New types of stable whistler nonlinear waveguides
The investigation of stationary whistler wave propagation in the framework of generalized nonlinear Shrödinger equation (GNSE) including fourth-order dispersive effects and saturable nonlinearity is performed. A novel class of whistler waveguides with a curved wave front is predicted. Necessary cond...
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| Zitieren: | New types of stable whistler nonlinear waveguides / T.A. Davydova, Yu.A. Zaliznyak // Вопросы атомной науки и техники. — 2000. — № 6. — С. 27-28. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-785042015-03-19T03:02:15Z New types of stable whistler nonlinear waveguides Davydova, T.A. Zaliznyak, Yu.A. Magnetic confinement The investigation of stationary whistler wave propagation in the framework of generalized nonlinear Shrödinger equation (GNSE) including fourth-order dispersive effects and saturable nonlinearity is performed. A novel class of whistler waveguides with a curved wave front is predicted. Necessary condition for different nonlinear structures formation, their spatial profile and stability properties are studied both analytically and numerically. 2000 Article New types of stable whistler nonlinear waveguides / T.A. Davydova, Yu.A. Zaliznyak // Вопросы атомной науки и техники. — 2000. — № 6. — С. 27-28. — Бібліогр.: 7 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/78504 533.9 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Magnetic confinement Magnetic confinement |
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Magnetic confinement Magnetic confinement Davydova, T.A. Zaliznyak, Yu.A. New types of stable whistler nonlinear waveguides Вопросы атомной науки и техники |
| description |
The investigation of stationary whistler wave propagation in the framework of generalized nonlinear Shrödinger equation (GNSE) including fourth-order dispersive effects and saturable nonlinearity is performed. A novel class of whistler waveguides with a curved wave front is predicted. Necessary condition for different nonlinear structures formation, their spatial profile and stability properties are studied both analytically and numerically. |
| format |
Article |
| author |
Davydova, T.A. Zaliznyak, Yu.A. |
| author_facet |
Davydova, T.A. Zaliznyak, Yu.A. |
| author_sort |
Davydova, T.A. |
| title |
New types of stable whistler nonlinear waveguides |
| title_short |
New types of stable whistler nonlinear waveguides |
| title_full |
New types of stable whistler nonlinear waveguides |
| title_fullStr |
New types of stable whistler nonlinear waveguides |
| title_full_unstemmed |
New types of stable whistler nonlinear waveguides |
| title_sort |
new types of stable whistler nonlinear waveguides |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2000 |
| topic_facet |
Magnetic confinement |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78504 |
| citation_txt |
New types of stable whistler nonlinear waveguides / T.A. Davydova, Yu.A. Zaliznyak // Вопросы атомной науки и техники. — 2000. — № 6. — С. 27-28. — Бібліогр.: 7 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT davydovata newtypesofstablewhistlernonlinearwaveguides AT zaliznyakyua newtypesofstablewhistlernonlinearwaveguides |
| first_indexed |
2025-07-06T02:34:52Z |
| last_indexed |
2025-07-06T02:34:52Z |
| _version_ |
1836863237942935552 |
| fulltext |
UDC 533.9
Problems of Atomic Science and Technology. 2000. № 6. Series: Plasma Physics (6). p. 28-32 27
NEW TYPES OF STABLE WHISTLER NONLINEAR WAVEGUIDES
T.A. Davydova and Yu. A. Zaliznyak,
Institute for Nuclear Research, Nauki Ave., 47, Kiev 03680, Ukraine
The investigation of stationary whistler wave propagation in the framework of generalized nonlinear Shrödinger
equation (GNSE) including fourth-order dispersive effects and saturable nonlinearity is performed. A novel class of
whistler waveguides with a curved wave front is predicted. Necessary condition for different nonlinear structures
formation, their spatial profile and stability properties are studied both analytically and numerically.
Spatial structures and self-focusing effects of
electromagnetic beams propagating in media with
saturable nonlinearity are usually described by nonlinear
Shrödinger equation [1,2]. However in magnetized
plasma different wave branches occur very anisotropic in
a space of wave vectors. As shown in [3] for stationary
whistler wave propagation along the external magnetic
field the forth order dispersion effects in the perpendicular
direction become crucial when wave frequency is
approximately equal to a half of electron cyclotron
frequency. Here we show that saturation of nonlinearity is
also very essential because taking into account both
effects one can find wider class of nonlinear structures. In
the case of whistler wave propagation in F-layer of
ionosphere the main nonlinear effect is connected with
plasma heating and additional ionization is strong wave
field [3,4].
In this work we study spatial structure of plane
electromagnetic beams propagating in z direction which is
described by the GNSE of the form
042
4
4
2
2
=+−
∂
∂+
∂
∂+
∂
∂ ψψψψψψψ K
xxz
i (1)
In Eq (1) ψ is the dimensionless field envelope, z and x
are dimensionless coordinates in direction of propagation
and in perpendicular direction. The normalization is
chosen in such a way that solutions of our nonlinear
problem depend on one parameter K. For saturable
nonlinearity the condition 12
max <ψK is assumed.
Note that Eq. (1) may also describe ultrashort and
ultraintense laser pulses in monomode optical fibers
propagating in z direction with group velocity (when x
corresponds to time variable [6]).
The Eq. (1) has three integrals: number of quanta
dxN ∫=
∞
∞−
2ψ , (2)
Hamiltonian
∫
−+−=
∞
∞−
dxK
dx
d
dx
dH 64
2
2
22
32
1 ψψψψ
(3)
and momentum
∫=
∞
∞−
dxsI ,
dx
d
dx
d
dx
dis ϕψψψψψ 2*
*
2
=
−= ,
( ) ( )zxzx ,arg, ψϕ = (4)
We have found an exact stationary soliton solution for
which momentum (4) is zero in the form
( ) ( ) ( )×= −− xzzhtx )(cosh2, 12/1 µψ
( ) ( )[ ]xzzizi )(coshln)(exp µγϕ + (5)
for fixed value of the coefficient K=7/18. Then the
soliton parameters are: µ2=1/8, γ2=5, λ=-9/16, h=3/2.
In application to nonlinear optics the phase of this
soliton changes nonlinearly with time and we shall
refer solitons of this type as chirped solitons. For
stationary waveguides this nonlinear phase
dependence corresponds to a curved wave front. Using
the profile ψ(x,z) defined by (5) as a trial function, the
approximate evolution of wave – packets was
investigated by means of variational approach [6,7]. It
predicts that the stationary solutions (solitons or
waveguides) and their stability properties follows
from the qualitative behavior of curves
( ) 2
0
2
0
2
00
0
00 µ
µ
µ
µ bdpbN ++−±= and
( ) 2
2
2
c
c
c
cc
bdpbN
µ
µ
µ
µ +−±= for ordinary (γ=0,
µ=µ0) and chirped (γ≠0, µ=µc) solitons. Here N is a
number of quanta, b=15/(16|K|), d=5/K, p=4/K,
-2.0 -1.5 -1.0 -0.5
0
5
10
15
20
25
30
35 K=0.1
K=0.3
K=0.5
N
um
be
r o
f q
ua
nt
a
λ
Fig. 1. An energy dispersion diagram for different
values of K: ordinary solitons. Circles indicate points
for which soliton profiles are presented at Fig. 2
28
p0=21p/4, d0=3d/2.
We also integrated stationary and nonstationary GNSE
numerically in order to check the predictions of
variational analysis with trial function (5). An energy
dispersion diagram (EDD) for ordinary single – humped
solitons (dependence N(λ)) is shown at Fig.1. Dashed
lines correspond to the dependencies predicted by
variational approach. It is clearly seen that: (i)
approximate variational results are in excellent qualitative
agreement with the results of numerical simulations, no
matter that all these solitons have oscillating tails while
the trial function is smooth, (ii) when K increases, the
region where solitons exist decreases (when K>1.5
solitons disappear, as it follows from the linear
asymptotical behavior of any localized solutions: λ<-
0.25). All single – humped ordinary solitons were found
to be stable. Examples of these solitons are shown at
Fig.2. Fig. 3 presents EDD for chirped solitons. They are
stable when 0>∂∂ λN and unstable in the opposite
case. Dashed line corresponds to the prediction of
variational approach. It is clear that variational analysis of
chirped solitons with trial function (5) is not as accurate
as for ordinary solitons, and numerical simulation is only
the way to investigate this novel class of soliton solutions.
The frequently employed trial function with Gaussian
chirp dependence would give the completely wrong
results. An example of stable stationary chirped soliton is
presented at Fig. 4.
Following [3,4] we have found that for whistler wave
propagation in the F-layer in ionosphere the main
parameter K in equation (1) is approximately equal to
( )222 41 ωeΩ− (where ω is wave frequency and Ωe is
electron cyclotron frequency) in the case when
2eΩ≈ω . Thus the necessary condition (K<1.5) for an
existence of novel class of nonlinear whistler waveguides
(ducts) is fulfilled. Our analysis may be also of interest in
application to nonlinear fiber optics where nonstationary
chirped pulses are widely used for pulse compression [5].
In conclusion, we have investigated both analytically
and numerically an existence and stability of localized
soliton solutions of Eq. (1) with the fourth – order
dispersive term of the same sign as the second one and
acc
hav
cha
1.
2.
3.
4.
5.
6.
7.
-15 -10 -5 0 5 10 1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
|Ψ|
2
t
Fig. 2. Single – humped ordinary solitons found for
K=0.3, λ=-0.65. Both solitons are stable
-1.0
-0.5
0 10 20
0.0
0.5
1.0
1.5
2.0
2.5
3.0
|Ψ|
2
Re Ψ
ΨIm
t
Fig. 4. Stable chirped soliton found for
K=0.3, λ=-0.684
-0.8 -0.7 -0.6 -0.5
12
13
14
15
16
17 K=0.3
K=7/18
N
um
be
r o
f q
ua
nt
a
λ
Fig. 3. An energy dispersion diagram for different
values of K: chirped solitons. Circle indicates point
for which soliton profile is presented at Fig 4
ounting for the saturable local nonlinearity. We
e found different stable solitons with nonlinearly
nging phase as well as with constant phase.
References
V.E. Zakharov, V.V. Sobolev, V.S. Synach,
ZETF, 60 (1971), 171.
F.V. Vidal, T.W. Johnston, Phys Rev. Lett., 77
(1996), 282.
N.A. Zharova, A.M. Sergeev, Fizika Plazmy, 15
(1989), 1175.
A.V. Gurevich, A.B. Schwarburg Nonlinear
theory of radiowave propagation in ionosphere,
Moscow, Nauka, 1973.
G.P. Agrawal, Nonlinear Fiber Optics, Acad.
Press, 1995.
D. Anderson, Phys. Rev. E 27 (1983), 3135
T.A. Davydova, Yu. A. Zaliznyak, Physica
Scripta 61 (2000), 476.
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