Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation in the Optical Fiber with Stochastic Dispersion and

Stochastic Soliton Solutions of the High-Order Nonlinear Schrödinger Equation in the Optical Fiber with Stochastic Dispersion and

Nonlinearity

Hui Zhong, Bo Tian, Hui-Ling Zhen, and Wen-Rong Sun

State Key Laboratory of Information Photonics and Optical Communications, and School of Science, 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.2013-0071

Received 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 white noise, 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, stochastic one- and two-soliton solutions for the stochastic HNLS equation are obtained. For the stochastic one soliton, the energy and shape keep unchanged along the soliton propagation, but the velocity and phase shift change randomly because of the effects of Gaussian white noise. Ranges of the changes increase with the increase in the intensity of Gaussian white noise, and the direction of velocity is inverted along the soliton propagation. For the stochastic two solitons, the effects of Gaussian white noise on the interactions in the bound and unbound states are discussed: In the bound state, periodic oscillation of the two solitons is broken because of the existence of the Gaussian white noise, and the oscillation of stochastic two solitons forms randomly. In the unbound state, interaction of the stochastic two solitons happens twice because of the Gaussian white noise. With the increase in the intensity of Gaussian white noise, the region of the interaction enlarges.

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

1. Introduction

Nonlinear Schrödinger (NLS) equations describe the 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 the interaction between the nonlinearity and dispersion [5, 6]. In reality, such models in the nonlinear media in- volve certain uncertainties [7]. Having considered the effects of stochastic coefficients or initial conditions on the soliton solutions, people are also able to work on the 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 dispersion which 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 birefringence have been discussed numerically [11].

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

∂U

∂z +i

3 k≥2

∑

i^kβ_k(z) k!

∂^kU

∂t^k =iγ(z)

1+ i ω₀

∂

∂t

·

U(z,t) Z _t

−∞

R t⁰

|U z,t−t⁰

|²dt⁰

,

(1)

whereU(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 the retarded time,t⁰is the formal variable,ω₀is the center frequency of the ultrashort pulse spectrum,R(t)is the Raman response function, while thekth-order group- velocity-dispersion (GVD) coefficientβ_k(z)and non-

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

linearity coefficient γ(z) are considered as stochastic functions:

β_k(z) =β_k⁰

1+m_βk(z)

, γ(z) =γ₀

1+m_γ(z) , whereβ_k⁰ andγ0 are respectively the mean values of βk(z) and γ(z), m_β

k(z) and m_γ(z) are the zero-mean stochastic processes of the Gaussian white noise, hm_β

ki=hm_γi=0, hm_β

k(z)m_β

k(z⁰)i=2σ_β²

kδ(z−z⁰), hm_γ(z)m_γ(z⁰)i=2σ_γ²δ(z−z⁰),

where h· · · i denotes the statistical average, z⁰ is the formal variable, and σ_βk and σ_γ are respectively the variances of stochastic processes m_β

k and m_γ. 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 has been 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 the case when the pulse envelope evolves slowly along the fiber, (1) can be approximately written as [1]

∂U

∂z +iβ₂(z) 2

∂²U

∂t² −β₃(z) 6

∂³U

∂t³ = iγ(z)

|U|²U+ i ω0

∂

∂t |U|²U

−iτ_RU∂|U|²

∂t

, (2)

where τR is the Raman resonant time constant. Al- though the periodic-like solutions [18] and numerical simulation [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 and dispersion, have not been obtained as yet. Effects of the noises on the optical solitons have been discussed in the optical fiber communication systems [20], fiber lasers [21,22], and fiber optical parametric ampli- fiers [23], so that the stochastic optical solitons, which we will obtain based on (2), might mirror the effects of the Gaussian white noise on those studies with the high-order dispersion and nonlinearity.

In this paper, we will work on (2) via the white noise functional 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 Section2, 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 be transformed into an equation under certain conditions to obtain the solutions of (2). In Section3, stochas- tic one- and two-soliton solutions will be obtained via the symbolic computation and inverse Hermite trans- form, and the effects of Gaussian white noise on the dynamic properties of the solitons will be discussed.

In Section4, the stabilities of stochastic solitons will be studied through the numerical simulation. Finally, our conclusions will be given in Section5.

2. White Noise Analysis of Equation (2) In (2), the Gaussian-white-noisem_β

2(z),m_β

3(z), and m_γ(z)have the following properties:

hm_β2(z)i=hm_β3(z)i=hmγ(z)i=0, m_β2(z) =h₁W(z), m_β3(z) =h₂W(z), mγ(z) =h3W(z),

whereh₁,h₂, andh₃are all non-zero constants, which represent the intensity coefficients of the Gaussian white noises, respectively,W(z)is the standard Gaus- sian white noise,W(z) =^dB(z)_dz , andB(z)is the standard Brownian motion [24].

The white noise functional approach to study the stochastic partial differential equations in the Wick versions has been given in [24]. For (2),U=U(z,t,W) is the generalized stochastic process andt∈R^d. Let (S(R^d))and(S(R^d))^∗be the Hide test function space and Hide function space onR^d, respectively [33]. Let h_n(x) be the dth-order Hermite polynomials. Setting ξ_n(x) =e^−12x2h_n(√

2x)/(π(n−1)!)¹², we denoteα= (α₁,· · ·,αd) being the d-dimensional multi-indices withα⁰_js(j=1,· · ·,d)∈N(Ndenotes the set of the natural numbers), and letα⁽ⁱ⁾= (α₁⁽ⁱ⁾,· · ·,α_d⁽ⁱ⁾)be the ith multi-index number in some fixed ordering of all the d-dimensional multi-indices α = (α₁,· · ·,α_d)∈ N^d[34–36]. Forα, we define

H_α(ω) =

∞

∏

i=1

h_αi(hω,η_ii), ω∈ S

R^d ∗

ηi=ξ_α(i) =ξ

α₁⁽ⁱ⁾⊗ · · · ⊗ξ

α_d⁽ⁱ⁾, where⊗is the tensor product.

Fix n∈N. Let(S)ⁿ₁ consist of thosex=∑αcαHα

with cα ∈ Rⁿ such that kxk =∑αc²_α(α!)²(2N)^kα,

∀k ∈ N with c²_α = |cα|² = ∑ⁿ_k=1(c^(k)_α )² if cα = (c⁽¹⁾_α ,· · ·,c⁽ⁿ⁾_α )∈Rⁿ.

The Wick product can be defined as [24,37,38]

FG=

∑

α,β

(a_α,b_β)H_α+β, (3) where β is the d-dimensional multi-indices, F =

∑αa_αH_α,G=∑βb_βH_β∈(S)ⁿ₋₁witha_α,b_β∈Rⁿ. By interpreting Wick versions, we can write (2) as

∂U

∂z +i

2H₁(z)∂²U

∂t² −1

6H₂(z)∂³U

∂t³ = iH₃(z)

"

|U|²U+ i ω0

∂

∂t |U|²U

−iτ_RU∂|U|²

∂t

#

, (4)

where H₁(z) = β₂⁰[1+h₁W(z)], H₂(z) = β₃⁰[1+

h2W(z)], andH3(z) =γ0[1+h3W(z)].

ForF=∑αa_αH_α ∈(S)ⁿ₋₁witha_α ∈Rⁿ, the Her- mite transform ofFis defined as [24,37,38]

F(z) =e H(F) =

∑

α

aαω^α∈Cⁿ, (5) whereω∈Cⁿis a complex variable. ForF,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 Wick products are turned into the ordinary products and (4) is written as

∂eu(z,t,ω)

∂z +i

2Hf₁(z,ω)∂²eu(z,t,ω)

∂t²

−1

6Hf₂(z,ω)∂³eu(z,t,ω)

∂t³ = ifH3(z,ω)

|ue(z,t,ω)|²eu(z,t,ω) + i

ω₀

∂

∂t |ue(z,t,ω)|²ue(z,t,ω)

−iτ_Rue(z,t,ω)∂|eu(z,t,ω)|²

∂t

,

(7)

where u(z,e t,ω) = H[U(z,t,W)], Hf1(z,ω) = β₂⁰[1 +h₁We(z,ω)], Hf₂(z,ω) = β₃⁰[1 +h₂We(z,ω)], and Hf₃(z,ω) =γ₀[1+h₃We(z,ω)] with We(z,ω) = H[W(z)] =∑^∞_k=1η_k(z)ω. Once the solutionsu(z,e t,ω) for (7) is obtained, the solutions U(z,t,W) for (2) can be obtained by the inverse Hermite transform of u(z,t,e ω).

3. Stochastic Soliton Solutions and Discussions 3.1. Bilinear Forms and Stochastic Soliton Solutions

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

ue(z,t,ω) =g(z,t,ω)

f(z,t,ω), (8)

where g(z,t,ω) is a complex differentiable function andf(z,t,ω)is a real one. Withh₂=h₃andβ₃⁰=τ_R=

1

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

D_z+i

2Hf₁(z,ω)D²_t −1

6Hf₂(z,ω)D³_t

g·f =0, (9) Hf₁(z,ω)D²_t f·f=Hf₃(z,ω)g·g^∗, (10) with “*” representing the complex conjugate and the Hirota operatorsDzandDt defined by [31,32]

D^m_z Dⁿ_t(a·b) = ∂

∂z− ∂

∂z⁰ m

∂

∂t− ∂

∂t⁰ n

·a(z,t)b(z⁰,t⁰) z⁰=z,t⁰=t

,

(11)

z⁰andt⁰being the formal variables,a(z,t)as the func- tion of z andt, b(z⁰,t⁰) as the function of z⁰ andt⁰, m=0,1,2,· · ·andn=0,1,2,· · ·.

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

g(z,t,ω) =εg1(z,t,ω) +ε³g3(z,t,ω)

+ε⁵g₅(z,t,ω) +· · ·, (12) f(z,t,ω) =1+ε²f₂(z,t,ω) +ε⁴f₄(z,t,ω)

+ε⁶f₆(z,t,ω) +· · ·, (13) withε is a small parameter.g_n(z,t,ω)and f_n(z,t,ω) (n =1,2,· · ·) are the functions to be determined.

Substituting Expressions (12) and (13) into Bilinear Forms (9) and (10) and equating the coefficients of the same powers ofεto zero yield the recursion relations forg_n(z,t,ω)and f_n(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) and

f₂(z,t,ω), respectively. Setting

g₁(z,t,ω) =n₁e^θ, f₂(z,t,ω) =m₁e^θ+θ∗, (14) whereθ=k(z,ω) +b₁t=k(z,ω) + (b₁₁+ib₁₂)tand n₁=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 _z

−∞ b³₁−ib²₁

Hf2(ξ,ω)dξ, m1=− n²₁₁+n²₁₂

8b²₁₁ , α=2γ₀

β₂⁰ h3=h1. Withε=1, the one-soliton solutions for (7) can be expressed as

eu(z,t,ω) = n₁e^b1t+k(z,ω) 1−(n²₁₁+n²₁₂)^γ0

4β₂⁰b²₁₁ e(b1+b^∗₁)^{t+k(z,ω)+k∗(z,ω)}

= n₁e^b1t+

R_z

0(b³₁−ib²₁)fH₂(ξ,ω)dξ

1−(n²₁₁+n²₁₂)γ₀

4β₂⁰b²₁₁ e(b₁+b^∗₁)t+^R₀^z(b³₁−ib²₁)Hf₂(ξ,ω)dξ+^R₀^z(b^∗3₁ +ib^∗2₁ )Hf₂(ξ,ω)dξ

.

(15)

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

g₁(z,t) =m₁e^θ1+m₂e^θ2,

f₂(z,t) =n₁e^θ1+θ1∗+n₂e^θ2+θ2∗+n₃e^θ1+θ2∗ +n₄e^θ2+θ1∗,

g₃(z,t) =m₃e^{θ1+θ2+θ1∗}+m₄e^{θ1+θ2+θ2∗}, f4(z,t) =O1e^{θ1+θ2+θ1∗+θ2∗},

whereO₁=O₁₁+iO₁₂,m_j=m_j1+im_j2,n_j=n_j1+ in_j2, andθ_l=k_l(z,ω) +b_lt=k_l(z,ω) + (b_l1+b_l2)t withO11,O12,mj1,mj2,nj1,nj2, b_l1, andb_l2 as the real constants, and k_l(z,ω)as the complex functions (j=1,2· · ·4,l=1,2). Substituting them into Bilinear Forms (9) and (10), we can get the constraints on the parameters,

k1(z,ω) = Z _z

−∞

(b₁₁+ib12)²(b₁₁+ib12−iα1)

·Hf₂(ξ,ω)dξ,

k₂(z,ω) = Z z

−∞

(b₂₁+ib₂₂)²(b₂₁+ib₂₂−iα₁)

·fH₂(ξ,ω)dξ, n1=− m²₁₁+m²₁₂

α2

8b²₁₁ , n2=− m²₂₁+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 m²₁₁+m²₁₂

α₂(b₁−b₂)² 8b²₁₁ b₁+b^∗₂2 , m₄=−m₁ m²₂₁+m²₂₂

α2(b₁−b₂)² 8b²₂₁ b₁+b^∗₂2 , α₁=3β₂⁰

β₃⁰ α₂=6γ₀

β₃⁰ h₁=h₂,

O₁=

m²₁₁+m²₁₂

m²₂₁+m²₂₂ α₂²

h

(b₁₁−b₂₁)²+ (b₁₂−b₂₂)²i2

64b²₁₁b²₂₁ h

(b₁₁+b21)²+ (b₁₂−b22)²i2 .

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

eu(z,t,ω) = m₁e^b1t++m₂e^θ2+m₃e^{θ1+θ2+θ1∗}+m₄e^{θ1+θ2+θ2∗} n₁e^θ1+θ1∗+n₂e^θ2+θ2∗+n₃e^θ1+θ2∗+n₄e^θ2+θ1∗+O₁e^{θ1+θ2+θ1∗+θ2∗}

=

m1e^{b1t+b21(b1−iα1)R−∞z Hf2(ξ,ω)dξ}+m₂e^{b2t+b22(b2−iα1)R−∞z Hf2(ξ,ω)dξ} +m₃e^(b2+2b11)t+[2b11(b²₁₁+2α₁b12−3b²₁₂)+b²₂(b₂−iα₁)]^R−∞^z Hf2(ξ,ω)dξ

+m₄e^(b1+2b21)t+[2b₂₁(b²₂₁+2α₁b₂₂−3b²₂₂)+b²₁(b₁−iα₁)]^R−^z∞Hf₂(ξ,ω)dξ

n₁e^{2b11t+2b11(b211+2α1b12−3b212)}

R_z

−∞Hf₂(ξ,ω)dξ

+n₂e^{2b21t+2b21(b221+2α1b22−3b222)R−∞z fH2(ξ,ω)dξ} +n₃e^(b1+b∗2)t+[b²₁(b₁−iα₁)+b^∗2₂ (b^∗₂+iα₁)]^R−^z∞Hf2(ξ,ω)dξ

+n₄e^(b2+b∗1)t+[b²₂(b₂−iα₁)+b^∗2₁ (b^∗₁−iα₁)]^R−∞^z Hf₂(ξ,ω)dξ

+O₁e^{2(b11+b21)t+2}[b³₁₁+b³₂₁+b₁₁(2α₁−3b₁₂)b₁₂+b₂₁(2α₁−3b₂₂)b₂₂]^R−∞^z Hf₂(ξ,ω)dξ . (16) Taking the inverse Hermite transformation of solutions (15), the one-soliton solutions for (4) are obtained as

U(z,t) = n₁e

R_z

0(b³₁−ib²₁)H₂(ξ)dξ+b₁t

1−(n²₁₁+n²₁₂)γ₀

4β₂⁰b²₁₁ e[^R₀^z(b³₁−ib²₁)H₂(ξ)dξ+^R₀^z(b^∗3₁ +ib^∗2₁ )H₂(ξ)dξ]+(b₁+b^∗₁)t

= n₁e(b³₁−ib²₁)β₃⁰z+b₁te^β30h2(b³₁−ib²₁)^R−∞^z W(ξ)dξ+b₁t

1−(n²₁₁+n²₁₂)γ₀

4β₂⁰b²₁₁ e(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )β₃⁰z+(b₁+b^∗₁)te^β30h2(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )^R−∞^z W(ξ)dξ

= n₁e(b³₁−ib²₁)β₃⁰z+b₁te^β30h2(b³₁−ib²₁)B(z)

1−(n²₁₁+n²₁₂)γ₀

4β₂⁰b²₁₁ e(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )β₃⁰z+(b₁+b^∗₁)te^β30h2(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )B(z)

. (17)

Since e^B(z)=e^B(z)−12z2 withrepresenting the Wick products andB(z)representing the standard Brownian motion [24], the stochastic one-soliton solutions for (2) can be expressed as

U(z,t) = n1e(b³₁−ib²₁)β₃⁰z+(b³₁−ib²₁)β₃⁰h₂B(z)−¹₂(b³₁−ib²₁)β₃⁰h₂z²+b₁t

1−(n²₁₁+n²₁₂)γ₀

4β₂⁰b²₁₁ e(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )β₃⁰z+β₃⁰h₂(b³₁−ib²₁+b^∗3₁ +ib^∗2₁ )[^B(z)−1₂z²]+(b₁+b^∗₁)t

. (18)

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

m₁e^{b1t+b21(b1−iα1)β30z}e^{b21(b1−iα1)β30h2B(z)} +m₂e^{b2t+b22(b2−iα1)β30z}e^{b22(b2−iα1)β30h2B(z)}

+m₃e^(b2+2b11)t+[2b₁₁(b²₁₁+2α₁b₁₂−3b²₁₂)+b²₂(b₂−iα₁)]β₃⁰z

·e[2b11(b²₁₁+2α₁b12−3b²₁₂)+b²₂(b₂−iα₁)]^β₃⁰h2B(z)

+m4e^(b1+2b21)t+[2b21(b²₂₁+2α1b22−3b²₂₂)+b²₁(b1−iα₁)]β₃⁰z

·e[2b₂₁(b²₂₁+2α₁b₂₂−3b²₂₂)+b²₁(b₁−iα₁)]β₃⁰h₂B(z)

n₁e^{2b11t+2b11(b211+2α1b12−3b212)β30z}e^{2b11(b211+2α1b12−3b212)β30h2B(z)} +n₂e^{2b21t+2b21(b221+2α1b22−3b222)β30z}e^{2b21(b221+2α1b22−3b222)β30h2B(z)}

+n3e^(b1+b∗2)t+[b²₁(b1−iα₁)+b^∗2₂ (b^∗₂+iα1)]β₃⁰ze[b²₁(b1−iα₁)+b^∗2₂ (b^∗₂+iα1)]β₃⁰h2B(z)

+n₄e^(b2+b∗1)t+[b²₂(b₂−iα₁)+b^∗2₁ (b^∗₁−iα₁)]β₃⁰ze[b²₂(b₂−iα₁)+b^∗2₁ (b^∗₁−iα₁)]β₃⁰h₂B(z)

+O₁e^{2(b11+b21)t+2}[b³₁₁+b³₂₁+b₁₁(2α₁−3b₁₂)b₁₂+b₂₁(2α₁−3b₂₂)b₂₂]^β₃⁰z

·e²[b³₁₁+b³₂₁+b11(2α1−3b₁₂)b12+b21(2α1−3b₂₂)b22]β₃⁰h2B(z) . (19) The stochastic two-soliton solutions for (2) can be expressed as

U(z,t) =

m₁e^{b1t+b21(b1−iα1)β30z}e^{b21(b1−iα1)β30h2}[^B(z)−1₂z²] +m₂e^{b2t+b22(b2−iα1)β30z}e^{b22(b2−iα1)β30h2}[^B(z)−1₂z²] +m3e^(b2+2b11)t+[2b11(b²₁₁+2α1b12−3b²₁₂)+b²₂(b2−iα₁)]β₃⁰z

·e[2b₁₁(b²₁₁+2α₁b₁₂−3b²₁₂)+b²₂(b₂−iα₁)]β₁h₂[^B(z)−1₂z²] +m₄e^(b1+2b21)t+[2b21(b²₂₁+2α₁b22−3b²₂₂)+b²₁(b₁−iα₁)]β₃⁰z

·e[2b₂₁(b²₂₁+2α₁b₂₂−3b²₂₂)+b²₁(b₁−iα₁)]β₃⁰h₂[^B(z)−1₂z²]

n₁e^{2b11t+2b11(b211+2α1b12−3b212)β30z}e^{2b11(b211+2α1b12−3b212)β30h2}[^B(z)−1₂z²] +n2e^{2b21t+2b21(b221+2α1b22−3b222)β30z}e^{2b21(b221+2α1b22−3b222)β30h2}[^B(z)−1₂z²]

+n₃e^(b1+b∗2)t+[b²₁(b₁−iα₁)+b^∗2₂ (b^∗₂+iα₁)]β₃⁰ze[b²₁(b₁−iα₁)+b^∗2₂(b^∗₂+iα₁)]β₃⁰h₂[B(z)−¹₂z²] +n4e^(b2+b∗1)t+[b²₂(b2−iα₁)+b^∗2₁ (b^∗₁−iα₁)]β₃⁰ze[b²₂(b2−iα₁)+b^∗2₁(b^∗₁−iα₁)]β₃⁰h2[^B(z)−1₂z²] +O₁e^{2(b11+b21)t+2}[b³₁₁+b³₂₁+b₁₁(2α₁−3b₁₂)b₁₂+b₂₁(2α₁−3b₂₂)b₂₂]β₃⁰z

·e²[b³₁₁+b³₂₁+b₁₁(2α₁−3b₁₂)b₁₂+b₂₁(2α₁−3b₂₂)b₂₂]^β₃⁰h2[^B(z)−1₂z²] . (20)

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

For the effects of the Gaussian white noise on the stochastic one soliton, solution (18) is rewritten as U(z,t) =Asech

η₁z+η₂t+η₃B(z)−1

2η₃z²+C

·e^{iη4z+iη5t+iη6B(z)−2iη6z2}, with

A=2n₁C, η₁= b³₁₁+2b₁₁b₁₂−3b₁₁b²₁₂ β₃⁰, η₂=b₁₁,

η3= b³₁₁+2b₁₁b₁₂−3b₁₁b²₁₂ β₃⁰h₂, η4= −b²₁₁+3b²₁₁b12+b²₁₂−b³₁₂

β₃⁰, η₅=b12,

η₆= −b²₁₁+3b²₁₁b₁₂+b²₁₂−b³₁₂ β₃⁰h₂, C=

s −β₂⁰b²₁₁ 4 n²₁₁+n²₁₂

γ0

,

where, of the stochastic one soliton,A andCare the amplitude and initial position, η2and η₅are respec- tively related to the pulsewidth and frequency,η1and

50 250

−2 2

z

Amplitude

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

η3are both related to the velocity, andη4andη₆are related to the phase shifting. Therefore, the velocity and phase shifting of the stochastic one soliton are ob- tained asη1+η₃B(z)˙ −η₃zandη4z+η6B(z)−¹₂η6z². 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). Parameters are:n₁=1−i,b₁=1−2i,α=−20,β₃⁰=1, andh₂=0.

Fig. 3. (a) Stochastic one soliton via Solutions (18). (b) Energy of the stochastic one soliton via Solutions (18). Parameters are:n₁=1−i,b₁=1−2i,α=−20,β₃⁰=1, andh₂=0.2.

Fig. 4. (a) Stochastic one soliton via Solutions (18) withh2=0.5. (b) Stochastic one soliton via Solutions (18) withh2=0.8.

Parameters are:n1=1−i,b1=1−2i,α=−20, andβ₃⁰=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 white noise. The average value of the velocity can be ob- tained asη₁−η₃z. It means that the acceleration is ex-