Stochastic Soliton Solutions of the High-Order Nonlinear SchrödingerEquation in the Optical Fiber with Stochastic Dispersion andNonlinearityHui Zhong, Bo Tian, Hui-Ling Zhen, and Wen-Rong Sun

State Key Laboratory of Information Photonics and Optical Communications, and School ofScience, Beijing University of Posts and Telecommunications, Beijing 100876, China

Reprint requests to B. T.; E-mail: tian_bupt@163.com

Z. Naturforsch. 69a, 21 – 33 (2014) / DOI: 10.5560/ZNA.2013-0071Received June 3, 2013 / revised September 7, 2013 / published online December 18, 2013

In this paper, the high-order nonlinear Schrödinger (HNLS) equation driven by the Gaussian whitenoise, which describes the wave propagation in the optical fiber with stochastic dispersion and non-linearity, is studied. With the white noise functional approach and symbolic computation, stochasticone- and two-soliton solutions for the stochastic HNLS equation are obtained. For the stochastic onesoliton, the energy and shape keep unchanged along the soliton propagation, but the velocity andphase shift change randomly because of the effects of Gaussian white noise. Ranges of the changesincrease with the increase in the intensity of Gaussian white noise, and the direction of velocity isinverted along the soliton propagation. For the stochastic two solitons, the effects of Gaussian whitenoise on the interactions in the bound and unbound states are discussed: In the bound state, periodicoscillation of the two solitons is broken because of the existence of the Gaussian white noise, andthe oscillation of stochastic two solitons forms randomly. In the unbound state, interaction of thestochastic two solitons happens twice because of the Gaussian white noise. With the increase in theintensity of Gaussian white noise, the region of the interaction enlarges.

Key words: Stochastic Solitons; Gaussion White Noise; High-Order Nonlinear SchrödingerEquation; Symbolic Computation; White Noise Functional Approach.

1. Introduction

Nonlinear Schrödinger (NLS) equations describethe wave propagation in different nonlinear media,such as the nonlinear fibers [1], photonic crystals [2],Bose-Einstein condensates [3], and ion plasmas [4].Some NLS solitons have been studied to analyse theinteraction between the nonlinearity and dispersion [5,6]. In reality, such models in the nonlinear media in-volve certain uncertainties [7]. Having considered theeffects of stochastic coefficients or initial conditions onthe soliton solutions, people are also able to work onthe NLS equations driven by the noises [8 – 14]. Evo-lution of the NLS solitons with stochastic cubic non-linearity has been investigated via the numerical sim-ulations [9]. NLS solitons with white noise dispersionwhich describe the propagation of a signal in an opti-cal fiber with dispersion management have been stud-ied [10], and dispersion-managed vector solitons in the

birefringent optical fibers with stochastic birefringencehave been discussed numerically [11].

Propagation of the soliton pulse in a single-modefiber with delayed Raman response and random param-eters is described by the stochastic NLS equation withhigh-order nonlinearity and dispersion [1, 15 – 17],

∂U∂ z

+ i3

∑k≥2

ikβk(z)k!

∂ kU∂ tk = iγ(z)

(1+

iω0

∂

∂ t

)·[U (z, t)

∫ t

−∞

R(t ′)|U(z, t− t ′

)|2 dt ′

],

(1)

where U(z, t) is the slowly varying normalized enve-lope of the ultrashort pulse, z is the normalized dis-tance along the direction of the propagation, t is theretarded time, t ′ is the formal variable, ω0 is the centerfrequency of the ultrashort pulse spectrum, R(t) is theRaman response function, while the kth-order group-velocity-dispersion (GVD) coefficient βk(z) and non-

© 2014 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com

22 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

linearity coefficient γ(z) are considered as stochasticfunctions:

βk(z) = β′k

[1+mβk

(z)], γ(z) = γ0

[1+mγ(z)

],

where β ′k and γ0 are respectively the mean values ofβk(z) and γ(z), mβk

(z) and mγ(z) are the zero-meanstochastic processes of the Gaussian white noise,

〈mβk〉= 〈mγ〉= 0 , 〈mβk

(z)mβk(z′)〉= 2σ

2βk

δ (z− z′) ,

〈mγ(z)mγ(z′)〉= 2σ2γ δ (z− z′) ,

where 〈· · · 〉 denotes the statistical average, z′ is theformal variable, and σβk

and σγ are respectively thevariances of stochastic processes mβk

and mγ . Mod-ulational instability of the periodic pluse arrays de-scribed by (1) in the optical fiber with stochastic pa-rameters of high-order nonlinearity and dispersion hasbeen studied [15]. In the nonlocal focusing and defo-cusing Kerr media with stochastic dispersion and non-linearity, effects of the noises on the modulational in-stability for (1) have been discussed [16, 17]. In thecase when the pulse envelope evolves slowly along thefiber, (1) can be approximately written as [1]

∂U∂ z

+iβ2(z)

2∂ 2U∂ t2 −

β3(z)6

∂ 3U∂ t3 =

iγ(z)[|U |2U +

iω0

∂

∂ t

(|U |2U

)− iτRU

∂ |U |2

∂ t

],

(2)

where τR is the Raman resonant time constant. Al-though the periodic-like solutions [18] and numericalsimulation [19] of (2) without high-order nonlinear-ity and dispersion have been obtained, to our knowl-edge, stochastic soliton solutions for (2), in the opti-cal fiber with stochastic high-order nonlinearity anddispersion, have not been obtained as yet. Effects ofthe noises on the optical solitons have been discussedin the optical fiber communication systems [20], fiberlasers [21, 22], and fiber optical parametric ampli-fiers [23], so that the stochastic optical solitons, whichwe will obtain based on (2), might mirror the effectsof the Gaussian white noise on those studies with thehigh-order dispersion and nonlinearity.

In this paper, we will work on (2) via the white noisefunctional approach1 and symbolic computation [5,

1With the white noise functional approach [24], certain ana-lytic solutions for the stochastic Korteweg–de Vries (KdV) [25 – 27],KdV–Burgers [28, 29], and (2 + 1)-dimensional Broer–Kaup equa-tions [30] have been obtained.

31, 32]. In Section 2, the Wick product, Hida test func-tion space, and Hida distribution space will be con-structed, then (2) will be transformed into the Wick-type stochastic NLS equation. Via the Hermite trans-formation, wick-type stochastic NLS equation will betransformed into an equation under certain conditionsto obtain the solutions of (2). In Section 3, stochas-tic one- and two-soliton solutions will be obtained viathe symbolic computation and inverse Hermite trans-form, and the effects of Gaussian white noise on thedynamic properties of the solitons will be discussed.In Section 4, the stabilities of stochastic solitons willbe studied through the numerical simulation. Finally,our conclusions will be given in Section 5.

2. White Noise Analysis of Equation (2)

In (2), the Gaussian-white-noise mβ2(z), mβ3

(z), andmγ(z) have the following properties:

〈mβ2(z)〉= 〈mβ3

(z)〉= 〈mγ(z)〉= 0 ,

mβ2(z) = h1W (z) , mβ3

(z) = h2W (z) ,mγ(z) = h3W (z) ,

where h1, h2, and h3 are all non-zero constants, whichrepresent the intensity coefficients of the Gaussianwhite noises, respectively, W (z) is the standard Gaus-sian white noise, W (z) = dB(z)

dz , and B(z) is the standardBrownian motion [24].

The white noise functional approach to study thestochastic partial differential equations in the Wickversions has been given in [24]. For (2), U =U(z, t,W )is the generalized stochastic process and t ∈ Rd . Let(S(Rd)) and (S(Rd))∗ be the Hide test function spaceand Hide function space on Rd , respectively [33]. Lethn(x) be the dth-order Hermite polynomials. Setting

ξn(x) = e−12 x2

hn(√

2x)/(π (n−1)!)12 , we denote α =

(α1, · · · ,αd) being the d-dimensional multi-indiceswith α ′js ( j = 1, · · · ,d) ∈ N (N denotes the set of the

natural numbers), and let α(i) = (α(i)1 , · · · ,α(i)

d ) be theith multi-index number in some fixed ordering of allthe d-dimensional multi-indices α = (α1, · · · ,αd) ∈Nd [34 – 36]. For α , we define

Hα(ω) =∞

∏i=1

hαi (〈ω,ηi〉) , ω ∈(

S(Rd))∗

ηi = ξα(i) = ξ

α(i)1⊗·· ·⊗ξ

α(i)d

,

where ⊗ is the tensor product.

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 23

Fix n ∈ N. Let (S)n1 consist of those x = ∑α cα Hα

with cα ∈ Rn such that ‖x‖ = ∑α c2α (α!)2 (2N)kα ,

∀k ∈ N with c2α = |cα |2 = ∑

nk=1(c

(k)α )2 if cα =

(c(1)α , · · · ,c(n)

α ) ∈ Rn.The Wick product can be defined as [24, 37, 38]

F �G = ∑α,β

(aα ,bβ )Hα+β , (3)

where β is the d-dimensional multi-indices, F =∑α aα Hα , G = ∑β bβ Hβ ∈ (S)n

−1 with aα ,bβ ∈Rn. Byinterpreting Wick versions, we can write (2) as

∂U∂ z

+i2�H1(z)�

∂ 2U∂ t2 −

16�H2(z)�

∂ 3U∂ t3 =

i�H3(z)�

[|U |�2 �U +

iω0� ∂

∂ t

(|U |�2 �U

)− iτR �U � ∂ |U |�2

∂ t

], (4)

where H1(z) = β ′2 [1+h1W (z)], H2(z) = β ′3[1+h2W (z)], and H3(z) = γ0 [1+h3W (z)].

For F = ∑α aα Hα ∈ (S)n−1 with aα ∈ Rn, the Her-

mite transform of F is defined as [24, 37, 38]

F(z) =H(F) = ∑α

aα ωα ∈ Cn, (5)

where ω ∈Cn is a complex variable. For F,G ∈ (S)n−1,

by the Wick product definition, we have

H(F(W )�G(W )) = ˜F(W )�G(W )

= F(ω) · G(ω) .(6)

With the Hermite transformation of (4), the Wickproducts are turned into the ordinary products and (4)is written as

∂ u(z, t,ω)∂ z

+i2

H1 (z,ω)∂ 2u(z, t,ω)

∂ t2

− 16

H2 (z,ω)∂ 3u(z, t,ω)

∂ t3 =

iH3 (z,ω)[|u(z, t,ω) |2u(z, t,ω)

+i

ω0

∂

∂ t

(|u(z, t,ω) |2u(z, t,ω)

)− iτRu(z, t,ω)

∂ |u(z, t,ω) |2

∂ t

],

(7)

where u(z, t,ω) = H[U (z, t,W )], H1(z,ω) =β ′2[1 + h1W (z,ω)], H2(z,ω) = β ′3[1 + h2W (z,ω)],and H3(z,ω) = γ0[1 + h3W (z,ω)] with W (z,ω) =H[W (z)] = ∑

∞k=1 ηk(z)ω . Once the solutions u(z, t,ω)

for (7) is obtained, the solutions U(z, t,W ) for (2)can be obtained by the inverse Hermite transform ofu(z, t,ω).

3. Stochastic Soliton Solutions and Discussions

3.1. Bilinear Forms and Stochastic Soliton Solutions

To obtain the bilinear forms for (7), we introduce thedependent variable transformation,

u(z, t,ω) =g(z, t,ω)f (z, t,ω)

, (8)

where g(z, t,ω) is a complex differentiable functionand f (z, t,ω) is a real one. With h2 = h3 and β ′3 = τR =1

ω0, the bilinear forms for (7) can be obtained as[

Dz +i2

H1 (z,ω) D2t −

16

H2 (z,ω) D3t

]g · f = 0 , (9)

H1 (z,ω)D2t f · f = H3 (z,ω)g ·g∗, (10)

with “*” representing the complex conjugate and theHirota operators Dz and Dt defined by [31, 32]

Dmz Dn

t (a ·b) =(

∂

∂ z− ∂

∂ z′

)m (∂

∂ t− ∂

∂ t ′

)n

·a(z, t)b(z′, t ′)∣∣∣∣z′=z,t ′=t

,

(11)

z′ and t ′ being the formal variables, a(z, t) as the func-tion of z and t, b(z′, t ′) as the function of z′ and t ′,m = 0,1,2, · · · and n = 0,1,2, · · · .

Via the Hirota method [31, 32], g(z, t,ω) andf (z, t,ω) can be expanded as

g(z, t,ω) = ε g1(z, t,ω)+ ε3 g3(z, t,ω)

+ ε5 g5(z, t,ω)+ · · · ,

(12)

f (z, t,ω) = 1+ ε2 f2(z, t,ω)+ ε

4 f4(z, t,ω)

+ ε6 f6(z, t,ω)+ · · · ,

(13)

with ε is a small parameter. gn(z, t,ω) and fn(z, t,ω)(n = 1,2, · · ·) are the functions to be determined.

24 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

Substituting Expressions (12) and (13) into BilinearForms (9) and (10) and equating the coefficients of thesame powers of ε to zero yield the recursion relationsfor gn(z, t,ω) and fn(z, t,ω) (n = 1,2, · · ·).

To obtain the one-soliton solutions for (7), we trun-cate Expressions (12) and (13) for g1(z, t,ω) andf2(z, t,ω), respectively. Setting

g1(z, t,ω) = n1 eθ , f2(z, t,ω) = m1 eθ+θ∗ ,(14)

where θ = k (z,ω)+b1t = k (z,ω)+(b11 + ib12)t andn1 = n11 + in12 with n11, n12, b11, b12, and m1 as

the real constants, k (z,ω) as the complex function.Substituting Expressions (14) into Bilinear Forms (9)and (10), we can get the constraints on the parame-ters,

k (z,ω) =∫ z

−∞

(b3

1− ib21

)H2 (ξ ,ω) dξ ,

m1 =−(n2

11 +n212

)8b2

11

, α =2γ0

β ′2h3 = h1 .

With ε = 1, the one-soliton solutions for (7) can beexpressed as

u(z, t,ω) =n1 eb1t+k(z,ω)

1− (n211+n2

12)γ0

4β ′2b211

e(b1+b∗1)t+k(z,ω)+k∗(z,ω)

=n1 eb1t+

∫ z0(b3

1−ib21)H2(ξ ,ω)dξ

1− (n211+n2

12)γ0

4β ′2b211

e(b1+b∗1)t+∫ z

0(b31−ib2

1)H2(ξ ,ω)dξ+∫ z

0(b∗31 +ib∗21 )H2(ξ ,ω)dξ

.

(15)

To obtain the two-soliton solutions for (7), we as-sume that

g1(z, t) = m1 eθ1 +m2 eθ2 ,

f2(z, t) = n1 eθ1+θ∗1 +n2 eθ2+θ∗2 +n3 eθ1+θ∗2

+n4 eθ2+θ∗1 ,

g3(z, t) = m3 eθ1+θ2+θ∗1 +m4 eθ1+θ2+θ∗2 ,

f4(z, t) = O1 eθ1+θ2+θ∗1 +θ∗2 ,

where O1 = O11 + iO12, m j = m j1 + im j2, n j = n j1 +in j2, and θl = kl (z,ω) + blt = kl (z,ω) + (bl1 +bl2) twith O11, O12, m j1, m j2, n j1, n j2, bl1, and bl2 as thereal constants, and kl (z,ω) as the complex functions( j = 1,2 · · ·4, l = 1,2). Substituting them into BilinearForms (9) and (10), we can get the constraints on theparameters,

k1 (z,ω) =∫ z

−∞

(b11 + ib12)2 (b11 + ib12− iα1)

· H2 (ξ ,ω) dξ ,

k2 (z,ω) =∫ z

−∞

(b21 + ib22)2 (b21 + ib22− iα1)

· H2 (ξ ,ω) dξ ,

n1 =−(m2

11 +m212

)α2

8b211

,

n2 =−(m2

21 +m222

)α2

8b221

,

n3 =−(m11 + im12)(m21− im22)α2

2(b11 + ib12 +b21− ib22)2 ,

n4 =−(m11− im12)(m21 + im22)α2

2(b11− ib12 +b21 + ib22)2 ,

m3 =−m2(m2

11 +m212

)α2 (b1−b2)

2

8b211

(b1 +b∗2

)2 ,

m4 =−m1(m2

21 +m222

)α2 (b1−b2)

2

8b221

(b1 +b∗2

)2 ,

α1 =3β ′2β ′3

α2 =6γ0

β ′3h1 = h2 ,

O1 =

(m2

11 +m212

)(m2

21 +m222

)α2

2

[(b11−b21)

2 +(b12−b22)2]2

64b211b2

21

[(b11 +b21)

2 +(b12−b22)2]2 .

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 25

With ε = 1, the two-soliton solutions for (7) can be expressed as

u(z, t,ω) =m1 eb1t+ +m2 eθ2 +m3 eθ1+θ2+θ∗1 +m4 eθ1+θ2+θ∗2

n1 eθ1+θ∗1 +n2 eθ2+θ∗2 +n3 eθ1+θ∗2 +n4 eθ2+θ∗1 +O1 eθ1+θ2+θ∗1 +θ∗2

={

m1 eb1t+b21(b1−iα1)

∫ z−∞ H2(ξ ,ω)dξ +m2 eb2t+b2

2(b2−iα1)∫ z−∞ H2(ξ ,ω)dξ

+m3 e(b2+2b11)t+[2b11(b211+2α1b12−3b2

12)+b22(b2−iα1)]

∫ z−∞ H2(ξ ,ω)dξ

+m4 e(b1+2b21)t+[2b21(b221+2α1b22−3b2

22)+b21(b1−iα1)]

∫ z−∞ H2(ξ ,ω)dξ

}/{n1 e2b11t+2b11(b2

11+2α1b12−3b212)∫ z−∞ H2(ξ ,ω)dξ

+n2 e2b21t+2b21(b221+2α1b22−3b2

22)∫ z−∞ H2(ξ ,ω)dξ

+n3 e(b1+b∗2)t+[b21(b1−iα1)+b∗22 (b∗2+iα1)]

∫ z−∞ H2(ξ ,ω)dξ

+n4 e(b2+b∗1)t+[b22(b2−iα1)+b∗21 (b∗1−iα1)]

∫ z−∞ H2(ξ ,ω)dξ

+O1 e2(b11+b21)t+2[b311+b3

21+b11(2α1−3b12)b12+b21(2α1−3b22)b22]∫ z−∞ H2(ξ ,ω)dξ

}. (16)

Taking the inverse Hermite transformation of solutions (15), the one-soliton solutions for (4) are obtained as

U(z, t) =n1 e�

∫ z0(b3

1−ib21)H2(ξ )dξ+b1t

1− (n211+n2

12)γ0

4β ′2b211

e�[∫ z

0(b31−ib2

1)H2(ξ )dξ+∫ z

0(b∗31 +ib∗21 )H2(ξ )dξ ]+(b1+b∗1)t

=n1 e(b3

1−ib21)β ′3z+b1t e�β

′3h2(b3

1−ib21)∫ z−∞ W (ξ )dξ+b1t

1− (n211+n2

12)γ0

4β ′2b211

e(b31−ib2

1+b∗31 +ib∗21 )β ′3z+(b1+b∗1)t e�β′3h2(b3

1−ib21+b∗31 +ib∗21 )

∫ z−∞ W (ξ )dξ

=n1 e(b3

1−ib21)β ′3z+b1t e�β

′3h2(b3

1−ib21)B(z)

1− (n211+n2

12)γ0

4β ′2b211

e(b31−ib2

1+b∗31 +ib∗21 )β ′3z+(b1+b∗1)t e�β′3h2(b3

1−ib21+b∗31 +ib∗21 )B(z)

. (17)

Since e�B(z) = eB(z)− 12 z2

with � representing the Wick products and B(z) representing the standard Brownianmotion [24], the stochastic one-soliton solutions for (2) can be expressed as

U(z, t) =n1 e(b3

1−ib21)β ′3z+(b3

1−ib21)β ′3h2B(z)− 1

2 (b31−ib2

1)β ′3h2z2+b1t

1− (n211+n2

12)γ0

4β ′2b211

e(b31−ib2

1+b∗31 +ib∗21 )β ′3z+β ′3h2(b31−ib2

1+b∗31 +ib∗21 )[B(z)− 12 z2]+(b1+b∗1)t

. (18)

Similarly, through the inverse Hermite transformation, Solutions (16) for (4) is transformed into

U(z, t) ={

m1 eb1t+b21(b1−iα1)β ′3z e�b

21(b1−iα1)β ′3h2B(z)

+m2 eb2t+b22(b2−iα1)β ′3z e�b

22(b2−iα1)β ′3h2B(z)

+m3 e(b2+2b11)t+[2b11(b211+2α1b12−3b2

12)+b22(b2−iα1)]β ′3z

· e�[2b11(b211+2α1b12−3b2

12)+b22(b2−iα1)]β ′3h2B(z)

+m4 e(b1+2b21)t+[2b21(b221+2α1b22−3b2

22)+b21(b1−iα1)]β ′3z

· e�[2b21(b221+2α1b22−3b2

22)+b21(b1−iα1)]β ′3h2B(z)}/{

n1 e2b11t+2b11(b211+2α1b12−3b2

12)β ′3z e�2b11(b211+2α1b12−3b2

12)β ′3h2B(z)

+n2 e2b21t+2b21(b221+2α1b22−3b2

22)β ′3z e�2b21(b221+2α1b22−3b2

22)β ′3h2B(z)

26 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

+n3 e(b1+b∗2)t+[b21(b1−iα1)+b∗22 (b∗2+iα1)]β ′3z e�[b

21(b1−iα1)+b∗22 (b∗2+iα1)]β ′3h2B(z)

+n4 e(b2+b∗1)t+[b22(b2−iα1)+b∗21 (b∗1−iα1)]β ′3z e�[b

22(b2−iα1)+b∗21 (b∗1−iα1)]β ′3h2B(z)

+O1 e2(b11+b21)t+2[b311+b3

21+b11(2α1−3b12)b12+b21(2α1−3b22)b22]β ′3z

· e�2[b311+b3

21+b11(2α1−3b12)b12+b21(2α1−3b22)b22]β ′3h2B(z)} . (19)

The stochastic two-soliton solutions for (2) can be expressed as

U(z, t) ={

m1 eb1t+b21(b1−iα1)β ′3z eb2

1(b1−iα1)β ′3h2[B(z)− 12 z2]

+m2 eb2t+b22(b2−iα1)β ′3z eb2

2(b2−iα1)β ′3h2[B(z)− 12 z2]

+m3 e(b2+2b11)t+[2b11(b211+2α1b12−3b2

12)+b22(b2−iα1)]β ′3z

· e[2b11(b211+2α1b12−3b2

12)+b22(b2−iα1)]β1h2[B(z)− 1

2 z2]

+m4 e(b1+2b21)t+[2b21(b221+2α1b22−3b2

22)+b21(b1−iα1)]β ′3z

· e[2b21(b221+2α1b22−3b2

22)+b21(b1−iα1)]β ′3h2[B(z)− 1

2 z2]}/{n1 e2b11t+2b11(b2

11+2α1b12−3b212)β ′3z e2b11(b2

11+2α1b12−3b212)β ′3h2[B(z)− 1

2 z2]

+n2 e2b21t+2b21(b221+2α1b22−3b2

22)β ′3z e2b21(b221+2α1b22−3b2

22)β ′3h2[B(z)− 12 z2]

+n3 e(b1+b∗2)t+[b21(b1−iα1)+b∗22 (b∗2+iα1)]β ′3z e[b

21(b1−iα1)+b∗22 (b∗2+iα1)]β ′3h2[B(z)− 1

2 z2]

+n4 e(b2+b∗1)t+[b22(b2−iα1)+b∗21 (b∗1−iα1)]β ′3z e[b

22(b2−iα1)+b∗21 (b∗1−iα1)]β ′3h2[B(z)− 1

2 z2]

+O1 e2(b11+b21)t+2[b311+b3

21+b11(2α1−3b12)b12+b21(2α1−3b22)b22]β ′3z

· e2[b311+b3

21+b11(2α1−3b12)b12+b21(2α1−3b22)b22]β ′3h2[B(z)− 12 z2]} . (20)

3.2. Discussions for Solutions (18) and (20)

For the effects of the Gaussian white noise on thestochastic one soliton, solution (18) is rewritten as

U(z, t) = Asech

[η1z+η2t+η3B(z)− 1

2η3z2 +C

]· eiη4z+iη5t+iη6B(z)− i

2 η6z2,

with

A = 2n1C , η1 =(b3

11 +2b11b12−3b11b212

)β′3 ,

η2 = b11,

η3 =(b3

11 +2b11b12−3b11b212

)β′3h2 ,

η4 =(−b2

11 +3b211b12 +b2

12−b312

)β′3 ,

η5 = b12,

η6 =(−b2

11 +3b211b12 +b2

12−b312

)β′3h2 ,

C =

√−β ′2b2

11

4(n2

11 +n212

)γ0

,

where, of the stochastic one soliton, A and C are theamplitude and initial position, η2 and η5 are respec-tively related to the pulsewidth and frequency, η1 and

50 250

−2

2

z

Amplitude

Fig. 1. Plot of the stochastic function B(z) via stochasticnumber sequence.

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 27

η3 are both related to the velocity, and η4 and η6 arerelated to the phase shifting. Therefore, the velocityand phase shifting of the stochastic one soliton are ob-tained as η1 +η3B(z)−η3z and η4z+η6B(z)− 1

2 η6z2.It shows that the Gaussian white noise affects the ve-

Fig. 2. (a) Stochastic one soliton via Solutions (18). (b) Energy of the stochastic one soliton via Solutions (18). Parametersare: n1 = 1− i, b1 = 1−2i, α =−20, β ′3 = 1, and h2 = 0.

Fig. 3. (a) Stochastic one soliton via Solutions (18). (b) Energy of the stochastic one soliton via Solutions (18). Parametersare: n1 = 1− i, b1 = 1−2i, α =−20, β ′3 = 1, and h2 = 0.2.

Fig. 4. (a) Stochastic one soliton via Solutions (18) with h2 = 0.5. (b) Stochastic one soliton via Solutions (18) with h2 = 0.8.Parameters are: n1 = 1− i, b1 = 1−2i, α =−20, and β ′3 = 1.

locity and phase shifting of the stochastic one soliton,while the amplitude, energy, and shape of the stochas-tic one soliton are unaffected by the Gaussian whitenoise. The average value of the velocity can be ob-tained as η1−η3z. It means that the acceleration is ex-

28 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

istent in the stochastic one soliton. When η3 > 0, thevalue of acceleration is negative. The velocity directionof the stochastic one soliton is inverted.

The stochastic function B(z) via stochastic num-ber sequence is generated by the stochastic algorithmsimulator in MATLAB, as shown in Figure 1. Withsuch stochastic function, compared with the solitonwithout stochastic perturbation, effects of the Gaus-sian white noise on the solitons are discussed. In Fig-ures 2 and 3, the energy and shape of the stochasticone soliton are not affected by the Gaussian whitenoise and are the same as the ones of one solitonwithout the stochastic perturbation, but the velocityof the stochastic one soliton is affected by the Gaus-sian white noise, and the effect is related to the in-tensity of the Gaussian white noise. When the inten-sity h2 of the Gaussian white noise is zero, the veloc-ity and phase shift of the one soliton are only relatedto β1, and the energy and velocity of the one solitonkeep unchanged, as seen in Figure 2. When the inten-sity h2 of the Gaussian white noise is not zero, veloc-ity change of the stochastic one soliton is positivelycorrelated with the intensity h2, as seen in Figures 3and 4.

Similarly, effects of the Gaussian white noise on thedynamics of the stochastic two solitons are discussed.Solutions (20) are rewritten as

U(z, t) =1Q

[χ1sech(a2 + ln

√n2 + iψ1) eiφ1

+ χ2sech(a1 + ln√

n1 + iψ2) eiφ2

] (21)

with

Q = κ1cosh(a1 +a2 +κ1)

+κ2cosh

(a1−a2 + ln

√n1

n2

)+κ3cos(a3−a4)+κ4cos

(a3−a4 +

π

2

),

a1 = b11t +b11(b2

11−3b212 +2b12α1

)·[

z+h2B(z)−h2z2

2

],

a2 = b21t +b21(b2

21−3b222 +2b22α1

)·[

z+h2B(z)−h2z2

2

],

a3 = b12t +[3b2

11b12−b312 +

(b2

12−b211

)α1]

·[

z+h2B(z)−h2z2

2

],

a4 = b22t +[3b2

21b22−b322 +

(b2

22−b221

)α1]

·[

z+h2B(z)−h2z2

2

],

χ1 =√

n2(m2

11 +m212

)(w2

1 +w22

) ,

χ2 =√

n1(m2

21 +m222

)(w2

1 +w22

) ,

ψ1 = ψ2 =12

arcsinw2√

w21 +w2

2

+π

2,

κ1 =(b11−b21)2 +(b12−b22)2

(b11 +b21)2 +(b12−b22)2

√n1n2 ,

κ2 =√

n1n2 ,

κ3 =−α2

(b2

21−b222

)(m11m21 +m12m22)+2(m11m22−m12m21)b21b22

2[(b11 +b21)

2 +(b12−b22)2]2

+2b12 (b22m11m21 +b21m12m21−b21m11m22 +b22m12m22)

+(b2

11−b212

)(m11m21 +m12m22)

,

κ4 = α2−2b21b22m11m21−b2

21m12m21 +b222m12m21 +b2

21m11m22−b222m11m22

2[(b11 +b21)

2 +(b12−b22)2]2

+b212 (m12m21−m11m22)+b2

11 (−m12m21 +m11m22)−2b21b22m12m22

+2b12 (b21m11m21−b22m12m21 +b22m11m22 +b21m12m22)

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 29

+2b11 [(b12−b22)(m11m21 +m12m22)−b21m12m21 +b21m11m22],

φ1 = a1 + arcsinm12√(

m211 +m2

12

) − 12

arcsinw2√

w21 +w2

2

,

φ2 = a2 + arcsinm12√(

m211 +m2

12

) − 12

arcsinw2√

w21 +w2

2

,

w1 =b4

11 +b412 +b4

21−4b312b22−6b2

21b222 +b4

22−6b212

(b2

21−b222

)+4b12

(3b2

21b22−b322

)[(b11 +b21)

2 +(b12−b22)2]2

+2b211

(b2

12−b221−2b12b22 +b2

22

),

w2 =4b21 (b12−b22)

(b2

11 +b212−b2

21−2b12b22 +b222

)[(b11 +b21)

2 +(b12−b22)2]2 ,

where the two hyperbolic secant functions in solu-tions (21) respectively represent the stochastic twosolitons, and Q reflects the interaction of the stochas-tic two solitons. Because of the nonlinear form ofQ, the interaction of the two solitons is nonlinear.The interaction of two optical solitons can be classi-fied into the different statuses, according to the dif-ferent initial optical pulses [39]. When the two ini-tial optical pulses have the equal initial phases, inter-action of the stochastic two solitons in a bound statehas the property of the period oscillation. Period ofthe oscillation is related to the cosine functions ofQ. For the effects of Gaussian white noise, the pe-riod changes randomly with the propagation distancez. The function form of the period can be obtained

Fig. 5. (a) Two solitons via Solutions (19) with h2 = 0. (b) Two stochastic solitons via Solutions (19) with h2 = 0.2. Parametersare: n1 = 1+ i, n2 = 1+ i, b1 = 1+ i, b2 = 2

√6+4i, α1 =−20, α2 = 3, and β ′3 = 1.

as

P(z) =[3b2

11b12−b312 +3b2

21b22−b322

+(b2

12−b211 +b2

22−b221

)α1][

1+h2B(z)−h2z2

].

When the intensity of the Gaussian white noise is zero,the periodic oscillation of the optical solitons in thebound state is stable, as shown in Figure 5a. When theintensity of the Gaussian white noise is not zero, the os-cillation of the stochastic two solitons is also stochas-tic, as shown in Figure 5b.

When the two initial optical pulses have differentinitial phases, the two optical solitons form the un-bound state. When the intensity of the Gaussian whitenoise is zero, the velocities of the two solitons have

30 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

Fig. 6. (a) Stochastic two solitons via Solutions (19). (b) Energy of the stochastic two solitons via Solutions (19). Parametersare: n1 =−1+ i, n2 =−1+ i, b1 = 0.5− i, b2 =−0.5+ i, α1 =−20, α2 = 3, β ′3 = 1, and h2 = 0.

Fig. 7. (a) Stochastic two solitons via Solutions (19). (b) Energy of the stochastic two solitons via Solutions (19). Parametersare: n1 =−1+ i, n2 =−1+ i, b1 = 0.5− i, b2 =−0.5+ i, α1 =−20, α2 = 3, β ′3 = 1, and h2 = 0.4.

opposite directions, as shown in Figure 6. When theintensity of the Gaussian white noise is not zero, thevelocity functions of two stochastic solitons can be ob-tained as

V1(z) = b11(b2

11−3b212 +2b12α1

)·[1+h2B(z)−h2

z2

],

V2(z) = b21(b2

21−3b222 +2b22α1

)·[1+h2B(z)−h2

z2

].

The velocities V1(z) and V2(z) change randomly be-cause of the effects of Gaussian white noise, and thevelocity directions of stochastic two solitons are in-verted along the soliton propagation. After the inter-

action of stochastic two solitons, the interaction hap-pens again because of the inverted velocity directions,as shown in Figure 7, while the region of the interac-tion changes with the increase of h2, and the distancebetween the two solitons changes randomly. When theintensity h2 of the Gaussian noise increases, the regionof interaction enlarges, as shown in Figures 6 and 7.

4. Stability Analysis of the Stochastic SolitonSolutions

For NLS equations in nonlinear optical fibers, thereare two numerical-simulation methods for the pulse-propagation problems, which are the finite differencemethod [9] and the split-step Fourier method [40 – 42].

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 31

Fig. 8 (colour online). (a) Numerical simulation on the one soliton of (2) without Gaussian white noise. (b) The case of (a)with Gaussian white noise.

Fig. 9 (colour online). (a) Numerical simulation on the two solitons of (2) without the Gaussian white noise. (b) The case of(a) with the Gaussian white noise.

In this paper, we will make use of the split-stepFourier method in the MATLAB software environ-ment to get the stability of stochastic soliton solu-tions for (2). For the stability analysis, dispersion andnonlinearity of (2) are added to the Gaussian whitenoise which is generated by the stochastic algorithmsimulator. In the region of t, 1024 points are simu-lated. The 2000 iterative times along the z-axis aretaken with the step size of 0.0001. The one- andtwo-soliton solutions are tested. Figure 8a shows theone soliton for (2) without the Gaussian white noise,and the one for (2) with the Gaussian white noise isshown in Figure 8b. Comparing Figure 8a and Fig-ure 8b, we find that the solitons with the Gaussianwhite noise show the smaller changes compared withthe solitons without the Gaussian white noise alongthe propagation. Similarly, the stability of two soli-tons is discussed, as seen in Figure 9. With the numeri-

cal simulation, one- and two-soliton solutions for (2)with the Gaussian white noise can propagate sta-bly.

5. Conclusions

Equation (2), the HNLS equation driven by theGaussian white noise, describes the wave propagationin the optical fiber with stochastic dispersion and non-linearity. In this paper, the stochastic solitons for (2)have been investigated analytically. With symboliccomputation and white noise functional approach, thestochastic one- and two-soliton solutions (18) and (20)have been obtained. For the stochastic one soliton, thevelocity and phase shift change randomly because ofthe effects of Gaussian white noise, and the ranges ofthe changes increase with the increase in the intensityh2 of Gaussian white noise. The energy and shape keep

32 H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation

unchanged along the soliton propagation, as shown inFigures 2, 3, and 4, but the direction of velocity forthe stochastic one soliton is inverted. For the stochastictwo solitons, the interactions in the bound and unboundstates have been discussed. The periodic oscillation ofthe two solitons in the bound state is broken because ofthe existence of Gaussian white noise, and the oscilla-tion of the stochastic two solitons forms randomly, asshown in Figure 5. In the unbound state, interaction ofthe stochastic two solitons happens twice because ofthe Gaussian white noise, as shown in Figure 7. Withthe increase in the intensity of Gaussian white noise,the region of the interaction enlarges, as shown in Fig-ures 6 and 7. Through the numerical simulation per-formed via the split-step Fourier method, the stochastic

one- and two-soliton solutions for (2) have been shownto be stable in Figures 8 and 9.

Acknowledgement

We express our sincere thanks to all the members ofour discussion group for their valuable comments. Thiswork has been supported by the National Natural Sci-ence Foundation of China under Grant No. 11272023,by the Open Fund of State Key Laboratory of In-formation Photonics and Optical Communications(Beijing University of Posts and Telecommunica-tions), and by the Fundamental Research Funds forthe Central Universities of China under Grant No.2011BUPTYB02.

[1] G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. Acad.Press, Boston 2007.

[2] Y. S. Kivshar and G. P. Agrawal, Optical Solitons:From Fibers to Photonic Crystals, Acad. Press, SanDiego 2003.

[3] K. Nakamura, D. Babajanov, D. Matrasulov, and M.Kobayashi, Phys. Rev. A 86, 053613 (2012).

[4] N. C. Lee, Phys. Plasmas 19, 082303 (2012).[5] B. Tian and Y. T. Gao, Phys. Lett. A 342, 228 (2005).[6] B. Tian and Y. T. Gao, Phys. Lett. A 359, 241 (2006).[7] V. V. Konotop and L. Vazquez, Nonlinear Random

Waves, World Sci., Salem 1994.[8] W. B. Cardoso, S. A. Leao, A. T. Avelar, D. Bazeia, and

M. S. Hussein, Phys. Lett. A 374, 4594 (2010).[9] W. B. Cardoso, A. T. Avelara, and D. Bazeiab, Phys.

Lett. A 374, 2640 (2010).[10] A. D. Bouard and A. Debussche, J. Funct. Anal. 259,

1300 (2010).[11] F. K. Abdullaev, B. A. Umarov, M. R. B. Wahiddin, and

D. V. Navotny, J. Opt. Soc. Am. B 17, 1117 (2000).[12] F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, and

M. P. Sorensen, J. Opt. Soc. Am. B 14, 27 (1997).[13] M. Karlsson, J. Opt. Soc. Am. B 15, 2269 (1998).[14] M. A. Molchan, Symmetry Integr. Geom. 3, 083

(2007).[15] C. G. L. Tiofack, A. Mohamadou, and T. C. Kofane,

Opt. Commun. 283, 1096 (2010).[16] E. V. Doktorov and M. A. Molchan, Int. Soc. Opt. Pho-

tonics 672513, 751290 (2007).[17] E. V. Doktorov and M. A. Molchan, Phys. Rev. A 75,

053819 (2007).[18] B. Chen and Y. C. Xie, J. Comput. Appl. Math. 203,

249 (2007).[19] G. P. Flessas, P. G. L. Leach, and A. N. Yannacopoulos,

J. Opt. B 6, S161 (2004).

[20] Y. Chung and A. Peleg, Phys. Rev. A 77, 063835(2008).

[21] E. Donkor, M. Noman, and P. D. Kumavor, Opt. Eng.45, 024202 (2006).

[22] C. Spiegelberg, J. H. Geng, Y. D. Hu, Y. Kaneda, S. B.Jiang, and N. Peyghambarian, J. Lightwave Technol.22, 57 (2004).

[23] P. Kylemark, M. Karlsson, T. Torounidis, and P. A. An-drekson, J. Lightwave Technol. 25, 612 (2007).

[24] H. Holden, B. Oksendal, J. Uboe, and T. S. Zhang,Stochastic Partial Differential Equations: A Modeling,White Noise Functional Approach, 2th ed. Acad. Press,New York 2009.

[25] S. Zhang and H. Q. Zhang, Phys. Lett. A 374, 4180(2010).

[26] C. Q. Dai and J. F. Zhang, Europhys. Lett. 86, 40006(2009).

[27] Q. Liu, D. L. Jia, and Z. H. Wang, Appl. Math. Comput.215, 3495 (2010).

[28] X. Han and Y. C. Xie, Chaos Soliton. Fract. 39, 1715(2009).

[29] C. M. Wei and Z. Q. Xia, Chaos Soliton. Fract. 26, 329(2005).

[30] C. Q. Dai and J. L. Chen, Phys. Lett. A 373, 1218(2009).

[31] R. Hirota, J. Math. Phys. 14, 805 (1973).[32] R. Hirota, Phys. Rev. Lett. 27, 1192 (1971).[33] C. F. Sun and H. J. Gao, Commun. Nonlin. Sci. 14,

1551 (2009).[34] W. Y. Jiang and H. Q. Zhang, Commun. Theor. Phys.

44, 981 (2005).[35] B. Chen and Y. C. Xie, Chaos Soliton. Fract. 23, 243

(2005).[36] B. Chen and Y. C. Xie, J. Comput. Appl. Math. 197,

345 (2006).

H. Zhong et al. · Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation 33

[37] Q. Liu, Chaos Soliton. Fract. 36, 1037 (2009).[38] Q. Liu, Europhys. Lett. 74, 377 (2006).[39] J. R. Taylor, Optical Solitons: Theory and Experiment,

Cambridge Univ. Press, New York 1992.[40] G. M. Muslu and H. A. Erbay, Math. Comput. Simul.

67, 581 (2005).

[41] O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R.Menyuk, J. Lightwave Technol. 21, 61 (2003).

[42] L. Zhao, Z. Sui, Q. H. Zhu, Y. Zhang, and Y. L. Zuo,Acta Phys. Sin. 58, 4731 (2009).