Features of weakly nonlinear wave interaction
Regular and stochastic dynamics of three wave interaction and dynamics of cascades of such three wave processes were investigated. It was shown that matrix elements of wave interaction are inversely proportional to square of frequency of LF wave participating in three wave interaction. It was shown,...
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Buts, V.A. Kovalchuk, I.K. Tolstoluzhsky, A.P. 2023-12-03T14:43:18Z 2023-12-03T14:43:18Z 2019 Features of weakly nonlinear wave interaction / V.A. Buts, I.K. Kovalchuk, A.P. Tolstoluzhsky // Problems of atomic science and technology. — 2019. — № 4. — С. 86-90. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 52.35.Mw https://nasplib.isofts.kiev.ua/handle/123456789/195172 Regular and stochastic dynamics of three wave interaction and dynamics of cascades of such three wave processes were investigated. It was shown that matrix elements of wave interaction are inversely proportional to square of frequency of LF wave participating in three wave interaction. It was shown, that thresholds of arising of regimes with dynamics chaos are proportional to cube of LF wave. By numerical methods dynamics of cascades of interacting waves was investigated in detail. Досліджена регулярна та стохастична динаміка трихвилевої взаємодії, а також динаміка каскадів таких трихвилевих процесів. Показано, що матричні елементи взаємодії хвиль обернено пропорційні квадрату частоти НЧ-хвилі, що бере участь у трихвилевій взаємодії. Показано, що пороги виникнення режимів з динамічним хаосом пропорційні кубу частоти НЧ-хвилі. Чисельними методами детально досліджена динаміка каскадів хвиль, що взаємодіють. Исследована регулярная и стохастическая динамика трехволнового взаимодействия, а также динамика каскадов таких трехволновых процессов. Показано, что матричные элементы взаимодействия волн обратно пропорциональны квадрату частоты НЧ-волны, участвующей в трехволновом взаимодействии. Показано, что пороги возникновения режимов с динамическим хаосом пропорциональны кубу частоты НЧ-волны. Численными методами детально исследована динамика каскадов взаимодействующих волн. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Nonlinear processes Features of weakly nonlinear wave interaction Особливості слабонелінійної взаємодії хвиль Особенности слабонелинейного взаимодействия волн Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Features of weakly nonlinear wave interaction |
| spellingShingle |
Features of weakly nonlinear wave interaction Buts, V.A. Kovalchuk, I.K. Tolstoluzhsky, A.P. Nonlinear processes |
| title_short |
Features of weakly nonlinear wave interaction |
| title_full |
Features of weakly nonlinear wave interaction |
| title_fullStr |
Features of weakly nonlinear wave interaction |
| title_full_unstemmed |
Features of weakly nonlinear wave interaction |
| title_sort |
features of weakly nonlinear wave interaction |
| author |
Buts, V.A. Kovalchuk, I.K. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Kovalchuk, I.K. Tolstoluzhsky, A.P. |
| topic |
Nonlinear processes |
| topic_facet |
Nonlinear processes |
| publishDate |
2019 |
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English |
| container_title |
Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Особливості слабонелінійної взаємодії хвиль Особенности слабонелинейного взаимодействия волн |
| description |
Regular and stochastic dynamics of three wave interaction and dynamics of cascades of such three wave processes were investigated. It was shown that matrix elements of wave interaction are inversely proportional to square of frequency of LF wave participating in three wave interaction. It was shown, that thresholds of arising of regimes with dynamics chaos are proportional to cube of LF wave. By numerical methods dynamics of cascades of interacting waves was investigated in detail.
Досліджена регулярна та стохастична динаміка трихвилевої взаємодії, а також динаміка каскадів таких трихвилевих процесів. Показано, що матричні елементи взаємодії хвиль обернено пропорційні квадрату частоти НЧ-хвилі, що бере участь у трихвилевій взаємодії. Показано, що пороги виникнення режимів з динамічним хаосом пропорційні кубу частоти НЧ-хвилі. Чисельними методами детально досліджена динаміка каскадів хвиль, що взаємодіють.
Исследована регулярная и стохастическая динамика трехволнового взаимодействия, а также динамика каскадов таких трехволновых процессов. Показано, что матричные элементы взаимодействия волн обратно пропорциональны квадрату частоты НЧ-волны, участвующей в трехволновом взаимодействии. Показано, что пороги возникновения режимов с динамическим хаосом пропорциональны кубу частоты НЧ-волны. Численными методами детально исследована динамика каскадов взаимодействующих волн.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/195172 |
| citation_txt |
Features of weakly nonlinear wave interaction / V.A. Buts, I.K. Kovalchuk, A.P. Tolstoluzhsky // Problems of atomic science and technology. — 2019. — № 4. — С. 86-90. — Бібліогр.: 15 назв. — англ. |
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2025-11-25T02:42:36Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2019. №4(122) 86
FEATURES OF WEAKLY NONLINEAR WAVE INTERACTION
V.А. Buts1,2.3, I.K. Kovalchuk1, A.P. Tolstoluzhsky1
1National Science Center “Kharkiv Institute of Physics and Technology”, Kharkiv, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkiv, Ukraine;
3Institute of Radio Astronomy of NAS of Ukraine, Kharkiv
E-mail: tolstoluzhsky@kipt.kharkov.ua
Regular and stochastic dynamics of three wave interaction and dynamics of cascades of such three wave pro-
cesses were investigated. It was shown that matrix elements of wave interaction are inversely proportional to square
of frequency of LF wave participating in three wave interaction. It was shown, that thresholds of arising of regimes
with dynamics chaos are proportional to cube of LF wave. By numerical methods dynamics of cascades of interact-
ing waves was investigated in detail.
PACS: 52.35.Mw
INTRODUCTION
Decay processes in plasma are well studied example
of weakly nonlinear interaction of wave. Their results
are widely presented in the scientific literature. The
fullest they are presented, for example, in [1 - 3]. In
works [4, 5] it was shown that there are conditions when
decay process may transfer into stochastic regime. Sto-
chastic decays were investigated as numerically as ex-
perimentally. Results of these investigations are pre-
sented in works [6 - 12].
This process may be continued. As result, the cas-
cade of decays is received. One of the variants of such
process is presented in [13]. It may talk about cascades
with different scenarios. In one of such cases the low
frequency wave is common for all decays. In other case
in each of cascade decay different low frequency waves
participate [14].
The one three wave decay is well have investigated.
Hower, as result of one three wave decay the waves
may appear that self may decay. In some cases there can
be many such elementary wave processes. There is nat-
ural question about redistribution wave energy in such
cascade of decays. Essential interest for plasma physics
in this case is question about energy value that is trans-
ferred into low frequency part of spectra that well inter-
acts with ion component of plasma. Besides, it is im-
portant to find conditions when dynamics of these low
frequency waves is chaotic.
It should be noted that in the overwhelming majority
of studies of the processes of nonlinear interaction of
waves, dimensionless both dependent and independent
variables are mainly used. In many cases it is conven-
ient. However, often with this consideration, the func-
tional dependence of the studied processes on the sys-
tem parameters is lost.
1. DYNAMICS OF ISOLATED THREE-
WAVE INTERACTION
Dynamics of amplitudes of interacting waves at
three wave interaction is described by set of equations
that contained in works [1 - 3].
* *31 2
2 3 1 3 1 2, , ,i t i t i taa aVa a e Va a e Va a e
t t t
δ δ δ− −∂∂ ∂
= = − = −
∂ ∂ ∂
(1)
where ia – slowly varying amplitudes of interacting
mode; 1 2 3δ ω ω ω= − − – detuning; 1,2ω – frequencies
of HF waves; 3ω – frequency of LF wave; V – matrix
elements of nonlinear wave interaction.
In this section we will investigate strict synchro-
nism. In this case is convenient to use dimensionless
time Vtt = . Set of equation (1) is universal. Hower, it
is important to define dependence of matrix element
versus LF wave frequency
To find this dependence from the Maxwell equa-
tions, we write out the following relationship of the
electric field strength with a non-linear current:
*
1 11 1
2
1
4 nle jE
t c A
πω∂
= −
∂
, (2)
where 1E – amplitude of decaying wave in the CGS
system (in contrast of 1a – which is normalized corre-
spondingly method that is presented in [1]); c –speed of
light; 1e – polarization vector of decaying wave; 1nlj
–
density of nonlinear current; 1A – value that functional-
ly depend versus frequency and wave vector of decay-
ing wave.
Note that equation (2) already is shortened. The ex-
pression for density of nonlinear current can be obtained
by means equation of hydrodynamics for electron and
ions and has next form:
( ) ( )(1)
1 2 3 3 2nl e e e ej e n v e n v= − −
, (3)
where e – electron charge; ejn , ejv – disturbance of
electron density and velocity correspondingly in the
field of j-th wave and have next form:
0 3 2
3 3 2 22
3 2
,z z
e z ez z
e e
n ek een i E v i E
m mω ω
= − = − . (4)
Substituting expressions (4) into (3) and last in (2) it
is obtained expression that is analogous to the first
equation of set (1)
1
1 2 3
E V E E
t
∂
=
∂
, (5)
where 1V is:
2
3
1 2 2
1 3
1pe
e
ek
V
m c A
ω
ω
= . (6)
Formula (6) explicitly expresses dependence of ma-
trix element versus parameters of low frequency wave.
Namely, explicit dependence versus frequency of LF
wave is visible ( 2
31/V ω≈ ).
mailto:tolstoluzhsky@kipt.kharkov.ua
ISSN 1562-6016. ВАНТ. 2019. №4(122) 87
1.1. CONDITIONS OF ARISING REGIMES
WITH DYNAMIC CHAOS
We will show in this subsection that the conditions
of the transition to the regime with dynamic chaos de-
pend anomalously on the frequency of the low-
frequency wave participating in the interaction. Namely,
the criterion is inversely proportional to the cube of this
frequency. Indeed, it was shown in [4, 5] that the decay-
ing process can go into the stochastic regime if the in-
crement of the decaying instability ( 10Va , 10a is the ini-
tial amplitude of the decaying wave) exceeds the fre-
quency of the most low-frequency wave. This criterion
is:
10 10
3
3 3
1
Va a
ω ω
≈ > . (7)
From (7) it follows that stochastic regimes in decays
can most easily be realized with the participation of
waves with low frequencies. This circumstance can be
used, for example, to heat the ion component of the
plasma.
1.2. DEPENDENCE OF THE WAVES DYNAMICS
INTERACTION ON THE VALUE OF DETUNING
Above, a rather idealized case was considered when
the characteristics of interacting waves strictly satisfy
the synchronism conditions. If these conditions are not
satisfied that it is necessary to analyze the system of
equations (1).
At the initial stage of decay, we will assume the am-
plitude of the pumping wave 1 10a a const= = . Repre-
senting a solution for the low-frequency wave in the
form 3 30
i ta a e ω= we find the dispersion equation
22
10( ) V aω ω δ+ = − , (8)
whose solution has the form
222
1,2 10 2V aω δ δ = − ± −
. (9)
Thus, in the presence of detuning, at the initial stage,
the unstable dynamics of decay occurs under the condi-
tion
222
10 0V aδ − < that it is of a threshold nature.
With strict synchronism ( 0δ = ), the dynamics have the
character of instability with an increment
10Im( ) V aγ ω= = .
1.3. ACCOUNTING WAVE PROPERTIES
OF LOW-FREQUENCY WAVE
In the previous subsection the case was studied
when the wave amplitude dynamics was considered
only on the basis of shortened equations for all the
waves involved in the interaction. In many cases, the
frequency of the low-frequency wave can be quite
small, so that the characteristic times of non-linear in-
teraction of waves are on the order of the low-frequency
wave period. In this case, the wave properties of low-
frequency waves can significantly affect the dynamics
of the interaction of waves. In this subsection, we will
take into account this possibility.
In this case, the system of equations (1) should be
rewritten:
*0 1
1 0
2
2 *
1 02
, ,
2 ,
i t i t
i t
da dai Vba e i Vb a e
dt dt
d b b Va a e
dt
∆ − ∆
−
− ∆
−
= =
+Ω = − Ω
(10)
where 0 1ω ω−∆ = − , 3ωΩ = .
At the initial stage of decay, we will also assume the
amplitude of the pump wave 0a const= . Writing the
solution for the low-frequency wave we write in the
form 0
i tb b e ω= , we obtain the dispersion equation:
2 22 2( )( ) 2 ( 0 ) 0V aω ω−Ω ∆ + − Ω = . (11)
The analytical representation of solutions of the dis-
persion equation is bulky, therefore the numerical anal-
ysis of the solutions of this equation was carried out.
The graph below presents the main features of the solu-
tions of this equation (Fig. 1).
Fig. 1. The dependence of increment of the three-wave
interaction on the parameters ∆ and Ω for the pump
wave amplitude 0 0.25a =
The maximum value of the increment is directly
proportional to the amplitude of the pumping wave.
With decreasing of the amplitude of the pumping
wave, the region of frequencies and detunings, at which
there exist instability-like regimes, narrows and for
small values of the pumping wave amplitude 0 0.1a ≤ ,
the instability develops under conditions close to reso-
nant: 0∆ −Ω ≅ .
The transition to stochasticity has the form
( )10
3
1
Va
ω
> . (12)
Note that the frequencies and wave vectors of the in-
teracting waves must satisfy the synchronism conditions
( 0δ = ).
2. DYNAMICS OF LIMITED NUMBER OF
ELEMENTARY THREE-WAVE PROCESSES
For the analysis of Dynamics of the limited cascade
of decays, it is convenient to use the following system
of equations:
2
1
*
1 1 1 1
*
1 1
,
.
i t i tn
n n n n
N
i t
n n n
n N
da
i V ba e V b a e
dt
dbi V a a e
dt
δ δ
δ
−
− − + +
−
− −
=−
= +
= ∑
(13)
Here 1n nδ ω ω −= − −Ω is detuning, nω are frequen-
cies of the high-frequency wave, Ω is frequency of the
low-frequency wave, nV are the matrix elements of the
nonlinear interaction of the waves. At the same time, we
assume that the frequencies and wave vectors of high-
ISSN 1562-6016. ВАНТ. 2019. №4(122) 88
frequency waves are much higher than the frequencies
and wave vectors of low-frequency waves
, kω k>> Ω >> .
Under this condition, the matrix elements nV weakly
depend on the number n . We will consider them the
same and equal to one.
In real systems, the number of high-frequency inter-
acting waves, although large, but is finite. Therefore, we
will assume that the amplitudes of the waves 0na = , if
n it is outside the range of values 1 2N n N− ≤ ≤ ( 1N is
number of red satellites, 2N is number of blue satel-
lites). The system of equations (14) up to terms of order
0( , )kω kΩ has the integrals of motion:
1
2
2 2| | | |
N
n n
n N
a bω ε
=−
+Ω =∑ , (14)
2| |na I=∑ . (15)
The relation (14) represents the total energy of the
interacting waves, and the relation (15) can be interpret-
ed as the number of quanta of the interacting high-
frequency waves. Besides, it is possible to show that
2
1 2
1
* 2 2
1 (| | | | )
N
n n N N
n N
d a a a a
dt − −
=−
= −∑ . (16)
And in the case when the number of "red" and
"blue" satellites is equal 1 2( )N N= , under the initial
conditions ( 0
0 0 ( 0)a a t= = , 0 ( 0)b b t= = ) one more in-
tegral of motion takes place:
2
1
*
1
N
n n
n N
a a const−
=−
=∑ . (17)
2.1. NUMERICAL ANALYSIS
Analytical study of system (13) is possible only for
an infinite number of interacting waves (see [1]). To
analyze the dynamics of a finite number of interacting
waves, it is necessary to use numerical methods. The
system of equations (13) was solved numerically under
various initial conditions for the fields and different
detuning parameters δ . We investigated the temporal
dynamics of wave interaction, the power spectra ( ( )S ω )
of realizations, their autocorrelation functions (Cf), and
the maximum Lyapunov exponent of the system (σ ).
At the initial moment of time, the real values of the
fields were chosen: 0
0 0Re ( 0)a t a= = ; Rea-1(t=0)=a-1
0;
( 0)b t = =b0. The imaginary parts of the fields and the
amplitudes of other waves at 0t = were selected equal
to zero. The accuracy of the calculations was controlled
by means of integral (16). The magnitude of the devia-
tion from the integral
2 2 9
int | ( ) | | ( 0) | 10n na t a t −∆ = − = ≤∑ ∑
in all conducted numerical researches.
In Fig. 2 results of numerical calculations of system
(13) are shown for an asymmetric case
( 1 27, 0N N= = ) with the initial conditions of the beat-
wave type 0
0a =1 and 0
1a− = 0.1, when synchronism con-
dition ( 0δ = ) is fulfilled.
a b
c d
e
Fig. 2. Dependences of magnitudes of amplitude
on time: a) amplitude 0a is red line, amplitude 1a−
is blue line; b) amplitude b ; c, d) spectrum and auto-
correlation function of amplitude;
e) σ is the maximum Lyapunov index
From these graphs it can be seen that the amplitudes
of high-frequency oscillations have the form corre-
sponding to the analytical representation of oscillations
[14] at time intervals up to *t ≅ 5.1, after which a com-
plicated dynamics of the amplitude of interacting waves
occurs.
A peculiarity of this type of nonlinear wave interac-
tion is the fact that the number of low-frequency wave
quanta increases over the time interval when the dis-
turbance did not reach the last waves participating in the
interaction, after which oscillations of the number of
low-frequency wave quanta occur. The maximum num-
ber of low-frequency quanta 2
bN b= in this case ex-
ceeds the number of quanta of each of the high-
frequency waves 2
k kN a= by more than six times. The
maximum Lyapunov index of the system does not ex-
ceed in order of magnitude 22 10σ −< ⋅ ; therefore, in
this case, we can speak about regular dynamics of the
decay cascade.
As the number of interacting waves increases
1 211, 0N N= = , as shown by numerical analysis, the
maximum amplitude of the low-frequency wave in-
creases and time increases somewhat *t ≅ 6.0. In this
case, if the initial amplitudes of the waves are increased
0
0 1a = and 0
1 0.5a− = , the maximum amplitude of the
low-frequency wave increases and time decreases
*t ≅ 4.2. Other characteristics of the decay process of
the waves at the same time qualitatively change a little.
The results of numerical calculations for the sym-
metric case ( 1 2 7N N= = ) with initial conditions of the
beat-wave type 0
0a =1 and 0
1a− =0.1, when the synchro-
nism condition ( 0δ = ) is satisfied, show that the ampli-
tudes of high-frequency oscillations also have the form
corresponding to the analytical representation oscilla-
tions [14] on the time intervals necessary for the dis-
turbance to reach the last waves involved in the interac-
tion.
ISSN 1562-6016. ВАНТ. 2019. №4(122) 89
The spectrum of the low-frequency wave line, the
correlation function is periodic. The maximum Lyapun-
ov exponent σ decreases with time and in order of
magnitude 0.02σ < , so that one can speak to the regu-
lar dynamics of the waves.
A feature of this type of nonlinear wave interaction
is the fact that the number of low-frequency wave quan-
ta increases linearly at time intervals *t ≅ 10 when the
disturbance does not reach the last waves participating
in the interaction, after which it falls linearly to zero and
the dynamics of quanta of low-frequency waves repeats.
The maximum number of low-frequency quanta does
not exceed the number of high-frequency quanta.
3. DYNAMICS OF THE LIMITED CASCADE
TAKING INTO ACCOUNT WAVE
PROPERTIES OF THE LOW-FREQUENCY
WAVE
Since the frequency of the low-frequency wave is ra-
ther small ( nω >> Ω ) when averaging its equation over
a high frequency, it is necessary to use not a shortened
equation, but a second-order equation.
At the same time the system of the equations, neces-
sary for the analysis, takes a form:
2
1
*
1 1
2
2 *
12
,
2 ,
i t i tn
n n
N
i t
n n
n N
da
i ba e b a e
dt
d b b a a e
dt
∆ − ∆
− +
− ∆
−
=−
= +
+Ω = − Ω ∑
(18)
whеre 1n nδ ω ω −∆ = +Ω = − .
3.1. NUMERICAL ANALYSIS
For a finite number n , the system of equations (18),
also as well as in the previous case, was solved numeri-
cally under different initial conditions for the fields and
different values of detuning ∆ and frequencyΩ . At the
same time, at the initial moment of time, the real values
of the fields were chosen: Rea0(t=0)=a0
0; Rea-1(t=0)=a-
1
0; ( 0)b t = =b0. The imaginary parts of the fields and the
amplitudes of the other waves at 0t = were chosen to
be zero. The value of the deviation from the integral
(15) 9
int 10−∆ ≤ .
Below are the results of a numerical analysis of the
system of equations (18).
In Fig. 3 results of numerical calculations of system
(18) for an asymmetric case ( 1 27, 0N N= = ) with ini-
tial conditions of the beat-wave type 0
0a =1 and 0
1a−
=0.1– are shown, at 0.3Ω = , 0.3∆ = .
From these graphs it can be seen that at the begin-
ning of the interaction, the amplitudes of high-frequency
oscillations also have the form corresponding to the
analytical representation oscillations [14] on the time
intervals necessary for the disturbance to reach the last
waves involved in the interaction. After that, the ampli-
tudes of high-frequency oscillations have a shape simi-
lar to irregular oscillations with different intensities,
whose frequencies also change irregularly.
The spectra of these waves take the form of a “table”
type; the correlation function rapidly decreases with
time to zero value and has irregular oscillations near this
value. Therefore, we can speak about random amplitude
and frequency modulation of HF waves. The spectrum
of the low-frequency wave is is a little widened, correla-
tion function slowly decreases remaining periodic, so
that the dynamics of the low-frequency wave is regular.
The maximum Lyapunov exponent eventually reaches
almost a constant level ~ 0.2σ (see Fig. 3,g), that con-
firms the chaotic nature of the process. Consequently,
when a low-frequency wave interacts with a system of
high-frequency waves with chaotic dynamics, the dy-
namics of the low-frequency wave remains regular.
a b
c d
e f
g
Fig. 3. The magnitudes of the amplitude on time are
a) amplitude 0a is red line, amplitude 1a− is blue line;
b) amplitude b ; c, d) spectrum and autocorrelation
function of the amplitude of the low frequency wave;
e, f) spectrum and autocorrelation function of the ampli-
tude of the high frequency wave; g) is the maximum
Lyapunov exponent σ
The number of quanta of the low-frequency wave
grows on time interval when the perturbation has not
reached the last waves participating in the interaction,
after which oscillations of the number of low-frequency
quanta occur. The maximum number of low-frequency
quanta 2
bN b= exceeds the number of quanta of each
of the high-frequency waves 2
k kN a= by more than an
order of magnitude.
Increase the number of waves involved in the cas-
cade does not lead to significant change in the dynamics
of the interaction of waves.
In the symmetric case ( 1 2 7N N= = ) with 0
0a =1 and
0 ( 0) 0.1b b t= = = , when 0.3Ω = and 0.3∆ = the low-
frequency wave spectrum is line, the autocorrelation
function changes periodically with time, the pump wave
spectrum has linear character, the autocorrelation func-
tion slowly changes with time. The maximum Lyapunov
index of the wave system σ decreases with time to
values 21 10σ −< ⋅ , therefore in this case it is possible to
say that dynamics of amplitudes as high frequency
waves and low-frequency wave has regular character.
ISSN 1562-6016. ВАНТ. 2019. №4(122) 90
CONCLUSIONS
Let’s note the most important results:
1. The effective transformation of the energy of the
high-frequency wave into the energy of the low-
frequency wave is possible at nonlinear wave interac-
tion.
2. The magnitude of the matrix element of the non-
linear interaction of the waves is inversely proportional
to the square of the low frequency. As a result, the pro-
cess of decay to low frequencies occurs at lower field
strengths of the decaying wave.
3. Modes with stochastic dynamics arise, primarily
with the participation of the wave with a minimum fre-
quency of the low-frequency waves. This is due to the
fact that the threshold of stochastic instability is inverse-
ly proportional to the cube of low frequency.
4. The feature of the cascade of decays in which were
took into account the wave properties of a low-frequency
wave is that the maximum number of low-frequency wave
quanta exceeds the number of quanta of each of the RF
waves by more than an order of magnitude.
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Article received 04.06.2019
ОСОБЕННОСТИ СЛАБОНЕЛИНЕЙНОГО ВЗАИМОДЕЙСТВИЯ ВОЛН
В.А. Буц, И.К. Ковальчук, А.П. Толстолужский
Исследована регулярная и стохастическая динамика трехволнового взаимодействия, а также дина-
мика каскадов таких трехволновых процессов. Показано, что матричные элементы взаимодействия
волн обратно пропорциональны квадрату частоты НЧ-волны, участвующей в трехволновом взаимо-
действии. Показано, что пороги возникновения режимов с динамическим хаосом пропорциональны
кубу частоты НЧ-волны. Численными методами детально исследована динамика каскадов взаимодей-
ствующих волн.
ОСОБЛИВОСТІ СЛАБОНЕЛІНІЙНОЇ ВЗАЄМОДІЇ ХВИЛЬ
В.О. Буц, І.К. Ковальчук, О.П. Толстолужський
Досліджена регулярна та стохастична динаміка трихвилевої взаємодії, а також динаміка каскадів
таких трихвилевих процесів. Показано, що матричні елементи взаємодії хвиль обернено пропорційні
квадрату частоти НЧ-хвилі, що бере участь у трихвилевій взаємодії. Показано, що пороги виникнення
режимів з динамічним хаосом пропорційні кубу частоти НЧ-хвилі. Чисельними методами детально
досліджена динаміка каскадів хвиль, що взаємодіють.
http://arxiv.org/
References
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