appears, in our opinion,
because of there is no the conservation of NSE integrals
in self-similar variables, which used in [6]. In our opin-
ion the listed above lacks can be removed by the correct
choice of self-similar variables which take into account
internal properties of the system and provides the con-
versation of NSE integrals.
In the present paper the conditions of origin and
temporary dynamics of the two-dimensional and three-
dimensional WC are investigated on the basis NSE and
consequences of its integrals.
1. THE NONLINEAR SCHRÖDINGER EQU-
ATION AND ITS INTEGRALS
NSE for the wave function in dimensionless vari-
ables has the form [1, 5]:
2 0t + Δ + =&ιψ ψ ψ ψ , (1)
where: Δ − is an Laplas operator, the lower index marks
a time partial derivative.
The equation (1) is a convertible: at replacements
and ( , ) *( , )r t r t− →
r r
ψ ψt →−t , where the sign
marks a complex conjugation, the form of the equation
does not change.
(*)
Under the theoretical description and analysis of the
wave processes in a plasma, the equation (1) can be
received from nonlinear wave equation for potential by
averaging the initial system of the equations, describing
nonlinear media on fast oscillations of the frequency
( )0kω , corresponding to the centre of the wave pack-
age, where − is the wave number. The analogue of
NSE in hydrodynamics is the equation, describing
propagation of the wave package on the surface of a
water, which has been obtained by V.E. Zakharov and
A.B. Shabbat [7].
0k
From the equation (1) it is easy to get the first inte-
gral of the motion − a wave action or "number of the
particles"1, which is valid for the model of any dimen-
sions:
2 0N dr const
∞
−∞
= = >∫
rψ . (2)
( )R ξThis integral is proportional to the energy of the
wave package.
where the function coincides with the form of a
standard two-dimensional soliton.
The second motion integral can be received from the
equation (1) in two- and three-dimensional cases and is
a Hamiltonian of the system:
In the present paper we show the method of the
choice of the self-similar substitution, which allows us to
study the properties of the wave function ( ,r tr )ψ
238
2 41
2
H dr dr X Y c
∞ ∞
−∞ −∞
= ∇ − ≡ − =∫ ∫
r rψ ψ onst
on the
stage of WC. For this, consider the equation (6) on the
assumption that is a root of the right-hand part of (6):
. (3)
0tBesides, from equation (1) it is possible to get the
functional (22
04r dr H t t )2 2
∞
−
−∞
= ⋅ ⋅∫
rψ . (8)
22 1 2r N r d
∞
−
−∞
= ⋅ ⋅∫
rψ r , (4) The right-hand part of the eq. (8) takes positive
magnitudes at 0t t< only for the negative Hamiltonian
which defines the average value of the squared radius of
the wave package. 0H < . Consequently, as it was noted above, instability
develops for the negative Hamiltonian (under a
destruction of the wave package occurs). In this regards
it is convenient to rewrite the eq. (8) in such a view:
0H >However, dependency of this functional on a time
has its specifics for two- and three-dimensional NSE.
So, below, we study the temporary dynamics of multi-
dimensional WC. (22
04r dr H t t
∞
−∞
)2 2= ⋅ −∫
rψ . (9)
2. THE TWO-DIMENSIONAL To study the properties of the wave function we
choose such a radial self-similar variable, which satis-
fies the constancy of the average squared radius. These
requirements are satisfied by the substitution:
WAVE COLLAPSE
In 1971 for two-dimensional NSE the Vlasov-
Petrischev-Talanov criterion of the wave collapse have
been found [8]. It was the fundamental result in theories
of waves collapse. It was the first, rigorous result for
nonlinear wave systems with dispersion. It was shown
that a formation of a singularity for a finite time in me-
dia is possible, in spite of a linear dispersion of the
waves, which, for instance, in linear optics prevents
arising singularity points - focuses.
2 2
0
r
t t
=
−
ξ , ( ) ( ) 2 2
0,r t t t′ = −ψ ξ ψ , (10)
The criterion of a wave collapse follows from the re-
lation for the second derivative in time of the average
squared radius of a wave package:
2
22
2 8 . (5) d r dr H
dt
ψ
∞
= ⋅∫
r
H
−∞
Since the Hamiltonian is a constant, the equation (5)
can be twice integrated:
22 2
14r dr H t C t C
∞
⋅
−∞
= ⋅ ⋅ + +∫
rψ 2 . (6)
The constants , are additional integrals of the
motion, which are defined by initial data [8]:
1C 2C
2
2 0
0
t
C r
=
= > , 2
1
0t
C r
t =
∂
=
∂
.
From equation (6) one can obtain the criterion of
Vlasov-Petrischev-Talanov, which states that in the sys-
tem with the negative Hamiltonian , at arbitrary
and , average squared width of the wave pack-
age
0H <
1C 2C
2r vanishes for a certain finite time . This
indicates existence of a singularity in the wave function
0t
( ),r trψ .
In [9] on the basis of complicated numerical experi-
ments the automodel type of dependency of the field on
time ( , )r trψ in vicinity of the singularity point has
been obtained in the form
0 0
1( , ) ( )
( ) ( )
rr t R
f t t f t t
→
− −
ψ , , (7) (0) 0f =
where ξ − is a new self-similar variable and ( )′ψ ξ is
a new wave function in new variables. Then the eq. (9)
takes the form:
22 4d
∞
−∞
′ H= ⋅∫
r
ξ ψ ξ . (11)
Since and are constants, not dependent on
time values, from eq. (11) follows the constancy of the
average squared radius of a wave package
HN
2 4r H= N .
Let's find key parameters of the module of wave func-
tion ( )′ψ ξ starting from the expression (11).
For this purpose we suppose, that the module of
wave function is set in the kind of Gauss package:
( ) 2 2exp( )A B′ = ⋅ − ⋅ψ ξ ξ , (12)
Awhere and B − are the amplitude and radial width of
the wave package respectively. Then, substituting (12)
in equality (11), we shall receive the following ratio
between parameters of wave function:
1 1
2 / 2 (2 )4B A H= ⋅ ⋅ . (13)
In a Fig.1 the dependences of relative amplitudes of
wave functions on dimensionless time 0t t in various
points of space ( i =1, 2, 3) for parameters of wave
function
ir
1; 0,06; 0,85A H B= = − = are shown.
From Fig.1 one follows, that near the singularity
point (the curve 1) the initial amplitude of the wave
function grows in the explosive-like manner, i.e. WC
develops. At moving from the singularity point, the am-
plitude of the wave function grows slower in time (a
curve 2), and then vanishes. At the some distance from
the singularity (a curve 3) the wave function is a mo-
notonous decreasing function of a time.
0,0 0,2 0,4 0,6 0,8 1,0
0
5
10
15
1
2
|ψ(ri,t)|
|ψ(ri,0)|
t/t0
3
where and are constants of the integration. '
1C '
2C
The expression (17) is received in suggestion of the
small sprain of the space: the factor of the resemblance
is given in the kind of 1≡ +α ε ε, where − is a small
value, in the limit tending to zero ( 0→ε ), and ex-
pression (14) is decomposed with a small parameter
series by ε with the subsequent allocation of members
of the equal order. Here necessary to note that expres-
sion (17) does not contradict with more general inequal-
ity (15).
The view of the criterion (17) allows us to study the
three-dimensional WC in analogy with the two-
dimensional one. The following self-similar variables and
the amplitude satisfy to requirements of the conservation
of the average squared radius of the wave package:
Fig.1. The dependence of the relative amplitude of the
self-similar solution of two-dimensional NES on a time
in different points of the space: 1 – r = R1/R <<1; 1 0
2 – r = 10 R1/R ; 3 – r = 50 R1/R ,
here R – is a typical size of the wave package
239
2 0 3 0 1 2
2 2
0
3
7
r
t t
−
⎛ ⎞= ⎜ ⎟
⎝ ⎠ −
ξ , ( ) ( )( )
3 4
3 42 2
.0
3 ,
7
r t t tψ ξ ψ⎛ ⎞′ = −⎜ ⎟
⎝ ⎠
(18) 0
Thus, for the first time, the method of the choice of
self-similar variables for the analysis of the two-
dimensional WC development is offered. It is in choise
of self-similar variables in term of which requirement
the conservation of the average squared radius of the
wave package should hold. Such self-similar variables
provide transition to such a system of frame where WC
is absent.
The eq. (17) taking into account of (18) becomes:
22 4d Hξ ψ ξ
∞
−∞
′ =∫
r
, (19)
and completely coincides with the equation for a two-
dimensional WC.
Starting from an self-similar form of the wave func-
tion (18), it is possible to make a conclusion that for a
case 2D = the collapse develops faster, than for 3D = ,
where – is a dimension of the space. Besides the
chosen form of self-similar replacements for
3. THE SPHERICAL-SYMMETRIC WAVE
COLLAPSE D
2; 3D =
keeps convertibility of the equation (1).
Let’s consider a spherical-symmetric WC. In this
case the expression for the second derivative in time of
the average square of radius of the symmetric wave
package (wave function does not depend on polar coor-
dinates) takes the form [5]
As it can be seen from derived results the offered
self-similar substitution allows us to present the average
square of the radius of the wave package in universal
king: both for the two-dimensional and the three-
dimensional (spherical symmetric) case the average
square of radius of a wave package is constant and is
equally expressed through the Hamiltonian of the sys-
tems.
(
2
22
2 4 2d r dr H Y
dt
∞
−∞
= −∫
rψ ) , (14)
where Hamiltonian is an invariant. H
For the development of the spherical-symmetric WC
in time the evaluation was received [5]: Acting in the same way as it is made for two-
dimensional WC, we result the expression describing a
relation between the amplitude and width of a wave
package for the spherical-symmetric WC:
2 2
14N r H t C t C⋅⋅ < ⋅ + + 2
r
, (15)
where and are constants of integration. 1C 2C
The lack of the criterion (15) is in its representation
as an inequality. Let's receive more strict condition of
the spherical-symmetric WC development. For this we
enter the scale transformation (stretching) of the space
1 1
25 5
5(3 )
2 2
B H
−
⋅= ⋅
π A . (20)
Thus, the WC in the spherical-symmetric case, as
well as in two-dimensional one, is formed for negative
Hamiltonian, but with other values of the wave package
parameters (12).
r → ⋅α , where α − is a factor of resemblance. The
law of transformation of the wave function ( ),r trψ in
this case we define from condition of the conservation
of the integral :
The dependences of relative amplitudes of wave
functions on dimensionless time N
0t t in various points
of space ( i =1, 2, 3) for a case ( ) ( )
3
2,r t r t
−
⋅ =
r ,rψ α α ψ ir. (16) 1; 0,188;A H= = −
r → ⋅rα 0,69B = are similar to Fig.1, but amplitude in spheri-
cal-symmetric case is more in two times than in two-
dimensional one at
Then, supposing in (14) and considering
the law of the wave function transformation (16) it is
easy to get the criterion of the development spherical-
symmetric collapse in suitable for analysis king as a
result of simple calculation
10 →tt .
4. THREE-DIMENSIONAL AXIAL-
SYMMETRIC WAVE COLLAPSE 2 2 '
1
12
7
N r H t C t C⋅⋅ = ⋅ + + '
2
, (17)
For analysis of the three-dimensional axial-
symmetric wave package choose the element of the vol-
dr rdrdzd=
r ϕ Two possible scenarios of the WC development fol-
low from the expression (24).
ume in expressions (2)-(4) in the view
(the wave function depends also on the coordinate ).
In this case the radial self-similar variable and the self-
similar amplitude providing the requirements of the
constancy of the average square of the radius of the
wave package
z
The first scenario is realized in the case when the
characteristic longitudinal size of a wave package ex-
ceeds the radial (the sphere stretched along the axis ).
Then the dynamics of development of the axial WC
which described by expression (24) will be prolonged
until the axial half-width of WC will coincide with its
radius. In this case WC passes in a spherical-symmetric
stage and the degree of singularity increases from 1/2 up
to 3/4.
z
2r are defined by expressions:
2 2
0
r
t t
=
−
ξ ,
2 2
0
z
t t
=
−
η ,
( ) ( ) 2 2
0, , , (z r z t t t′ = ⋅ − )ψ ξ ψ . (21)
240
The eq. (1) in terms of the variables (21) takes the
form:
2 2 2 2
0 0
2
22 2
0 2
( )(( 2 )( ) 2 )
1 ( ) 0.
it t t t t
t t
ψ ψξ η ψ
ξ η
ψ ψξ ψ
ξ ξ ξ η
′ ′∂ ∂ ′− + − +
∂ ∂
′ ′∂ ∂ ∂ ′ ′+ − + +
∂ ∂ ∂
ψ
+
=
The second scenario can develop when the charac-
teristic longitudinal size of the wave package along the
− axis less then the radial one. In this case according
to expression (24) wave package having originally a
slightly compressed sphere will be transformed into a
«presolar disk» [11].
z
(22)
The above mentioned statement about development
of the WC in the axial direction proves to be true in the
next experimental facts on which we stop below.
Since in the chosen self-similar variable and ampli-
tude (21) the average square of radius of a wave pack-
age is a constant, it is possible to put ( , ) 0′∂ ∂ =ψ ξ η ξ
in the equation (22), i.e. to consider, that wave function
*( , ) ( , )′ ′=ψ ξ η ψ ξ η does not depend on a radial self-
similar variable ξ *=ξ ξ. Here is an arbitrary point in
the region of existence of the wave function.
Besides, from a constancy of integral (2) follows,
that in eq. (22) parameter will be equal to zero. t
Then the eq. (22) is converted into the one-dimensional
stationary nonlinear Schrödinger equation:
2
2*
* *2
( , ) ( , ) ( , ) 0.d
d
ψ ξ η ψ ξ η ψ ξ η
η
′
′ ′+ ⋅ = (23)
The solution of the equation (23) in general case is
expressed through the Weierstraß’s function of the im-
aginary argument. However for the analysis of the WC
development in an axial direction it is enough to find the
solution in a class of real functions. In this case the solu-
tion of the equation (23) in terms of variable is ex-
pressed through the elliptic Jacobi’s function
[10]:
Fig.2. Formation of the star system NGC 1333-IRAS 4B
The Fig.2 shows the artistic interpretation (in inverse
black and white color) of the embryonic star, called
NGC 1333-IRAS 4B (figured in center of image) [12],
which have been observed by the NASA's Spitzer Space
Telescope. NGC 1333-IRAS 4B is located in a pretty
star-forming region, which is approximately 1,000 light-
years away in the constellation Perseus. Its central stel-
lar embryon is still "feeding" off the material collapsing
around it and growing in size. The cold toroidal cocoon,
including ice, surrounds the embryonic star which is in a
state of the axi-symmetric WC.
z
( , )cn x k
* * * 2 2
0
1( , ) ( ,0) ( ( ,0) , )
2
zz cn
t t
′ ′ ′= ⋅ ⋅
−
ψ ξ ψ ξ ψ ξ , (24)
1 2k =where: *( ,0)′ψ ξ − the amplitude, -the module.
Choose the boundary conditions for the wave func-
tion (24) in axial direction, for instance, as a require-
ment of vanishing the amplitude of the wave function at
the value of the argument corresponding to the first pe-
riod of the elliptic Jacobi’s function:
Microwave Radiation [13] is quite corresponds to
the concept of the axial-symmetric three-dimensional
WC, when the radial rate of expansion of the Universe
differ from the axial one.
* *
1( , ) ( ,0) ( , ) 0,
2i iz cn hψ ξ ψ ξ′ ′= ⋅ = (25)
where: i 1,2; =
1,2
1,2 * 2 2
0
1( ,0) ( ) 1,8541
2
z
h K
t t
′= ⋅ = ± = ±
−
ψ ξ ; ( )K k is
the Legendre's elliptic integrals of the first kind.
Then from expression (25) follows, that the borders
of a wave package on z-axis (zeros of the Jacobi’s func-
tion) are shifted
with the time to the beginning of coordinates, i.e. the
axial WC develops.
1 2 2
* 01,8541 ( ( ,0)) ( )iz −′= ± ⋅ ⋅ −ψ ξ t t
In summary we shall note, that the time dynamics of
the WC in the offered self-similar variables (10), (18),
(21) is convertible, the same as the initial equation (1).
Therefore, it is quite probable, that WC can be not only
a final state of evolution of a wave package, but also the
initial one. Besides the anisotropy of the Relic In con-
clusion it is necessary to note that the present theory
gives only follow the dynamics of self-focusing singu-
larities until the condition of weak nonlinearity of NSE
is valid. It is possible, that the further research of the
formed system as a result of development three-
dimensional WC will serve as a key to understanding of
the formation of the star systems, galaxies and the origin
of the solar system [13]. But these questions require
special studies.
CONCLUSIONS
The method of a choice of self-similar variables for
the description of the development of multidimensional
WC is proposed. It consists of choice of the self-similar
variables which should provide preservation of an aver-
age square of radius of a wave package. Such self-
similar variables mean the transition to such a frame in
which WC is absent.
241
The suggested self-similar substitutions in two- and
three-dimensional cases (axi-symmetric and spherical –
symmetric collapses) allow us to write an average
square of a wave package radius in a universal kind.
Both for the two-dimensional and the three-dimensional
(spherical symmetric) case the average square of radius
of a wave package is constant and is equally expressed
through the Hamiltonian of the system. In physical vari-
ables (radius, time) the rate of development spherical -
symmetric WC is higher than for two –dimensional one.
At the development axial-symmetric three-dimensional
WC the explosive increase in amplitude of wave func-
tion occurs along an axis of the system. Thus two sce-
narios of the development of WC are possible.
The first scenario is realized in that case when the
characteristic longitudinal size of a wave package ex-
ceeds the radial (the sphere stretched along the axis ).
Then the dynamics of development of the axial WC will
be prolonged until axial half-width WC will coincide
with its radius. In this case WC passes in a spherical –
symmetric stage and the radial degree of singularity
increases from 1/2 up to 3/4.
z
The second scenario can develop when the character-
istic longitudinal size of the wave package less then the
radial (a sphere compressed along -axis). In this case
wave package having originally a form of a slightly com-
pressed sphere will be transformed into a «presolar disk».
z
The time dynamics of the WC in the offered self-
similar variables is convertible, the same as the initial
NSE equation. Therefore, it is quite probable, that WC
can be not only a final state of evolution of a wave
package, but also the initial one. Probably, the theory of
the Big Bang and the Expansion of the Universe are the
confirmation of it. Besides the anisotropy of the Relic
Microwave Radiation is quite corresponds to the con-
cept of the axial-symmetric three-dimensional WC,
when the radial rate of expansion of the Universe differ
from the axial one.
The present theory gives only follow the dynamics
of self-focusing singularities until the condition of weak
nonlinearity of NSE is valid. It is possible, that the fur-
ther research of the system formed as a result of the
development WC will serve as a key for understanding
of the formation of the star systems, galaxies and the
origin of the solar system. But these questions require
special studies.
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13. http://map.gsfc.nasa.gov/m_mm.html
Статья поступила в редакцию 05.06.2008 г.
АВТОМОДЕЛЬНЫЕ РЕШЕНИЯ МНОГОМЕРНЫХ НЕЛИНЕЙНЫХ УРАВНЕНИЙ ШРЕДИНГЕРА
С.Ф. Скоромная, В.И. Ткаченко
Предложен метод выбора автомодельных переменных для описания развития многомерного волнового
коллапса (ВК). Он заключается в том, что искомые автомодельные переменные должны обеспечивать
сохранение среднего квадрата радиуса волнового пакета. Предложенные автомодельные подстановки не
нарушают обратимости исходного уравнения и позволяют представить средний квадрат радиуса волнового
пакета в универсальном виде. В предложенных автомодельных переменных исследована временная
динамика сферически-симметричного, растянутого вдоль одного из направлений ВК.
АВТОМОДЕЛЬНІ РІШЕННЯ БАГАТОВИМІРНИХ НЕЛІНІЙНИХ РІВНЯНЬ ШРЕДИНГЕРА
С.Ф. Скоромна, В.І. Ткаченко
Запропоновано метод вибору автомодельних змінних для опису розвитку багатовимірного хвильового
колапсу (ХК). Він полягає в тім, що шукані автомодельні змінні повинні забезпечувати збереження
середнього квадрата радіуса хвильового пакета. Запропоновані автомодельні підстановки не порушують
оборотності вихідного рівняння й дозволяють представити середній квадрат радіуса хвильового пакета в
універсальному виді. У запропонованих автомодельних змінних розглянута часова динаміка сферично-
симетричного, розтягнутого уздовж одного з напрямків ХК.
http://map.gsfc.nasa.gov/m_mm.html
1. THE NONLINEAR SCHRÖDINGER EQUATION AND ITS INTEGRALS
2. THE TWO-DIMENSIONAL
WAVE COLLAPSE
3. THE SPHERICAL-SYMMETRIC WAVE COLLAPSE
4. THREE-DIMENSIONAL AXIAL-SYMMETRIC WAVE COLLAPSE
CONCLUSIONS
|
| id | nasplib_isofts_kiev_ua-123456789-110687 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:04:24Z |
| publishDate | 2008 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Skoromnaya, S.F. Tkachenko, V.I. 2017-01-06T07:57:20Z 2017-01-06T07:57:20Z 2008 Self-similar solutions of multi-dimensional nonlinear Schrödinger equations / S.F. Skoromnaya, V.I. Tkachenko // Вопросы атомной науки и техники. — 2008. — № 4. — С. 237-241. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 52.35.-g, 52.35.Fp https://nasplib.isofts.kiev.ua/handle/123456789/110687 The method of a choice of self-similar variables for the description of multi-dimension wave collapse (WC) evolution is offered. It is based on the requirement that self-similar variables should provide preservation of an average square of radius of a wave package. Proposed self-similar substitutions do not break convertibility of the initial equation and allow us to present an average square of width of a wave package in a universal kind. In offered self-similar variables a spherical-symmetric stretched in the one of directions WC is investigated. Запропоновано метод вибору автомодельних змінних для опису розвитку багатовимірного хвильового колапсу (ХК). Він полягає в тім, що шукані автомодельні змінні повинні забезпечувати збереження середнього квадрата радіуса хвильового пакета. Запропоновані автомодельні підстановки не порушують оборотності вихідного рівняння й дозволяють представити середній квадрат радіуса хвильового пакета в універсальному виді. У запропонованих автомодельних змінних розглянута часова динаміка сферично-симетричного, розтягнутого уздовж одного з напрямків ХК. Предложен метод выбора автомодельных переменных для описания развития многомерного волнового коллапса (ВК). Он заключается в том, что искомые автомодельные переменные должны обеспечивать сохранение среднего квадрата радиуса волнового пакета. Предложенные автомодельные подстановки не нарушают обратимости исходного уравнения и позволяют представить средний квадрат радиуса волнового пакета в универсальном виде. В предложенных автомодельных переменных исследована временная динамика сферически-симметричного, растянутого вдоль одного из направлений ВК. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы в плазменных средах Self-similar solutions of multi-dimensional nonlinear Schrödinger equations Article published earlier |
| spellingShingle | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations Skoromnaya, S.F. Tkachenko, V.I. Нелинейные процессы в плазменных средах |
| title | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations |
| title_full | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations |
| title_fullStr | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations |
| title_full_unstemmed | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations |
| title_short | Self-similar solutions of multi-dimensional nonlinear Schrödinger equations |
| title_sort | self-similar solutions of multi-dimensional nonlinear schrödinger equations |
| topic | Нелинейные процессы в плазменных средах |
| topic_facet | Нелинейные процессы в плазменных средах |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/110687 |
| work_keys_str_mv | AT skoromnayasf selfsimilarsolutionsofmultidimensionalnonlinearschrodingerequations AT tkachenkovi selfsimilarsolutionsofmultidimensionalnonlinearschrodingerequations |