Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations
The scheme for describing the kinetic behaviour of solitons in the integrable systems is considered for sine-Gordon and nonlinear Schrödinger equations. The collision integral for solitons is constructed without the assumption of the gas approximation. The kinetic equation for solitons in the case...
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irk-123456789-1202432017-06-12T03:04:47Z Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations Baryakhtar, I.V. The scheme for describing the kinetic behaviour of solitons in the integrable systems is considered for sine-Gordon and nonlinear Schrödinger equations. The collision integral for solitons is constructed without the assumption of the gas approximation. The kinetic equation for solitons in the case of small inhomogeneities in the soliton gas is proposed. Possible applications of the kinetic properties of solitons are discussed. Пропонується схема опису кінетичної поведінки солітонів у інтегровних системах на прикладі моделі синус-Гордона та нелінійної моделі Шредінгера. Побудовано інтеграл зіткнень, що не використовує газове наближення для солітонів. Пропонується кінетичне рівняння для солітонів у випадку невеликої неоднорідності. Обговорюється можливе використання кінетичних властивостей солітонів. 1999 Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations / I.V. Baryakhtar // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 227-234. — Бібліогр.: 21 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.227 PACS: 05.20.Dd http://dspace.nbuv.gov.ua/handle/123456789/120243 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The scheme for describing the kinetic behaviour of solitons in the integrable systems is considered for sine-Gordon and nonlinear Schrödinger
equations. The collision integral for solitons is constructed without the assumption of the gas approximation. The kinetic equation for solitons in the
case of small inhomogeneities in the soliton gas is proposed. Possible applications of the kinetic properties of solitons are discussed. |
| author |
Baryakhtar, I.V. |
| spellingShingle |
Baryakhtar, I.V. Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations Condensed Matter Physics |
| author_facet |
Baryakhtar, I.V. |
| author_sort |
Baryakhtar, I.V. |
| title |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations |
| title_short |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations |
| title_full |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations |
| title_fullStr |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations |
| title_full_unstemmed |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations |
| title_sort |
kinetic equation for solitons in sine-gordon and nonlinear schrödinger equations |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
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http://dspace.nbuv.gov.ua/handle/123456789/120243 |
| citation_txt |
Kinetic equation for solitons in sine-Gordon and nonlinear Schrödinger equations / I.V. Baryakhtar // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 227-234. — Бібліогр.: 21 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT baryakhtariv kineticequationforsolitonsinsinegordonandnonlinearschrodingerequations |
| first_indexed |
2025-07-08T17:31:41Z |
| last_indexed |
2025-07-08T17:31:41Z |
| _version_ |
1837100859241005056 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 227–234
Kinetic equation for solitons in
sine-Gordon and nonlinear Schr ödinger
equations
I.V.Baryakhtar
Institute for Low Temperature Physics and Engineering,
47 Lenin Ave., 310164 Kharkov, Ukraine
Received June 29, 1998
The scheme for describing the kinetic behaviour of solitons in the inte-
grable systems is considered for sine-Gordon and nonlinear Schrödinger
equations. The collision integral for solitons is constructed without the as-
sumption of the gas approximation. The kinetic equation for solitons in the
case of small inhomogeneities in the soliton gas is proposed. Possible ap-
plications of the kinetic properties of solitons are discussed.
Key words: soliton, collision integral, entropy
PACS: 05.20.Dd
1. Introduction
Soliton is one of the most actively investigated nonlinear objects. Soliton ideas
were successfully applied for explaining the physical effects in solids, hydrodynamics,
plasma [1,2] and biological systems [3]. One soliton solution of corresponding equa-
tions has been used in the most of soliton applications. However, in the integrable
systems soliton-soliton interaction demonstrates a very important property: solitons
conserve their velocities and therefore energies and get only the shifts of their posi-
tions (see for example [1]). This fact enables us to expect unusual kinetic behaviour
of solitons. First, kinetic properties of solitons were considered by Zakharov [4]. He
has constructed kinetic equation for solitons in Korteweg – de Vries equation, but
this equation describes only renormalization of solitons velocities and does not de-
scribe the diffusion processes. From the other point of view kinetic behaviour was
considered by Wada and Schrieffer [5] and Theodorakopoulos [6]. In [5,6] the dif-
fusion of kinks interacting with phonons in ϕ4 model was considered. This model
is not exactly an integrable model but potential for kink-phonons interaction has a
reflectionless form and the result of the interaction is the same as for soliton-soliton
collision in integrable models. Taking into account such kind of interaction in [5,6]
kink diffusion coefficient was calculated. A bit later the same calculations have been
c© I.V.Baryakhtar 227
I.V.Baryakhtar
done for sine-Gordon model [7]. Following the approach [5,6], diffusion coefficient
and renormalization of kink velocity due to kink-breather collision were calculated
in sine-Gordon model in [8]. In fact the regular method of investigating the soliton
kinetic properties has not been formulated. The problem of the entropy of the sys-
tem of solitons was considered in [9] and it was shown that entropy increases in the
case of inhomogeneous distribution of solitons in real space on the example of kinks
in sine-Gordon equation. It enables us to derive the transport equations for kinks
and to prove the existence of other kinetic coefficients, except the diffusion coeffi-
cient for the kinks in sine-Gordon equation. In [10] the kinetic equation for solitons
interaction with 2D phonons in the system close to the completely integrable one
has been proposed.
In the present paper the scheme for describing the kinetic behaviour of solitons
is considered for sine-Gordon and nonlinear Schrödinger equations. We follow the
Boltzmann approach for usual particles. The steps are the following: to analyze
the processes of soliton-soliton interaction; to introduce the distribution function of
solitons and to construct the collision integrals; to formulate the kinetic equation for
solitons and to prove the entropy production; to calculate the corresponding kinetic
coefficients and relaxation times.
2. Models
Among few physically reasonable models, covered by the inverse scattering meth-
od and described solitons we consider sine-Gordon (SG) and nonlinear Schrödinger
(NS) models.
SG equation can be written as
∂2u
∂t2
−
∂2u
∂t2
+ ω2
0
sinu = 0. (1)
SG equation is widely applied in analysing the nonlinear properties of magnets and
Josephson junction [1,3].
Two types of localized solutions are known for this equation: kink type soliton
and breather type soliton. Kink is characterized by its coordinate of the centre of
mass xk and its velocity vk, breather – by the coordinate of the centre of mass xb,
the phase ϕb , the velocity vb and the intrinsic frequency ωb. When ωb → 1 breather
can be interpreted as a bound state of two kinks. Kink is the topological soliton, the
topological charge of breather is equal to zero.
NS equation in dimensionless variables has the form:
i
∂ψ
∂t
+
∂2ψ
∂x2
+ 2ψ |ψ|2 = 0. (2)
NS solitons, often named as envelope solitons, are characterized by the coordinate
of the v centre of mass xe, the velocity ve, the phase ϕe and the frequency ωe. NS
soliton application see, for example, in [1–3].
228
Kinetic equation for solitons
In the integrable systems solitons clearly display the difference from the usual
particles and waves in soliton-soliton collisions. Solitons conserve their forms and
velocities and the only result of the collision is shifts of the soliton’s coordinates
and phases for breathers and envelope solitons [1]. It is necessary to emphasize that
many particle effects are missing, i.e. the total shift of soliton position is equal to
the sum of partial shifts [1].
The following formulae can be written for the two soliton collisions. For kink-kink
collision
v1,k = v′
1,k, v2,k = v′
2,k,
x′
1,k = x1,k +∆x1,k(v1, v2), x′
2,k = x2,k −∆x1,k(v1, v2). (3)
For breather-breather and envelope soliton-soliton collisions
v1,i = v′
1,i, v2,i = v′
2,i,
x′
1,i = x1,i +∆x1,i(v1, v2, ω1, ω2), x′
2,i = x2,i −∆x2,i(v1, v2, ω1, ω2);
ω1,i = ω′
1,i, ω2,i = ω′
2,i,
ϕ′
1,i = ϕ1,i +∆x1,i(v1, v2, ω1, ω2), ϕ′
2,i = ϕ2,i −∆x2,i(v1, v2, ω1, ω2). (4)
Here i = b, e for breather and envelope soliton; the values after collision are noted
with a prime. The general expression for coordinate and phase through the scattering
date see, for example, in [1].
3. Kinetic equations and entropy production
Let us introduce distribution function of solitons
fi(Γi, t), i = k, b, e; (5)
where Γk = (x, v) for kinks, Γi = (x, v, ϕ, ω) for breathers and envelope solitons.
Following the Boltzmann way the collision integral can be written as
L {f} =
∫
[
W (Γ1,i,Γ2,l; Γ
′
1,i,Γ
′
2,l)fi(Γ
′
1,i, t)fl(Γ
′
2,l, t)
−W (Γ′
1,i,Γ
′
2,l; Γ1,i,Γ2,k)fi(Γ1,i, t)fl(Γ2,l, t)
]
dΓ′
1,idΓ
′
2,ldΓ2,l , (6)
where W is probability of soliton transition per unit time.
For simplicity we will consider only the changes of soliton coordinates . Taking
into account the fact that the cross section of soliton-soliton collision is equal to 1
due to the one dimensionality of the problem, W can be written as
W = 1 · |v1,i−v2,l|δ(v1,i−v
′
1,i)δ(v2,l−v
′
2,l)δ(x
′
1,i−x1,i−∆x1,i)δ(x
′
2,l−x2,l−∆x2,l). (7)
Here the first two delta functions describe the solitons momentum conservation laws,
the two last delta functions describe the fact of solitons shifts due to the collision.
229
I.V.Baryakhtar
It is necessary to underline that the gas approximation has not been used in (6), (7)
because many particle effects for solitons are missing.
Assuming that f(x, V, t) is slow varying in scales comparing to ∆x and expand-
ing f(x, V, t) in powers of ∆x and keeping the leading terms we can rewrite the
expression for collision integral (6) in the form:
L{fi} = −δv1,i
∂fi
∂x
+D1,i
∂2fi
∂x2
, (8)
where
δv1,i =
∫
|v1,i − v2,l|∆x1,i(V1,i, V2,l)fi(x2,l, V2,l, t)dΓ2,l (9)
is the renormalization of soliton velocity due to collision and
D1,i =
∫
|v1,i − v2,l| (∆x1,i(V1,i, V2,l))
2 fi(x2,l, V2,l, t)dΓ2,l (10)
describes the soliton diffusion process. Here Vi = vk for kinks and Vi = (vi, ωi) for
breathers and envelope solitons.
The kinetic equation can be written as
∂fi
∂t
+ (v + δv1,i)
∂fi
∂x
= D1,i
∂2fi
∂x2
. (11)
From kinetic equation one can show that soliton-soliton collisions lead to entropy
production in kink gas. Really, defining entropy as
S = −
∫
f(x, V1,i, t) ln [f(x, V1,i, t)/e] dΓ1,i (12)
and using kinetic equation and formula (10) one can obtain that
dS
dt
=
∫
qdΓ1,i, (13)
where the source of entropy production is:
q =
∫
|v1,i − v2,l|[∆x1,i(V1,i, V2,l)]
2
fi
fl
(
∂fi
∂x
)2
dΓ2,l > 0. (14)
The expression (14) proves the Boltzmann entropy production theorem. The entropy
production connect only with the inhomogeneities in real space. In addition, when
the inhomogeneities in the phase space take place, the entropy production can be
shown similarly.
Applying the standard methods from (11) it is easy to derive the transport equa-
tions and to calculate kinetic coefficients. The transport equations can be written
in the following general form:
∂
∂t
ni < ai > +
∂
∂x
(
ji + jdi
)
= 0, (15)
230
Kinetic equation for solitons
where ni is density of i-type soliton and
ni < ai > =
∫
ai(x, Vi)f (x, Vi, t) dVi , (16)
ji =
∫
ai(x, Vi)[v + δvi]fi(x, Vi, t)dVi ,
jdi = −
d
dx
∫
ai(x, Vi)Difi(x, Vi, t)dVi . (17)
Substituting in equation (1), the velocity and the energy of solitons for variable a,
one can obtain the continuity equation, hydrodynamic equation and equation for
local energy density.
To calculate the specific kinetic coefficients it is necessary to assume the dis-
tribution function of solitons. For kinks in sine-Gordon equation in the case of
Maxwell-Boltzmann distribution function, the self-diffusion, thermal conductivity
and internal friction coefficients were calculated in [9].
4. Applications
Let us consider some applications that result from kinetic equation (11). NS
equation appears in the analysis of nonlinear electron plasma waves [1,2,11]. The
conditions of solitons creation [12] and collapse [13] and interpretation of Lengmuir
turbulence in soliton terms have been analyzed in the frame of NS equation (see [11]).
In the nonintegrable system solitons collide with momentum changing, but in
the case close to completely integrable model it is not difficult to estimate the
relaxation time due momentum exchange, 1/τex ∼ ∆E/E , where ∆E/E is relative
energy change due to collision. Thus we have two steps of the relaxation: the first
one deals with shifts of the soliton positions and the second deals with momentum
exchange [10]. The characteristic time τin of soliton instability in 2D space is finite
and τins ∼ L2 (see [11]), where L is a characteristic size in the direction perpendicular
to soliton propagation. In other words it is possible to conclude that the above
considered kinetic behaviour of solitons deals with shift of its positions and can be
realized as an intermediate regime before solitons collapse and can be interpreted
as soliton turbulence. Soliton turbulence phenomena have been considered in [2],
where kinetic equation for many particle distribution function of soliton like waves
has been proposed. In fact in [2] processes with momentum exchange only were
considered.
Other application of NS equation deals with soliton propagation in optic fibers
[14]. From the applied point of view the problem of relaxation of solitons interacting
with defects is one of the most urgent among the kinetic effects. The possible re-
sults of soliton-defect interaction are transmission, reflection or capturing of soliton
by defect (see [15]). Dynamic properties of solitons in optic fiber interacting with
dispersion-spectrum inhomogeneities have been studied in [16]. Obviously, the most
natural way of analyzing the relaxation of solitons is to formulate soliton-defect
231
I.V.Baryakhtar
collision integral. In general form it can be written as
Lsd =
∫
W (p1, p2)[f(x, p1, t)− f(x, p2, t)]dp2 . (18)
A specific expression for W (p1, p2) in sine-Gordon model was calculated in [17] for
the case of elastic scattering of soliton by impurity.
Here we estimate the relaxation time τsd in the case of low density of solitons
comparing with defects concentration Cd . Really from dimensional consideration
1
τsd
∼ Cd
∆Esd
E
, (19)
where ∆Esd is a relative energy loss due to interaction with defect.
The presense of defects can be induced by fiber irradiation. As a result, the
characteristics of light and sound propagation have changed under the influence of
the ionizing radiation. These effects have been applied for measuring the radiation
dose [18,19]. Obviously, the relaxation time and kinetic coefficients of solitons depend
on the intensity of fiber irradiation.
Let us consider the expression for the concentration of defects in the case of
γ-rays. The γ-rays can be generated in the processes of electron beam breaking on
heavy elements targets. The breaking radiation has a continuous spectrum with the
maximum energy of photons equal to the energy of electrons (see for example [20]).
In this case [21]
Cd = JeKt, (20)
where t is the time of irradiation, J is the current of electrons, coefficient K=2.1 for
Si [21] and energy of γ-rays Eγ > 20Mev. Therefore
Je ∼
1
τsdt(∆Esd/E)
. (21)
Consequently due to the stability of soliton and low damping in optical fiber this
effect can be applied for measuring the irradiation dose.
5. Acknowledgements
The author is grateful to V.G.Baryakhtar and E.N.Economou for discussions.
This work was supported by INTAS grants No 93–1324 and No 94–3754, and by
International Atomic Energy Agency, contract N 7996.
References
1. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskii L.P. Theory of Solitons.
Moscow, Nauka, 1980 (in Russian).
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(in Russian).
232
Kinetic equation for solitons
3. Davydov A.S. Solitons in Molecular Systems. Kiev, Naukova Dumka, 1988 (in Rus-
sian).
4. Zakharov V.E. Kinetic equation for solitons. // Zh. Eksp. Theor. Fiz., 1971, vol. 60,
p. 993–1000 (in Russian).
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// Phys. Rev. B, 1978, vol. 18, p. 3897–3912.
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Z. Phys. B, 1979, vol. 33, p. 385–390.
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// Z. Phys. B, 1979, vol. 39, p. 47–52.
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Rev. B, 1987, vol. 35, p. 257–262.
9. Baryakhtar I.V., Baryakhtar V.G., Economou E.N. Kinetic equation and transport
coefficients for kink solitons in the sine-Gordon equation. Phys. Lett. A, 1985, vol. 207,
p. 67–71.
10. Baryakhtar I.V. On the kinetic equation for solitons interaction with phonons. Preprint
ILT–1–96, Kharkov, 1996, 8 p.
11. Kadomtsev B.B. Collective Phenomena in Plasma, 2nd edn., Moscow, Nauka, 1988
(in Russian).
12. Vedenov A.A., Rudakov L.I. Wave interaction in continuous media // Doklady AN,
1964, vol. 159, p. 767–770 (in Russian).
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p. 1747–1759 (in Russian).
14. Hasegawa A., Tappert F. Transmission of stationary optical pulses. // Appl. Phys.
Lett., 1973, vol. 23, p. 142–144.
15. Bass F.G., Kivshar Yu.S., Konotop V.V., Sinitsin Yu.A. Dynamics of solitons under
random perturbations. // Phys. Reports, 1988, vol. 157, p. 63–181.
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dispersion-spectrum inhomogeneities. // Quantum Electronics, 1995, vol. 22, p. 1129–
1134.
17. Baryakhtar I.V., Baryakhtar V.G., Economou E.N. Additivity of diffusion coefficients
for solitons. // Europhys. Lett, 1996, vol. 36(2), p. 87–91.
18. Vassilopoulos C., Kortis A., Mantakas C., Conception of an ionising radiation detection
scheme. // IEE Proc. J, 1993, vol. 140, p. 267–272.
19. Poret J.C., Lindgren E., Rosen M., Suter J.J., Rifkind J. M. Laser-generated ultra-
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dat, 1979 (in Russian).
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p. 625–629 (in Russian).
233
I.V.Baryakhtar
Кінетичне рівняння для солітонів у моделі
синус-Гордона та нелінійній моделі Шредінгера
І.В.Бар’яхтар
Фізико-технічний інститут низьких температур ім. Б.І.Вєркіна НАН
України, 310164 Харків, просп. Леніна, 47
Отримано 29 червня 1999 р.
Пропонується схема опису кінетичної поведінки солітонів у інтегров-
них системах на прикладі моделі синус-Гордона та нелінійної моделі
Шредінгера. Побудовано інтеграл зіткнень, що не використовує га-
зове наближення для солітонів. Пропонується кінетичне рівняння для
солітонів у випадку невеликої неоднорідності. Обговорюється мо-
жливе використання кінетичних властивостей солітонів.
Ключові слова: солітон, ентропія
PACS: 05.20.Dd
234
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