statistical dynamics of optical solitons in a non-kerr law media

Int. J. Contemp. Math. Sciences, Vol. 1, 2006, no. 16, 777 - 800

Statistical Dynamics of Optical Solitons

in a Non-Kerr Law Media

Anjan Biswas

Department of Applied Mathematics and Theoretical Physics

Delaware State University

Dover, DE 19901-2277, USA

Swapan Konar

Department of Applied Physics

Birla Institute of Technology

Mesra, Ranchi-835215, India

Essaid Zerrad

Department of Physics and Pre-Engineering

Delaware State University

Dover, DE 19901-2277, USA

INVITED PAPER

Abstract

The statistical dynamics of optical solitons, in a non-Kerr law media,

is studied in this paper. The Langevin equations are derived and it

is proved that the solitons travel through a fiber with a fixed mean

velocity. The non-linearities that are considered here are the power law,

parabolic law and the dual-power law types.

778 Anjan Biswas, Swapan Konar and Essaid Zerrad

OCIS Codes: 060.2310; 060.4510; 060.5530; 190.4370

1 INTRODUCTION

The dynamics of pulses propagating in optical fibers has been a major area

of research given its potential applicability in all optical communication sys-

tems. It has been well established [1, 5, 15, 17-20] that this dynamics is de-

scribed, to first approximation, by the integrable Nonlinear Schrodinger Equa-

tion (NLSE). Here the global characteristics of the pulse envelope can be fully

determined by the method of Inverse Scattering Transform (IST) and in many

instances, the interest is restricted to the single pulse described by the one

soliton form of the NLSE. Typically though, distortions of these pulses arise

due to perturbations which are either higher order corrections in the model as

derived from the original Maxwell’s equations [13], physical mechanisms not

considered at first approximation like Raman effects or external perturbations

such as the lumped effect due to the addition of bandwidth limited amplifiers

in a communication line. Mathematically, these corrections are seen as per-

turbations of the NLSE and most of them have been studied thoroughly by

regular asymptotic, soliton perturbation or Lie tranform methods [13, 14, 16].

Besides the deterministic type perturbations one also needs to take into ac-

count, from practical considerations, the stochastic type perturbations. These

effects can be classified into three basic types [1]:

1. Stochasticity associated with the chaotic nature of the initial pulse due

to partial coherence of the laser generated radiation.

2. Stochasticity due to random nonuniformities in the optical fibers like the

fluctuations in the values of dielectric constant the random variations of

the fiber diameter and more.

3. The chaotic field caused by a dynamic stochasticity might arise from a

periodic modulation of the system parameters or when a periodic array

Statistical Dynamics of Optical Solitons 779

of pulses propagate in a fiber optic resonator.

Thus, stochasticity is inevitable in optical soliton communications [1-4]. Stochas-

ticity are basically of two types namely homogenous and nonhomogenous [11].

1. In the inhomogenous case the stochasticity is present in the input pulse

of the fiber. So the parameter dynamics are deterministic but however

the initial values are random.

2. In the homogenous case the stochasticity originates due to the random

perturbation of the fiber like the density fluctuation of the fiber material

or the random variations in the fiber diameter etc.

2 MATHEMATICAL FORMULATION

The dimensionless form of the Nonlinear Schrodinger’s Equation (NLSE) is

given by [5]

iqt +1

2qxx + F

(|q|2)q = 0. (1)

where F is a real-valued algebraic function and the smoothness of the complex

function F (|q|2) q : C �→ C is necessary. Considering the complex plane C as a

two-dimensional linear space R2, the function F (|q|2) q is k times continously

differentiable so that [5, 9]

F(|q|2)q ∈

∞⋃m,n=1

Ck((−n, n) × (−m,m);R2

)(2)

Equation (1) is a nonlinear partial differential equation (PDE) that is not

integrable, in general. The special case, F (s) = s, also known as the Kerr law

of nonlinearity, in which case it reduces to the cubic Schrodinger’s equation, is

integrable by the method of Inverse Scattering Transform (IST) [1, 13]. The

IST is the nonlinear analog of Fourier transform that is used for solving the

linear partial differential equations. Schematically, the IST and the technique

of Fourier transform are similar [13]. The solutions are known as solitons.

The general case F (s) �= s takes it away from the IST picture. Equation (1),


physically, represents the propagation of solitons through an optical fiber.

The three conserved quantities or integrals of motion [5, 9, 13] are the energy

(E) or L2 norm, linear momentum (M) and the Hamiltonian (H) that are

respectively given by

E =∫ ∞

−∞|q|2dx (3)

M =i

2

∫ ∞

−∞(q∗qx − qq∗x)dx (4)

H =∫ ∞

−∞

[1

2|qx|2 − f(I)

]dx (5)

where

f(I) =∫ I

0F (ξ)dξ (6)

and the intensity I is given by I = |q|2. The soliton solution of (1), although

not integrable, is assumed to be given in the form [5]

q(x, t) = A(t)g [B(t) {x− x(t)}] eiφ(x,t) (7)

where

dx

dt= v (8)

with

∂φ

∂x= −κ (9)

and

∂φ

∂t=B2

2

I0,0,2,0I0,2,0,0

− κ2

2+

1

I0,2,0,0

∫ ∞

−∞g2(s)F

(A2g2(s)

)ds (10)

Here, the following integral is defined

Il,m,n,p =∫ ∞

−∞τ lgm(τ)

(dg

dτ

)n (d2g

dτ 2

)pdτ (11)


for non-negative integers l, m, n and p with τ = B(t)(x − x(t)). In (7), the

function g represents the shape of the soliton described by the GNLSE and it

depends on the type of nonlinearity in (1). Also, φ(x, t) represents the phase

of the soliton while the width of the soliton B(t) is related to the amplitude

A(t) as B(t) = λ ((A(t)) where the functional form of λ is known if the law of

nonlinearity in (1) is given. Finally, x(t) gives the mean position of the soliton

so that v in (8) represents the velocity of the soliton. For such a general form

of the soliton given by (6), the integrals of motion, from (3), (4) and (5),

respectively reduce to

E =∫ ∞

−∞|q|2 dx =

A2

BI0,2,0,0 (12)

M =i

2

∫ ∞

−∞(qq∗x − q∗qx) dx = −κA

2

BI0,2,0,0 (13)

H =∫ ∞

−∞

[1

2|qx|2 − f

(|q|2)]

dx

=A2B

2I0,0,2,0 +

κ2A2

2BI0,2,0,0 −

∫ ∞

−∞

∫ I

0F (s)dsdx (14)

For the soliton assumed in (7), the parameters are now defined as [5]

κ(t) =i

2

∫∞−∞ (qq∗x − q∗qx) dx∫∞

−∞ |q|2 dx (15)

x(t) =

∫∞−∞ x |q|2 dx∫∞−∞ |q|2 dx (16)

The velocity of the soliton is given by

v =dx

dt= −κ =

∂φ

∂x(17)

From (12), (13) and (15), the parameter dynamics of the unperturbed soliton

is as follows

dE

dt= 0 (18)

dM

dt= 0 (19)


dκ

dt= 0 (20)

Here, (10) is obtained by differentiating (7) with respect to t and subtracting

from its conjugate while using (1). The parameter dynamics for the amplitude

and the width of the soliton individually can be obtained for the special cases

of F (s) once the functional form of F , that represents the type of nonlinearity,

is known.

2.1 PERTURBATION TERMS

The NLSE along with its perturbation terms is given by

iqt +1

2qxx + F

(|q|2)q = iεR[q, q∗] (21)

Here R is a spatio-differential operator while the perturbation parameter ε,

with 0 < ε � 1, represents the relative width of the spectrum in fiber optics

that arises due to quasi-monochromaticity [13]. In presence of perturbation

terms, as in (21), the integrals of motion are modified. In most instances, a

consequence of this is an adiabatic deformation of the soliton parameters like

its amplitude, width, frequency and velocity accompanied by small amounts

of radiation or small amplitude dispersive waves. The adiabatic parameter

dynamics and the evolution of energy, in presence of perturbation terms, ne-

glecting the radiation, are [5-8]

dE

dt= ε

∫ ∞

−∞(q∗R + qR∗) dx (22)

dM

dt= iε

∫ ∞

−∞(q∗xR− qxR

∗) dx (23)

Equations (22) and (23) lead to

dκ

dt=

ε

I0,2,0,0

B

A2

[i∫ ∞


∗) dx− κ∫ ∞

−∞(q∗R+ qR∗) dx

](24)

Equation (16), for the perturbed NLSE (21) gives

dx

dt= −κ+

ε

I0,2,0,0

B

A2

∫ ∞

−∞x (q∗R+ qR∗) dx (25)


while (10) extends to

∂φ

∂t=B2

2

I0,0,2,0I0,2,0,0

− κ2

2

+1

I0,2,0,0

∫ ∞

−∞g2(s)F

(A2g2(s)

)ds+

iε

I0,2,0,0

B

2A2

∫ ∞

−∞(qR∗ − q∗R) dx

(26)

In this paper, the perturbation terms that are going to be considered are given

by

R = δ|q|2mq + βqxx − γqxxx + λ(|q|2q

)x

+ ν(|q|2)xq + σ(x, t) (27)

For the perturbation terms, in (27), δ < 0 is the nonlinear damping coefficient

[1, 5, 9], β is the bandpass filtering term [10, 20]. Also, λ is the self-steepening

coefficient for short pulses [13] (typically ≤ 100 femto seconds), ν is the higher

order dispersion coefficient [13] and γ is the coefficient of the third order dis-

persion [5, 13, 20].

It is known that the NLSE, as given by (1), does not give correct prediction for

pulse widths smaller than 1 picosecond. For example, in solid state solitary

lasers, where pulses as short as 10 femtoseconds are generated, the approx-

imation breaks down. Thus, quasi-monochromaticity is no longer valid and

so higher order dispersion terms come in. If the group velocity dispersion is

close to zero, one needs to consider the third order dispersion for performance

enhancement along trans-oceanic distances. Also, for short pulse widths where

group velocity dispersion changes, within the spectral bandwidth of the sig-

nal cannot be neglected, one needs to take into account the presence of the

third order dispersion [17]. The perturbation terms due to α and β are of non

Hamiltonian or non conservative type while those due to λ, ν and γ are of

Hamiltonian or conservative type [5, 13].

The amplifiers, although needed to restore the soliton energy, introduces noise

originating from amplified spontaneous emmision (ASE). To study the impact

of noise on soliton evolution, the evolution of the mean free velocity of the


soliton due to ASE will be studied in this paper. In case of lumped amplifica-

tion, solitons are perturbed by ASE in a discrete fashion at the location of the

amplifiers. It can be assumed that noise is distributed all along the fiber length

since the amplifier spacing satisfies za � 1 [8, 15]. In (27), σ(x, t) represents

the Markovian stochastic process with Gaussian statistics and is assumed that

σ(x, t) [6-8, 11] is a function of t only so that σ(x, t) = σ(t). Now, the complex

stochastic term σ(t) can be decomposed into real and imaginary parts as

σ(t) = σ1(t) + iσ2(t) (28)

is further assumed to be independently delta correlated in both σ1(t) and σ2(t)

with

〈σ1(t)〉 = 〈σ2(t)〉 = 〈σ1(t)σ2(t′)〉 = 0 (29)

〈σ1(t)σ1(t′)〉 = 2D1δ(t− t′) (30)

〈σ2(t)σ2(t′)〉 = 2D2δ(t− t′) (31)

where D1 and D2 are related to the ASE spectral density. In this paper, it is

assumed that D1 = D2 = D. Thus,

〈σ(t)〉 = 0 (32)

and

〈σ(t)σ(t′)〉 = 2Dδ(t− t′) (33)

In soliton units, one gets [15],

D =FnFGNphza

(34)

where Fn is the amplifier noise figure, while

FG =(G− 1)2

G lnG(35)

is related to the amplifier gain G and finally Nph is the average number of

photons in the pulse propagating as a fundamental soliton [15].


Thus, in presence of these perturbation terms, the soliton parameter dynamics

changes, by virtue of (22) and (23) to

dE

dt=

2εA2

B

[δA2mI0,2m+2,0,0 + βB

(B2I0,1,0,1 − κ2I0,2,0,0

)]

+2εA∫ ∞

−∞g(τ) (σ1 cosφ+ σ2 sinφ) dx (36)

dM

dt=

2ε

B

[κA2

(2βAB3I0,0,3,0 − δA2mI0,2m+2,0,0

)+ βκA2

(κ2I0,2,0,0 − B2I0,1,0,1

)]

+2εA∫ ∞

−∞

[g(τ) (σ1 cos φ+ σ2 sinφ) − B

dg

dτ(σ2 cosφ− σ1 sinφ)

]dx

(37)

Equations (36) and (37) lead to

dκ

dt=

4εβκB

I0,2,0,0

(κ2I0,2,0,0 +B2I0,0,2,0 − B2I0,1,0,1

)

− 2εB

AI0,2,0,0

∫ ∞

−∞

[Bdg

dτ(σ2 cosφ− σ1 sinφ) + 2κg(τ) (σ1 cosφ+ σ2 sinφ)

]dx

(38)

These equations governing the variation of soliton parameters will be individ-

ually analysed for Kerr, power, parabolic and dual-power laws of nonlinearity

in the following sections.

3 KERR LAW NONLINEARITY

The Kerr law of nonlinearity originates from the fact that a light wave, in an

optical fiber, faces nonlinear responses. Even though the nonlinear responses

are extremely weak, their effects appear in various ways over long distance

of propagation that is measured in terms of light wavelength. The origin of

nonlinear response is related to the non-harmonic motion of bound electrons

under the influence of an applied field. As a result the Fourier amplitude of

the induced polarization from the electric dipoles is not linear in the electric

field, but involves higher terms in electric field amplitude [13].


For the case of Kerr law nonlinearity, F (s) = s so that f(s) = s2/2. Thus, the

NLSE given by (1) modifies to

iqt +1

2qxx + |q|2q = 0 (39)

whose 1-soliton solution, as obtained by IST, is given by [5, 9, 13, 15, 20]

q(x, t) =A

cosh [B(x− x(t))]ei(−κx+ωt+σ0) (40)

where

ω =B2 − κ2

2(41)

and

A = B (42)

On comparing (40) with (7)

g(τ) =1

cosh τ(43)

while the phase is given by

φ(x, t) = −κx + ωt+ σ0 (44)

where ω is the wave number and σ0 is the center of phase of the soliton.

3.1 MATHEMATICAL ANALYSIS

For Kerr law nonlinearity, the NLSE given by (36) has infinitely many integrals

of motion. By Noether’s theorem [13], this means that NLSE has an infinite

number of symmetries, corresponding to the conserved quantities, which leave

the NLSE invariant. The first two integrals of motion are

E =∫ ∞

−∞|q|2dx = 2A (45)

and

M =i

2

∫ ∞

−∞(qq∗x − q∗qx)dx = −2κA (46)


In presence of perturbation terms, the adiabatic variations of the soliton pa-

rameters A and κ that are obtained from (36) and (37) are

dA

dt=ε

2

∫ ∞

−∞(q∗R+ qR∗)dx (47)

and

dκ

dt=

ε

2A

[i∫ ∞

−∞(q∗xR − qxR

∗)dx− κ∫ ∞

−∞(q∗R + qR∗)dx

](48)

Now substituting the perturbation terms R from (27) and performing the

integrations in (47) and (48) yields

dA

dt= ε

⎡⎣A⎧⎨⎩δA2m

Γ(

12

)Γ (m+ 1)

Γ(m+ 3

2

) − 2

3βA2 − 3βκ2

⎫⎬⎭

+π

⎧⎨⎩σ1 cos(ωt+ σ0) − σ2 sin(ωt+ σ0)

cosh(πκ2η

)⎫⎬⎭⎤⎦ (49)

dκ

dt= −ε

⎡⎣4

3βA2κ+

π

2η

(2A2 + 2κ2 − κA

)⎧⎨⎩σ1 cos(ωt+ σ0) − σ2 sin(ωt+ σ0)

cosh(πκ2A

)⎫⎬⎭⎤⎦(50)

Equations (47) and (48), as it appears, are difficult to analyse. If the terms

with σ1 and σ2 are suppressed, the resulting dynamical system has a stable

fixed point, namely a sink, given by (A, κ) = (A, 0) where

A =

⎡⎣ 2σΓ

(m+ 3

2

)3δΓ

(12

)Γ(m+ 1)

⎤⎦

12m−2

(51)

Now, linearizing the dynamical system about this fixed point and simplifying

gives

dA

dt= −ε

(A2m+1 − ζ

A

)(52)

dκ

dt= −ε [κ− ζ (1 + A− κ)] (53)

where

ζ = π

⎧⎨⎩σ1 cos (ωt+ σ0) − σ2 sin (ωt+ σ0)

cosh(πκ2A

)⎫⎬⎭ (54)


Equations (52) and (53) are called the Langevin equations which will now

be analyzed to compute the soliton mean drift velocity of the soliton. If the

soliton parameters are chosen such that ζA is small, then (53) yields

dκ

dt= −ε [κ− ζ (1 − κ)] (55)

One can solve (55) for κ and eventually the mean drift velocity of the soliton

can be obtained. The stochastic phase factor of the soliton is defined by

ψ(t, y) =∫ t

yζ(s)ds (56)

where t > y. Assuming that σ is a Gaussian stochastic variable one arrives at

〈eψ(t,y)〉 = eD(t−y) (57)

〈e[ψ(t,y)+ψ(t′ ,y′)]〉 = eDθ (58)

where

θ = 2(t+ t′ − y − y′) − |t− t′| − |y − y′| (59)

and

〈ζ(y)e−ψ(t,y)〉 =∂

∂y〈e−ψ(t,y)〉 = DeD(t−y) (60)

〈ζ(y)ζ(y′)e[−ψ(t,y)−ψ(t′ ,y′)]〉 = 2Dδ(y − y′)eDθ +∂2

∂y∂y′eDθ (61)

Now solving (55) with the initial condition as κ(0) = 0 and using equations

(56)-(61) the soliton mean drift velocity is given by

〈κ(t)〉 = − D

1 −D

{1 − e−ε(1−D)t

}(62)

From (62), it follows that

limt→∞〈κ(t)〉 = − D

1 −D(63)

so that in the limiting case, for D < 1, the mean free velocity of the soliton is

given by

〈v〉 =D

1 −D(64)

Thus, for large t, 〈v(t)〉 is constant. If D > 1, 〈κ(t)〉 becomes unbounded for

large t. Similarly, the two point correlation computes out to be

〈κ(t)κ(t′)〉 = − D

1 − 2D

{e[−(1−D)|t−t′|] − e[−(1−2D)(t+t′)−D|t−t′|]}+O(D2) (65)


4 POWER LAW NONLINEARITY

Power law nonlinearity is exhibited in various materials including semiconduc-

tors. This law also occurs in media for which higher order photon processes

dominate at different intensities. This law is also treated as a generalization

to the Kerr law nonlinearity.

For the case of power law nonlinearity, F (s) = sp so that f(s) = sp+1/(p+ 1)

so that the NLSE, given by (1), modifies to

iqt +1

2qxx + |q|2pq = 0 (66)

In (66), it is necessary to have 0 < p < 2 to prevent wave collapse [5, 9] and,

in particular, p �= 2 to avoid self-focussing singularity [5]. The soliton solution

of (66) is given by [5, 9]

q(x, t) =A

cosh1p [B(x− x(t))]

ei(−κx+ωt+σ0) (67)

where

ω =B2

2p2− κ2

2(68)

and

B = Ap(

2p2

1 + p

) 12

(69)

On comparing (67) with (7), gives

g(τ) =1

cosh1p τ

(70)


The first two integrals of motion of the power law nonlinearity are

E =∫ ∞

−∞|q|2 dx

= A2−p(

1 + p

2p2

) 12 Γ

(12

)Γ(

1p

)Γ(

1p

+ 12

) = B2−p

p

(1 + p

2p2

) 1p Γ

(12

)Γ(

1p

)Γ(

1p

+ 12

) (71)


M =i

2

∫ ∞

−∞(q∗qx − qq∗x)dx

= 2κA2−p(

1 + p

2p2

) 12 Γ

(12

)Γ(

1p

)Γ(

1p

+ 12

)

= 2κB2−p

p

(1 + p

2p2

) 1p Γ

(12

)Γ(

1p

)Γ(

1p

+ 12

) (72)

Using these integrals of motion, one can obtain

dA

dt=

ε

2 − pAp−1

(2p2

1 + p

) p−12p Γ

(1p

+ 12

)Γ(

12

)Γ(

1p

) ∫ ∞

−∞(q∗R+ qR∗)dx (73)

dκ

dt= εB

p−2p

(2p2

1 + p

) 1p Γ

(1p + 1

2

)Γ(

12

)Γ(

1p

) [i ∫ ∞

−∞(q∗xR − qxR

∗)dx − κ

∫ ∞

−∞(q∗R + qR∗)dx

]

(74)

Now substituting the perturbation terms R from (27) and carrying out the


dA

dt=

2εδ

2 − pA2m+1

(1 + p

2p2

) 12p Γ

(1p + 1

2

)Γ(

1p

) Γ(

m+1p

)Γ(

m+1p + 1

2

)

+2εβ

2 − p

Ap−1

B

(2p2

p + 1

) p−12p Γ

(1p + 1

2

)Γ(

1p

)⎡⎣A2B2

p2

Γ(

p+1p

)Γ(

p+1p + 1

2

) − A2

p2

(κ2p2 + B2

) Γ(

1p

)Γ(

1p + 1

2

)⎤⎦

+2εA

2 − pAp−1

(2p2

1 + p

) p−12p Γ

(1p + 1

2

)Γ(

12

)Γ(

1p

) [σ1

∫ ∞

−∞

cos φ

cosh1p τ

dx + σ2

∫ ∞

−∞

sin φ

cosh1p τ

dx

](75)

dκ

dt=

4εβ

p2κA2B

2p−2p

(2p2

p+ 1

) 12(p− 2

p+ 2

)

+4εκABp−2

p

(2p2

1 + p

) 1p Γ

(1p

+ 12

)Γ(

12

)Γ(

1p

) ∫ ∞

−∞

[B

p

tanh τ

cosh1p τ

(σ2 cosφ− σ1 sinφ)

+2κ

cosh1p τ

(σ1 cos φ+ σ2 sinφ)

]dx (76)

Equations (75) and (76) are difficult to analyse. If the terms with σ1 and σ2 are

suppressed, the resulting dynamical system has a stable fixed point, namely a


sink, given by (A, κ) = (A, 0) where

A =

⎡⎣ 2σ

δ(p+ 1)

Γ(m+1p

+ 12

)Γ(m+1p

)⎧⎨⎩

Γ(p+1p

)Γ(p+1p

+ 12

) −Γ(

1p

)Γ(

1p

+ 12

)⎫⎬⎭⎤⎦

12(m−p)

(77)

Now, linearizing the dynamical system about this fixed point gives, after sim-

plification

dA

dt= −ε

⎛⎝A2m+1 − ζ

(1)1

A

⎞⎠ (78)

dκ

dt= −ε

[κ− ζ

(1)2 (1 + A− κ)

](79)

where

ζ(1)1 = σ1

∫ ∞

−∞cos φ

cosh1p τdx + σ2

∫ ∞

−∞sinφ

cosh1p τdx (80)

and

ζ(1)2 =

∫ ∞

−∞

[B

p

tanh τ

cosh1p τ

(σ2 cosφ− σ1 sinφ) +2κ

cosh1p τ

(σ1 cosφ+ σ2 sinφ)

]dx(81)

Similarly, as in the case of Kerr law nonlinearity, these Langevin eqiuations

lead to the mean drift velocity of the soliton as

〈v〉 =D

1 −D(82)

5 PARABOLIC LAW NONLINEARITY

There was little attention paid to the propagation of optical beams in the fifth

order nonlinear media, since no analytic solutions were known and it seemed

that chances of finding any material with significant fifth order term was slim.

However, recent developments have rekindled interest in this area. The optical

susseptibility of CdSxSe1−x-doped glasses was experimentally shown to have a

considerable χ(5), the fifth order succetibility. It was also demonstrated that

there exists a significant χ(5) nonlinearity effect in a transparent glass in intense

femtosecond pulses at 620 nm [5].


It is necessary to consider nonlinearities higher than the third order to obtain

some knowledge of the diameter of the self-trapping beam. It was recognized

in 1960s and 70s that saturation of the nonlinear refractive index plays a

fundamental role in the self-trapping phenomenon. Higher order nonlinearities

arise by retaining the higher order terms in the nonlinear polarization tensor.

For this law, F (s) = s+νs2 where ν is a constant, so that f(s) = s2/2+νs3/3.

The form of the GNLSE here is

iqt +1

2qxx +

(|q|2 + ν |q|4

)q = 0 (83)

The solution of (83) is now written as [5]

q(x, t) =A

[1 + a cosh {B (x− x(t))}] 12

ei(−κx+ωt+σ0) (84)

where

ω =A2

4− κ2

2(85)

and

B =√

2A (86)

with

a =

√1 +

4

3νA2 (87)


g(τ) =1

(1 + a cosh τ)12

(88)


The fist two integrals of motion of the parabolic law nonlinearity are [5]

E =∫ ∞

−∞|q|2dx =

⎧⎨⎩

√32ν

tan−1[2A√

ν3

]: 0 < ν <∞√

− 32ν

tanh−1[2A√−ν

3

]: − 3

4A2 < ν < 0(89)


M =i

2

∫ ∞

−∞(qq∗x − q∗qx) dx

=

⎧⎨⎩

−κ2

√32ν

tan−1[2A√

ν3

]: 0 < ν <∞

−κ2

√− 3

2νtanh−1

[2A√−ν

3

]: − 3

4A2 < ν < 0(90)

Using these integrals of motion, one obtains

dA

dt= εa2

√2

2

∫ ∞

−∞(q∗R+ qR∗)dx (91)

dκ

dt= ε

B

A2E

[i∫ ∞


∗)dx− κ∫ ∞

−∞(q∗R+ qR∗)dx

](92)

where E is the energy of the soliton. Now, substituting the perturbation terms

R from (27) and carrying out the integrations in (91) and (92) yields

dA

dt=εδA2m+1

2mam+1F(m+ 1, m+ 1, m+

3

2;a− 1

2a

)B(m+ 1,

1

2

)

+ε√

2βA

a

[κ2F

(1, 1,

3

2;a− 1

2a

)B(1,

1

2

)− 2A2F

(3, 1,

5

2;a− 1

2a

)B(1,

3

2

)]

+εa2A√

2

⎡⎣σ1

∫ ∞

−∞cos φ

(1 + a cosh τ)12

dx+ σ2

∫ ∞

−∞sinφ

(1 + a cosh τ)12

dx

⎤⎦ (93)

dκ

dt= −εβκA

2

2

F(3, 2, 3; a−1

2a

)F(1, 1, 3

2; a−1

2a

) B (2, 1)

B(1, 1

2

)

−ε√

2

AE

∫ ∞

−∞

⎡⎣aB

2

(σ2 cosφ− σ1 sinφ) sinh τ

(1 + a cosh τ)32

− 2κ (σ1 cos φ+ σ2 sinφ)

(1 + a cosh τ)12

⎤⎦ dx(94)

where in (93) and (94), F (a, b; c; z) is the Gauss’ hypergeometric function

defined as

F (a, b; c; z) =Γ(c)

Γ(a)Γ(b)

∞∑n=0

Γ(a+ n)Γ(b+ n)

Γ(c+ n)

zn

n!(95)

and B(l,m) is the beta function that is defined as

B(l,m) =∫ 1

0xl−1(1 − x)m−1dx (96)



plification

dA

dt= −ε

⎛⎝A2m+1 − ζ

(2)1

A

⎞⎠ (97)

dκ

dt= −ε

[κ− ζ

(2)2 (1 + A− κ)

](98)

where A is the fixed point of the amplitude while

ζ(2)1 = σ1

∫ ∞

−∞cosφ

(1 + a cosh τ)12

dx + σ2

∫ ∞

−∞sinφ

(1 + a cosh τ)12

dx (99)

and

ζ(2)2 =

∫ ∞

−∞

⎡⎣aB

2


(1 + a cosh τ)32

− 2κ (σ1 cosφ+ σ2 sinφ)

(1 + a cosh τ)12

⎤⎦ dx

(100)

Again, as in the case of Kerr law nonlinearity, these Langevin equations lead

to the mean drift velocity of the parabolic law soliton as

〈v〉 =D

1 −D(101)

6 DUAL-POWER LAW NONLINEARITY

An important aspect that has not been addressed with proper perspective is

the fact that due to its nonsaturable nature, Kerr nonlinearity is inadequate

to describe the soliton dynamics in the ultrahigh bit rate transmission. For

example, when transmission bit rate is very high, for soliton formation the

peak power of the incident field accordingly become very large. On the other

hand higher order nonlinearities may become significant even at moderate

intensities in certain materials such as semiconductor doped glass fibers. This

problem can be addressed by incorporating the dual-power law nonlinearity in

the NLSE.


For this law, F (s) = sp + νs2p where ν is a constant, so that f(s) = sp+1/(p+

1) + νs2p+1/(2p+ 1). The form of the GNLSE here is

iqt +1

2qxx +

(|q|2p + ν|q|4p

)q = 0 (102)

Equation (102) supports solitary waves of the form [5]

q(x, t) =A

[1 + b cosh {B (x− x(t))}] 12p

ei(−κx+ωt+σ0) (103)

where

ω =A2p

2p+ 2− κ2

2(104)

while

B = Ap(

2p2

1 + p

) 12p

(105)

with

b =

√1 +

νB2

2p2

(1 + p)2

1 + 2p(106)

The dual-power law case the solitons exist for

−2p2

B2

1 + 2p

(1 + p)2< ν < 0 (107)


g(τ) =1

(1 + b cosh τ)12p

(108)


The first two integrals of motion of the dual-power law nonlinearity are

E =∫ ∞

−∞|q|2dx =

2A2

B21p b

1p

F

(1

p,1

p;1

2+

1

p;b− 1

2b

)B

(1

p,1

2

)(109)

M =i

2

∫ ∞

−∞(qq∗x − q∗qx) dx =

2κA2

B21p b

1p

F

(1

p,1

p;1

2+

1

p;b− 1

2b

)B

(1

p,1

2

)(110)


Using these integrals of motion, it is possible to obtain

dA

dt=

ε

pLAp−1

(p+ 1

2p2

) 12p ∫ ∞

−∞(q∗R + qR∗)dx (111)

dκ

dt=

ε

E

[i∫ ∞


∗)dx− κ∫ ∞

−∞(q∗R+ qR∗)dx

](112)

where E is the energy, while

L =Γ(

12

)Γ(

1p

)Γ(

1p

+ 12

)[(b− 1)(2p+ 1)

2ν(1 + p)

] 1p

{2ν2

bp3

(p+ 1)3

(b− 1)(2p+ 1)2F

(1

2,1

p;1

2+

1

p;1 − b

1 + b

)

− 2

B2F

(1

2,1

p;1

2+

1

p;1 − b

1 + b

)

− 2ν

bp2

(p+ 1)2

(b− 1)2(p+ 2)(2p+ 1)F

(1

2,1

p;1

2+

1

p;1 − b

1 + b

)}(113)

Now substituting the perturbation terms R from (27) and carrying out the


dA

dt=

4εδA2m+2

Bbm+1

p 2m+1

p

F

(m+ 1

p,m+ 1

p,m+ 1

p+

1

2;b− 1

2b

)B

(m+ 1

p,1

2

)

− 4εβA2

Bb1p 2

1p

[B2F

(2 +

1

p,1

p,3

2+

1

p;b− 1

2b

)B

(1

p,3

2

)

+κ2F

(1

p,1

p,1

2+

1

p;b− 1

2b

)B

(1

p,1

2

)]

+2ε

pLAp−2

(p+ 1

2p2

) 12p

⎡⎣σ1

∫ ∞

−∞cosφ

(1 + a cosh τ)12p

dx + σ2

∫ ∞

−∞sinφ

(1 + a cosh τ)12p

dx

⎤⎦

(114)

dκ

dt= −εβκB

2

4p2A2

F(2 + 1

p, 1 + 1

p, 2 + 1

p; b−1

2b

)F(

1p, 1p, 1

2+ 1

p; b−1

2b

) B(1 + 1

p, 1)

B(

1p, 1

2

)

− ε

E

∫ ∞

−∞

⎡⎣aB

2p

(σ2 cos φ− σ1 sinφ) sinh τ

(1 + a cosh τ)2p+12p


(1 + a cosh τ)12p

⎤⎦ dx(115)



plification

dA

dt= −ε

⎛⎝A2m+1 − ζ

(3)1

A

⎞⎠ (116)

dκ

dt= −ε

[κ− ζ

(3)2 (1 + A− κ)

](117)

where A is the fixed point of the amplitude while

ζ(3)1 = σ1

∫ ∞

−∞cosφ

(1 + a cosh τ)12p

dx + σ2

∫ ∞

−∞sinφ

(1 + a cosh τ)12p

dx (118)

and

ζ(3)2 =

∫ ∞

−∞

⎡⎣aB

2p


(1 + a cosh τ)2p+12p


(1 + a cosh τ)12p

⎤⎦ dx

(119)

Once again, as in the case of Kerr law nonlinearity, one can derive the mean

drift velocity of the soliton as

〈v〉 =D

1 −D(120)

7 CONCLUSIONS

In this paper, the dynamics of optical solitons with non-Kerr law nonlinearities

in presence of perturbation terms, both deterministic as well as stochastic, are

studied. The Langevin equations were derived and the corresponding parame-

ter dynamics was studied. The mean drift velocity of the soliton was obtained.

In this study, it was assumed that the stochastic perturbation term σ is a

function of t only, for simplicity. However, in reality, σ is a function of both

x and t and thus making it a far more difficult system to analyze although

such kind of situations are being presently studied. Although, in this paper

the stochastic perturbation due to other non-Kerr law nonlinearities was not

studied, for example saturable law, triple power law of nonlinearity amongst

many others, the results of those studies are awaited at this time.


ACKNOWLEDGEMENT

This research, for the first author, was fully supported by NSF Grant No:

HRD-970668 and the support is genuinely and thankfully appreciated.

The research work of the third author is partially supported by AFOSR and

the grant number is FA9550-05-1-0377 and partially supported by US Army

Research and the award number is W911NF-05-1-0451. Therefore E.Z. would

like to thankfully acknowledge these supports.

References

[1] F. Abdullaev, S. Darmanyan, P. Khabibullaev. Optical Solitons. Springer

Verlag, New York, NY, USA. (1993).

[2] F. K. Abdullaev & J. Garnier. “Solitons in media with random dispersive

perturbations” Physica D. Vol 134, Issue 3, 303-315. (1999).

[3] F. K. Abdullaev, J. C. Bronski & G. Papanicolaou. “Soliton perturbations

and the random Kepler problem”. Physica D. Vol 135, Issue 3-4, 369-386.

(2000).

[4] F. K. Abdullaev & B. B. Baizakov. “Disintegration of soliton in a

dispersion-managed optical communication line with random parame-

ters”. Optics Letters. Vol 25, No 2, 93-95. (2000).

[5] A. Biswas. “Quasi-stationary non-Kerr law optical solitons” Optical Fiber

Technology. Vol 9, Issue 4, 224-259. (2003).

[6] A. Biswas. “Dynamics of stochastic optical solitons”. Journal of Electro-

magnetic Waves and Applications. Vol 18, Number 2, 145-152. (2004).

[7] A. Biswas. “Stochastic perturbation of optical solitons in Schrodinger-

Hirota equation”. Optics Communications. Vol 239, Issue 4-6, 457-462.

(2004).


[8] A. Biswas. “Stochastic perturbation of dispersion-managed optical soli-

tons”. Optical and Quantum Electronics. Vol 37, Number 7, 649-659.

(2005).

[9] A. Biswas & K. Porsezian. “Soliton perturbation theory for the modified

nonlinear Schrodinger’s equation”. To appear.

[10] K. J. Blow, N. J. Doran & D. Wood. “Suppression of the soliton self-

frequency shift by bandwidth-limited amplification” J. Opt Soc Am B.

Vol 5, No 6, 1301-1304. (1988).

[11] J. N. Elgin. “Stochastic perturbations of optical solitons”. Optics Letters.

Vol 18, No 1, 10-12. (1993).

[12] C. G. Goedde, W. L. Kath & P. Kumar. “Controlling soliton perturbations

with phase-sensitive amplification”. J. Opt Soc Am B. Vol 14, No 6, 1371-

1379. (1997).

[13] A. Hasegawa & Y. Kodama. Solitons in Optical Communications. Oxford

University Press, Oxford. (1995).

[14] D. J. Kaup. “Perturbation theory for solitons in optical fibers” Physical

Review A. Vol 42, No 9, 5689-5694. (1990).

[15] Y. Kivshar & G. P. Agarwal. Optical Solitons From Fiber Optics to Pho-

tonic Crystals. Academic Press. (2003).

[16] Y. Kodama & A. Hasegawa. “Generation of asymptotically stable optical

solitons and suppression of the Gordon-Haus effect” Optics Letters. Vol

17, 31-33. (1992).

[17] Y. Kodama & A. Hasegawa. “Theoretical Foundation of Optical-Soliton

concept in Fibers” Progress in Optics Vol XXX, No IV, 205-259. (1992).

[18] S. Konar & Soumendu Jana. “Linear and nonlinear propagation of sinh-

Gaussian pulses in dispersive media possessing Kerr nonlinearity”. Optics

Communications. Vol 236, Issues 1-3, 7-20. (2004).


[19] S. Konar, J. Kumar & P. K. Sen. “Suppression of soliton instability by

higher order nonlinearity in long haul optical communication systems”.

Journal of Nonlinear Optical Physics and Materials. Vol 8, Number 4,

497-502. (1999).

[20] S. Wabnitz, Y. Kodama. & A. B. Aceves “Control of Optical Soliton

Interactions” Optical Fiber Technology. Vol 1, 187-217. (1995).

Received: March 16, 2006

statistical dynamics of optical solitons in a non-kerr law media

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