Oscillations in the Interactions Among Multiple Solitons in an Optical Fibre

Wen-Qiang Hu; Yi-Tian Gao; Chen Zhao; Yu-Jie Feng; Chuan-Qi Su

doi:10.1515/zna-2016-0310

Article Publicly Available

Oscillations in the Interactions Among Multiple Solitons in an Optical Fibre

Wen-Qiang Hu , Yi-Tian Gao , Chen Zhao , Yu-Jie Feng and Chuan-Qi Su

Published/Copyright: October 10, 2016

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Zeitschrift für Naturforschung A Volume 71 Issue 12

Abstract

In this article, under the investigation on the interactions among multiple solitons for an eighth-order nonlinear Schrödinger equation in an optical fibre, oscillations in the interaction zones are observed theoretically. With different coefficients of the operators in this equation, we find that (1) the oscillations in the solitonic interaction zones have different forms with different spectral parameters of this equation; (2) the oscillations in the interactions among the multiple solitons are affected by the choice of spectral parameters, the dispersive effects and nonlinearity of the eighth-order operator; (3) the second-, fifth-, sixth-, and seventh-order operators restrain oscillations in the solitonic interaction zones and the higher-order operators have stronger attenuated effects than the lower ones, while the third- and fourth-order operators stimulate and extend the scope of oscillations.

Keywords: Higher-Order Dispersive Effects and Nonlinearity; Nonlinear Schrödinger Equation; Optical Fibre; Oscillations; Solitons

PACS: 05. 45. Yv; 47. 35. Fg; 02. 30. Jr

1 Introduction

Via the balance between the group velocity dispersion and self-phase modulation, optical solitons have been seen to propagate undistorted over a long distance and remain unaffected after the interaction with each other in an optical fibre [1], [2], [3], which have been attractive in the optical communication systems [4], [5]. Evolution of the pulses in an optical fibre has been modelled by the nonlinear Schrödinger (NLS) equation [6],

(1)iqξ+qττ+2|q|2q=0,

where q is the slowly varying electric field and the subscripts denote the partial derivatives with respect to the scaled distance ξ and time τ. Nonetheless, (1) has been seen to represent the lowest-order approximation describing the propagation process [7]. When the intensity of the incident light field increases and the pulses shorten, (1) fails in the physical description of the propagation of light pulses in fibres due to the negligence of the higher-order dispersion terms and the non-Kerr nonlinearity effects [7], [8], [9]. Thus, validity of the propagation modelling has been extended via the consideration on the higher-order dispersive effects as well as higher-order nonlinearity [10], [11], [12], [13], [14].

The NLS hierarchy, used to investigate the higher-order dispersive effects and nonlinearity, has appeared as [15]:

(2)iψx+Å2K2[ψ(x, t)] − iÅ3K3[ψ(x, t)]+Å4K4[ψ(x, t)] −iÅ5K5[ψ(x, t)]+…=0,

with

K2[ψ(x, t)]=ψtt+2ψ|ψ|2,K3[ψ(x, t)]=ψttt+ 6|ψ|2ψt,K4[ψ(x, t)]=ψtttt+ 6ψ∗ψt2+4ψ|ψt|2+ 8|ψ|2ψtt+2ψ2ψtt∗+ 6|ψ|4ψ,K5[ψ(x, t)]=ψttttt+ 10|ψ|2ψttt+ 30|ψ|4ψt+ 10ψψtψtt∗+ 10ψψt∗ψtt+ 20ψ∗ψtψtt+10ψt2ψt∗,⋮

where ψ represents a normalised complex amplitude of the optical pulse envelope, the subscripts x and t, respectively, denote the partial derivatives with respect to the scaled distance and time, the parameters Å₂, Å₃, … are all the real constants, and K_j[ψ(x, t)] (j=1, 2, …) is the j^th-order operator in the NLS hierarchy as seen in the Appendix of this article. Special cases of (2) have been seen as:

Truncating (2) at the second-order operator have yielded (1) for the steady nonlinear waves on the surface of an infinitely deep fluid [16];
Truncating (2) at the third-order operator have yielded the Hirota equation for the third-order dispersion and time-delay correction to the cubic nonlinearity in ocean waves [17], [18];
Truncating (2) at the fourth-order operator have yielded the Lakshmanan–Porsezian–Daniel equation for the ultrashort optical pulse propagation in a long-distance high-speed optical fibre [19], [20], [21], [22].

In this article, we will truncate (2) to an eighth-order NLS equation with the following form,

(3)iψx+Å2K2[ψ(x, t)]−iÅ3K3[ψ(x, t)]+Å4K4[ψ(x, t)] −iÅ5K5[ψ(x, t)]+Å6K6[ψ(x, t)] − iÅ7K7[ψ(x, t)] +Å8K8[ψ(x, t)]=0,

to investigate the higher-order dispersive effects on the interactions among multiple solitons.

The oscillations, which exist in the propagation of solitons, have been observed by the experiments conducted in [23], [24], [25], whose intention is to investigate the dynamics of long-lived dark and dark-bright solitons in the highly stable optically trapped Bose–Einstein condensates [23], described within the framework of the nonlinear Gross–Pitaevskii equation [24]. Via the investigation of the interactions among multiple solitons for (3), we have observed graphically that some oscillations arise in the solitonic interaction zones. To our knowledge, the oscillations in the interactions among multiple solitons for an eighth-order NLS equation and its characteristics have not been discussed yet. Motivated by that, in Section 2, we will give the Lax pair and infinitely many conservation laws to analyse the integrability of (3). In Section 3, we will present the analytic one-, two-, and multisoliton solutions via the Darboux transformation (DT). In Section 4, the effects of the parameters Å₂, Å₃, …, Å₈ on the propagation of one solitons will be analyzed, and the characteristics and origins of the oscillations among multiple solitons will be investigated as well. Section 5 will be our conclusions.

2 Lax Pair and Infinitely Many Conservation Laws for (3)

Lax pair can ensure the integrability of an NLS-type equation [26], [27] and can be used to derive the multisoliton solutions through the DT and inverse scattering transformation [28]. In addition, existence of the infinitely many conservation laws can be a definition of the complete integrability for an NLS-type equation [29], [30]. Hence, deriving the Lax pair and infinitely many conservation laws is a prerequisite for us to investigate the characteristics of soliton solutions for (3).

Lax pair of (3) can be derived as

(4)Φt=UΦ,

(5)Φx=VΦ,

where Φ=(ϕ₁, ϕ₂)^T is a vector eigenfunction (the superscript T denotes the transpose of the vector), ϕ₁ and ϕ₂ are the complex functions of x and t and the 2×2 matrices U and V are expressible in the forms

(6)U=(iλiψ∗iψ−iλ), V=(A(x, t)B(x, t)C(x, t)−A(x, t)),

λ is the isospectral parameter, i.e. λ_x=0, and A(x, t), B(x, t), and C(x, t) are the complex differentiable functions of x and t.

According to the compatibility condition Φ_xt=Φ_tx, from Lax Pair (4, 5), we can get

(7)Vt−Ux+VU−UV=0.

Substituting Expressions (4, 5) into (7), we get the following relations:

(8)At=i(ψ∗C−ψB),

(9)Bt=i(ψx∗+2λB−2ψ∗A),

(10)Ct=i(ψ−2λC+2Aψ).

Let

(11)A(x, t)=∑j = 08λjaj, B(x, t)=∑j = 08λjbj, C(x, t)=∑j = 08λjcj,

where a_j’s, b_j’s, and c_j’s are the functions of x and t to be determined.

Substituting Expressions (11) into (8–10), and equating the coefficients of the same powers of λ, we can obtain the expressions for a_j, b_j, and c_j, which are shown in Appendix for the sake of simplicity.

From the above results, (3) is integrable in the Lax sense. Then, we derive the infinitely many conservation laws for (3) based on Lax Pair (4).

Defining Γ=ϕ₂/ϕ₁, we can derive the following Riccati-type equation from Lax Pair (4) as

(12)Γt=i(ψ−2λΓ−ψ∗Γ2).

According to [30], we can induce the following expansion as

(13)Γ=∑i = 0∞χiλi,

where χ_i are functions of x and t to be determined. Substituting the Expansion (13) into (12) and equating the same powers of λ to zero, we can get the recurrence relations,

(14)χ1=12ψ,

(15)χ2=12iχ1,x,

(16)χ3=12(iχ2,x−ψ∗χ12),

(17)χ4=12(iχ3,x−2ψ∗χ1χ2),

(18)χ5=12(iχ4,x−2ψ∗χ1χ2−ψ∗χ22),⋮

(19)χn + 1=12(iχn,x−∑k = 1n − 1χkχn − k).

Using the compatibility condition (ϕ_1,x)_t=(ϕ_1,t)_x, we can obtain the following conservation form:

(20)(A(x, t)+B(x, t)Γ)t=i(λ+ψ∗Γ)x.

Substituting (12) and (14) into (20) and equating the same powers of λ, we obtain the infinitely many conservation laws for (3),

(21)∂∂tΩj+∂∂xϒj=0, (j=1, 2, 3, …),

with

Ωj=∑k = 08bkχj + k,

ϒj=iψ∗χj.

where Ω_j’s are the conserved densities and ϒ_j’s are the associated fluxes.

Existence of the Infinitely-Many Conservation Laws (21) can verify that (3) is completely integrable.

3 Darboux Transformation and Multi-soliton Solutions for (3)

Based on Lax Pair (4), the DT can be used to acquire the bright N-soliton solutions of (3) among various methods in soliton theory.

3.1 Darboux Transformation

To construct the DT, we assume that ψ^[0] is the solution of (3), which is called the seed solution, and (ϕ₁₁, ϕ₂₁)^T is a solution of Lax Pair (4) with λ=λ₁ and ψ=ψ^[0]. One can verify that (−ϕ21∗, ϕ11∗)T is a solution of Lax Pair (4) with λ=λ1∗ and ψ=ψ^[0]. The DT matrix D^[1] can be constructed as

(22)D[1]=λI−S[1], S[1]=H[1]L(H[1])−1,H[1]=(ϕ11−ϕ21∗ϕ21ϕ11∗), L=(λ100λ1∗),

where I denotes the 2×2 identity matrix, (H^[1])⁻¹ is the inverse matrix of H^[1]. The DT for (3) can be expressed as

(23)ψ[1]=ψ[0]−2(λ1−λ1∗)ϕ11∗ϕ21ϕ11ϕ11∗+ϕ21ϕ21∗.

Setting (ϕ₁₁, ϕ₂₁)^T, (ϕ₁₂, ϕ₂₂)^T, …, (ϕ_1,N, ϕ_2,N)^T to be the N distinct solutions for Lax Pair (4) at λ₁, λ₂, …, λ_N, respectively, where λ_k’s (k=1, 2, …, N) are the eigenvalues of Lax Pair (4), ϕ_1,k’s and ϕ_2,k’s are functions of x and t, we can derive the N-^th order DT for (3) as,

(24)ψ[N]=ψ[0]−2∑k = 1N(λk∗−λk)ϕ1,k[k − 1]ϕ2,k∗,[k − 1]ϕ1,k[k − 1]ϕ1,k∗,[k − 1]+ϕ2,k[k − 1]ϕ2,k∗,[k − 1],Φ[N]=D[N]D[N − 1]⋯D[1]Φ,

with

(ϕ1,1[0]ϕ2,1[0])=(ϕ1,1ϕ2,1), (ϕ1,k[k−1]ϕ2,k[k−1])=(D[k]D[k − 1]⋯D[1])|λ = λk(ϕ1,kϕ2,k),D[k]=λI−S[k], S[k]=H[k]L(H[k])−1,H[k]=(ϕ11[k]−ϕ21∗,[k]ϕ21[k]ϕ11∗,[k]), L=(λk00λk∗).

3.2 Multi-soliton Solutions

In this section, we apply the iterative algorithm of the DT to construct the soliton solutions for (3). Taking ψ^[0]=0 as the seed solutions and substituting it into Lax Pair (4) with λ=λ_k, we can get the eigenfunctions (ϕ_1,k, ϕ_2,k)^T as

(25)ϕ1,k=eθ1 + iλkt,ϕ2,k=eθ2 − iλkt,

with

θ1=ς 1−2ix(64λk8Å8+32λk7Å7−16λk6Å6−8λk5Å5+4λk4Å4+ 2λk3Å3−λk2Å2),θ2=ς 2+2ix(64λk8Å8+32λk7Å7−16λk6Å6−8λk5Å5+4λk4Å4+ 2λk3Å3−λk2Å2).

where ς₁ and ς₂ are arbitrary real constants. In general, we set ς₁=ς₂=0.

According to (23), the one-soliton solutions for (3) is given as

(26)ψ[1]=−2(λ1−λ1∗)eξ1 + η1e2iλ1∗t + ξ1+e2iλ1t + η1,

with

ξ1=4ixλ12(−Å2+2λ1Å3+4λ12Å4−8λ13Å5−16λ14Å6+32λ15Å7+ 64λ16Å8),η1=4ix(λ1∗)2[−Å2+2λ1∗Å3+4(λ1∗)2Å4−8(λ1∗)3Å5−16(λ1∗)4Å6+ 32(λ1∗)5Å7+64(λ1∗)6Å8].

We take two sets of basic solutions corresponding to two eigenvalues λ₁ and λ₂, respectively, and obtain the two-soliton solutions from(24) with N=2,

(27)ψ[2]=ψ[1]−GF,

with

G=2(λ2−λ2∗)[−λ1eξ1(eη1 + 2iλ2t+eξ2 + 2iλ1∗t)+ λ2eξ2(eη1 + 2iλ1t+eξ1 + 2iλ1∗t) + eη1λ1∗t(eξ1 + 2iλ2t− eξ2 + 2iλ1t)][λ1eξ1(eη2 + 2iλ1∗t−eη1 + 2iλ2∗t)+eη1λ1∗(eη2 + 2iλ1t+eξ1 + 2iλ2∗t)− eη2λ2∗(eη1 + 2iλ1t+eξ1 + 2iλ1∗t)],

F=(eη1 + 2iλ1t+eξ1 + 2iλ1∗t){λ1∗λ2∗(eη2 + 2iλ1∗t−eη1 + 2iλ2∗t)(eξ1 + 2iλ2t−eξ2 + 2iλ1t) + λ1[λ2(eη2 + 2iλ1∗t−eη1 + 2iλ2∗t)(eξ1 + 2iλ2t−eξ2 + 2iλ1t)−λ1∗(eη1 + 2iλ1t +eξ1 + 2iλ1∗t)(eη2 + 2iλ2t+eξ2 + 2iλ2∗t)+λ2∗(eη1 + 2iλ2t+eξ2 + 2iλ1∗t)(eη2 + 2iλ1t +eξ1 + 2iλ2∗t)]+λ2[λ1∗(eη1 + 2iλ2t+eξ2+2iλ1∗t)(eη2 + 2iλ1t+eξ1 + 2iλ2∗t) −λ2∗(eη1 + 2iλ1t+eξ1 + 2iλ1∗t)(eη2 + 2iλ2t+eξ2 + 2iλ2∗t)]},

where

ξ1=4ixλ12(−Å2+2λ1Å3+4λ12Å4−8λ13Å5−16λ14Å6+32λ15Å7+ 64λ16Å8),η1=4ix(λ1∗)2[−Å2+2λ1∗Å3+4(λ1∗)2Å4−8(λ1∗)3Å5−16(λ1∗)4Å6+ 32(λ1∗)5Å7 +64(λ1∗)6Å8],ξ2=4ixλ22(−Å2+2λ2Å3+4λ22Å4−8λ23Å5−16λ24Å6+32λ25Å7+ 64λ26Å8),η2=4ix(λ2∗)2[−Å2+2λ2∗Å3+4(λ2∗)2Å4−8(λ2∗)3Å5−16(λ2∗)4Å6+ 32(λ2∗)5Å7 +64(λ2∗)6Å8].

By taking N linearly independent solutions of the Lax Pair (4) with λ=λ_k(k=1, 2, …, N), the explicit N-soliton solutions of (3) can be generated from the N^th-iterated DT (24).

4 Soliton Propagations and Interactions of (3)

In this section, based on Solutions (24), (26), and (27), we analytically and graphically investigate the propagation and interaction of the solitons.

4.1 The Propagation of One Solitons

Setting λ₁=α+iβ, where α and β are real constants, We rewrite Solution (26) as

(28)|ψ[1]| =2|β|Sech(−2βt−2βℏx),

with

ℏ=4αÅ2+4Å3(β2−3α2)+32Å4α(β2−α2)+ 16Å5(5α4−10α2β2+β4) +64Å6α(3α4−10α2β2+3β4)− 64Å7(7α6−35α4β2+21α2β4−β6)− 1024Å8α(α6−7α4β2+7α2β4−β6).

According to Solution (28), the soliton amplitude Δ, frequency ω, and velocity v are, respectively, expressed as

(29)Δ=2|β|, ω=2|β|, v=1ℏ.

Expression (29) shows us that the soliton amplitude can only be affected by |β|, which is the module of the eigenvalue λ’s image part, while the velocity is a function of α, β, Å₂, Å₃, …, Å₈. For example, with the different values of eigenvalue λ and parameters Å₂, Å₃, …, Å₈, the solitons shown in Figures 1 and 2 have the same amplitude but different velocities and a certain degree of compressing or broadening.

$Figure 1: One solitons via Solution (28) with the parameters as Å2=Å3=Å4=Å5=Å6=Å7=Å8=0.005, (a) λ=1+i, (b) λ=12+i,$\lambda = {1 \over 2} + i,$ (c) λ=45+i.$\lambda = {4 \over 5} + i.$$

Figure 1:

One solitons via Solution (28) with the parameters as Å₂=Å₃=Å₄=Å₅=Å₆=Å₇=Å₈=0.005, (a) λ=1+i, (b) λ=12+i, (c) λ=45+i.

Figure 2:

One solitons via Solution (28) with the eigenvalue λ=1+i, (a) Å₂=Å₃=Å₄=Å₅=Å₆=Å₇=Å₈=0.00125; (b) Å₂=Å₃=Å₄=Å₅=Å₆=Å₇=Å₈=0.005; and (c) Å₂=Å₃=Å₄=Å₅=Å₆=Å₇=Å₈=0.02.

4.2 The Oscillations in the Interactions Between Two Solitons

In this part, we investigate the interactions between two solitons. Under observations, we find that some oscillations are arisen in the interaction zone of two solitons, which is shown in Figure 3, where the parameters are set as Å₂=Å₃=Å₄=Å₅=Å₆=Å₇=Å₈=0.05.

$Figure 3: (a) The propagations and interactions of two solitons via Solution (27) with λ1=1+i, λ2=12+i,${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} + i,$ (b) the local amplification figure of the interaction zone.$

Figure 3:

(a) The propagations and interactions of two solitons via Solution (27) with λ1=1+i, λ2=12+i, (b) the local amplification figure of the interaction zone.

In order to obtain more observations of the oscillations, we fix a series of t and get a sequence of x~|ψ| graphes, which are shown in Figure 4.

$Figure 4: The x~|ψ| graphs of two solitons via Solution (27) with λ1=1+i, λ2=12+i${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} + i$ at (a) t=−4, (b) t=−3, (c) t=−2, (d) t=−1, (e) t=0, (f) t=1, (g) t=2, (h) t=3, and (i) t=4.$

Figure 4:

The x~|ψ| graphs of two solitons via Solution (27) with λ1=1+i, λ2=12+i at (a) t=−4, (b) t=−3, (c) t=−2, (d) t=−1, (e) t=0, (f) t=1, (g) t=2, (h) t=3, and (i) t=4.

We can find that the oscillations with a slight amplitude exist at the beginning of interaction, and the amplitude of the oscillations get a sharp increase in a short period of time. At t=0, which represents the central moment of the two solitons interaction, the oscillations replace the original interaction structures completely and the whole energy of the two solitons decentralises into finite waves. With the elapse of time, the oscillations recede gradually and the two solitons propagate along each directions.

In addition, if we fix x and draw the t~|ψ| graphs of two solitons, we can obtain smooth curves without oscillations, which is shown in Figure 5. This results manifest that the oscillations are in spatial distribution and the amplitude of every point in the space varies continuously in the period of solitons propagations and interactions.

$Figure 5: The t~|ψ| graphs of two solitons via Solution (27) with λ1=1+i, λ2=12+i${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} + i$ at (a) x=−2, (b) x=0, and (c) x=2.$

Figure 5:

The t~|ψ| graphs of two solitons via Solution (27) with λ1=1+i, λ2=12+i at (a) x=−2, (b) x=0, and (c) x=2.

Then, we take different groups of eigenvalues λ₁ and λ₂ and can observe the oscillations in the interaction zone clearly, which are shown in Figure 6.

$Figure 6: The propagations and interactions of two solitons via Solution (27) with (a) λ1=1+i, λ2=12−13i,${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} - {1 \over 3}i,$ (b) λ1=512+512i, λ2=45+56,${\lambda _1} = {5 \over {12}} + {5 \over {12}}i,{\rm{ }}{\lambda _2} = {4 \over 5} + {5 \over 6},$ and (c) λ1=23+34i, λ2=56+56i.${\lambda _1} = {2 \over 3} + {3 \over 4}i,{\rm{ }}{\lambda _2} = {5 \over 6} + {5 \over 6}i.$$

Figure 6:

The propagations and interactions of two solitons via Solution (27) with (a) λ1=1+i, λ2=12−13i, (b) λ1=512+512i, λ2=45+56, and (c) λ1=23+34i, λ2=56+56i.

Meanwhile, when we take some groups of eigenvalues, we can also observe some peculiar interaction of two solitons without the oscillations that are called bound state solitons, which is shown in Figure 7.

$Figure 7: The propagations and interactions of two solitons via Solution (27) with (a) λ1=45+45i, λ2=56+56i,${\lambda _1} = {4 \over 5} + {4 \over 5}i,{\rm{ }}{\lambda _2} = {5 \over 6} + {5 \over 6}i,$ (b) λ1=23+34i, λ2=59+57i.${\lambda _1} = {2 \over 3} + {3 \over 4}i,{\rm{ }}{\lambda _2} = {5 \over 9} + {5 \over 7}i.$$

Figure 7:

The propagations and interactions of two solitons via Solution (27) with (a) λ1=45+45i, λ2=56+56i, (b) λ1=23+34i, λ2=59+57i.

The previous observation demonstrates that existence of the oscillations in the interaction zone may be not only caused by the selection of eigenvalues λ₁ and λ₂ but also caused by other factors. Hence, we make a conjecture and consider the effects of higher-order derivative terms as one of the origins of the oscillations.

Selecting the same value of the groups of eigenvalues in Figures 3 and 6, we set the coefficient of higher-order operator K_j[ψ(x, t)] to zero in sequence and observe the interaction of two solitons, which are shown in Figures 8–11. We can see that the oscillations in the interaction zones recede gradually with the vanish of eighth- and seventh-order operators and disappear in the sixth-order equation and other lower-order equations. Hence, the eighth- and seventh-order operators in (3) may be the origins of the oscillations.

Figure 8:

The propagations and interactions of two solitons via Solution (27) with the same parameters as Figure 3 besides (a) Å₈=0, (b) Å₇=Å₈=0, and (c) Å₆=Å₇=Å₈=0.

Figure 9:

The propagations and interactions of two solitons via Solution (27) with the same parameters as Figure 6(a) besides (a) Å₈=0, (b) Å₇=Å₈=0, and (c) Å₆=Å₇=Å₈=0.

Figure 10:

The propagations and interactions of two solitons via Solution (27) with the same parameters as Figure 6(b) besides (a) Å₈=0, (b) Å₇=Å₈=0, and (c) Å₆=Å₇=Å₈=0.

Figure 11:

The propagations and interactions of two solitons via Solution (27) with the same parameters as Figure 6(c) besides (a) Å₈=0, (b) Å₇=Å₈=0, (c) Å₆=Å₇=Å₈=0.

Next, we discuss the impacts on the oscillations of every order operator in (3). Without loss of generality, we set the eigenvalues λ1=1+i, λ2=12+i in the following discussion.

Firstly, we just keep the eighth-order operator in (3). From the analysis in Part 4.1, we can set the coefficient of eighth-order operator Å₈=0.005 when we consider the compressibility of the solitons and obtain a sequence of x~|ψ| graphes at different t, which are shown in Figure 12.

Figure 12:

The x~|ψ| graphs of two solitons via Solution (27) with Å₈=0.005 at (a) t=−2, (b) t=0, and (c) t=2.

By contrast, we just keep the seventh-order operator in (3) and set the Å₇=0.005. A sequence of x~|ψ| graphes at different t are shown in Figure 13. Comparing with these two groups of graphes and combining with the preceding analysis, the eighth-order operator is the cause of the oscillations at this group of eigenvalues.

Figure 13:

The x~|ψ| graphs of two solitons via Solution (27) with Å₇=0.005 at (a) t=−2, (b) t=0, and (c) t=2.

Secondly, we keep the j^th (j=2, 3, …, 7) order operator and eighth-order operator in (3), where the eighth-order operator provides the “normal” oscillations and the j^th order operator exerts influence on the oscillations. We set Å₈=0.005 and Å_j=0.05, and get a sequence of x~|ψ| graphes at t=0, when the oscillations replace the original interaction structures completely and can illustrate the peculiarity of the oscillations most, shown in Figure 14. We can find that the second-, fifth-, sixth-, and seventh-order operators restrain the oscillations, and the extent of the inhibiting effect increases with the order aggrandising. The third- and fourth-order operators stimulate the oscillations and expand the range of them.

Figure 14:

The x~|ψ| graphs of two solitons via Solution (27) at t=0 with Å₈=0.005 and (a) A₂=0.05, (b) Å₃=0.05, (c) Å₄=0.05, (d) Å₅=0.05, (e) Å₆=0.05, and (f) Å₇=0.05.

In conclusion, through selecting some groups of eigenvalues, the oscillations are arisen in the interactions of two solitons, which is caused by the eighth-order operator in (3). In other words, the eighth-order dispersive effects and nonlinearity generate the oscillations. The second-, fifth-, sixth-, and seventh-order operators restrain the oscillations and the higher-order operators have stronger attenuate effects than lower ones, while the third- and fourth-order operators stimulate the oscillations.

4.3 The Oscillations in the Interactions Among Multiple Solitons

In this part, we investigate the interactions among multiple solitons and observe the oscillations. Based on Expression (24), we obtain the multisoliton solutions for (3).

Interactions among the three solitons are displayed in Figure 15. We can find the oscillations in the interactions of the three solitons clearly. Comparing to the two solitons, the amplitude and range of the oscillations have an increase, which are more complex. Meanwhile, we can observe the bound-state solitons, as shown in Figure 16.

$Figure 15: The propagations and interactions of three solitons via Expression (24) with (a) λ1=1+i, λ2=12+i, λ3=45+i,${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} + i,{\rm{ }}{\lambda _3} = {4 \over 5} + i,$ (b) λ1=12+23i, λ2=34+56i, λ3=67+89i,${\lambda _1} = {1 \over 2} + {2 \over 3}i,{\rm{ }}{\lambda _2} = {3 \over 4} + {5 \over 6}i,{\rm{ }}{\lambda _3} = {6 \over 7} + {8 \over 9}i,$ (c) λ1=23+34i, λ2=59+57i, λ3=78+78i.${\lambda _1} = {2 \over 3} + {3 \over 4}i,{\rm{ }}{\lambda _2} = {5 \over 9} + {5 \over 7}i,{\rm{ }}{\lambda _3} = {7 \over 8} + {7 \over 8}i.$$

Figure 15:

The propagations and interactions of three solitons via Expression (24) with (a) λ1=1+i, λ2=12+i, λ3=45+i, (b) λ1=12+23i, λ2=34+56i, λ3=67+89i, (c) λ1=23+34i, λ2=59+57i, λ3=78+78i.

$Figure 16: The propagations and interactions of three solitons via Expression (24) with (a) λ1=45+45i, λ2=56+56i, λ3=34+34i,${\lambda _1} = {4 \over 5} + {4 \over 5}i,{\rm{ }}{\lambda _2} = {5 \over 6} + {5 \over 6}i,{\rm{ }}{\lambda _3} = {3 \over 4} + {3 \over 4}i,$ (b) λ1=23+34i, λ2=59+57i, λ3=56+56i.${\lambda _1} = {2 \over 3} + {3 \over 4}i,{\rm{ }}{\lambda _2} = {5 \over 9} + {5 \over 7}i,{\rm{ }}{\lambda _3} = {5 \over 6} + {5 \over 6}i.$$

Figure 16:

The propagations and interactions of three solitons via Expression (24) with (a) λ1=45+45i, λ2=56+56i, λ3=34+34i, (b) λ1=23+34i, λ2=59+57i, λ3=56+56i.

With reference to the analysis in Part 4.2, we can investigate the cause of the oscillations, and get the same conclusion as the case of two solitons, which have no need to rewrite it again.

Furthermore, beyond Expression (24), we also investigate the interaction among four solitons and obtain the oscillations and bound-state solitons graphs, which are displayed Figure 17. The higher-order dispersive effects and nonlinearity generate the oscillations in the case of the four solitons under the same analysis in Part 4.2, which is unnecessary to give a concrete derivation to demonstrate it again.

$Figure 17: The propagations and interactions of four solitons via Expression (24) (a) oscillations with λ1=1+i, λ2=12+i, λ3=45+i, λ4=910+910i,${\lambda _1} = 1 + i,{\rm{ }}{\lambda _2} = {1 \over 2} + i,{\rm{ }}{\lambda _3} = {4 \over 5} + i,{\rm{ }}{\lambda _4} = {9 \over {10}} + {9 \over {10}}i,$ (b) bound-state solitons with λ1=45+45i, λ2=56+56i, λ3=34+34i, λ4=78+78i.${\lambda _1} = {4 \over 5} + {4 \over 5}i,{\rm{ }}{\lambda _2} = {5 \over 6} + {5 \over 6}i,{\rm{ }}{\lambda _3} = {3 \over 4} + {3 \over 4}i,{\rm{ }}{\lambda _4} = {7 \over 8} + {7 \over 8}i.$$

Figure 17:

The propagations and interactions of four solitons via Expression (24) (a) oscillations with λ1=1+i, λ2=12+i, λ3=45+i, λ4=910+910i, (b) bound-state solitons with λ1=45+45i, λ2=56+56i, λ3=34+34i, λ4=78+78i.

From the above, under the observation and analysis of the interactions among multiple solitons, we find the oscillations in the interaction zone, which are arisen by the higher-order dispersive effects and nonlinearity, especially the eighth order. The eigenvalues λ_j and the coefficients of the operators Å₂, Å₃, …, Å₈ in (3) can effect the velocity and compressibility of the solitons, while the amplitude of the solitons can only be affected by the image parts of eigenvalues.

5 Conclusions

With ever-increasing intensity of the optical field and further shortening of pulses up, higher-order NLS equations have been constructed to describe the nonlinear soliton pulses in an optical fibre, i.e. (3). Lax Pair (4) and Infinitely-Many Conservation Laws (21) for (3) have been derived to ensure the integrability. Via the DT, the analytic one-, two-, and multisoliton solutions for (3) have been obtained.

Based on the Solution (28) and the demonstration of Figures 1 and 2, the eigenvalues λ_j and the coefficients of the operators Å₂, Å₃, …, Å₈ in (3) can effect the velocity and compressibility of the solitons, while the amplitude of the solitons can only be affected by the image part of eigenvalues.

The oscillations in the interactions among multiple solitons have been investigated. Figures have been presented to investigate the oscillation characteristics: Figures 3, 4, 6, 15, and 17 have shown different shapes of the oscillations when we take different groups of eigenvalues. By observing Figures 8–12, the conclusion has been obtained: (1) the eighth-order dispersive effects and nonlinearity generate the oscillations; (2) the second-, fifth-, sixth-, and seventh-order operators restrain the oscillations and the higher-order operators have stronger attenuate effects than lower ones, while the third- and fourth-order operators stimulate the oscillations; (3) even though there have existed the oscillations in the interactions among multiple solitons, solitons remain unaffected after interactions with each other except for some phase shifts.

Acknowledgements

The authors express our sincere thanks to the Editors, Referees, and members of our discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023 and by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant No. IPOC2013B008.

Appendix

The j^th order operator K_j[ψ(x, t)](j≥3) is defined in [15] as
Kj=∂jψ∂tj+2j|ψ|2∂j − 2ψ∂tj − 2+[1+(−1)j]ψ2∂j − 2ψ∗∂tj − 2+ j(j−3)ψψt∗∂j − 3ψ∂tj − 3+ ….
K₆, K₇, and K₈ have been given in [15] and we will show them as follows:
K6[ψ(x, t)]=ψtttttt+ψ2[60|ψt|2ψ∗+50ψtt(ψ∗)2+2ψtttt∗]+ ψ[12ψttttψ∗+8ψtψttt∗ +22|ψtt|2+ 18ψtttψt∗+ 70ψt2(ψ∗)2]+20ψt2ψtt∗ +10ψt(5ψttψt∗+3ψtttψ∗)+ 20ψtt2ψ∗+10ψ3[(ψt∗)2+2ψ∗ψtt∗]+20ψ|ψ|6,
K7[ψ(x, t)]=ψttttttt+70ψtt2ψt∗+112ψt|ψtt|2+ 98ψttt|ψt|2+ 70ψ2[ψt(ψt∗)2+2ψtψ∗ψtt∗ +ψ∗(2ψttψt∗+ψtttψ∗)]+ 28ψt2ψttt∗+14ψ[ψ∗(20|ψt|2ψt+ψttttt)+3ψtttψtt∗+ 2ψttψttt∗+2ψtψtttt∗+20ψtψtt(ψ∗)2]+140|ψ|6ψt+ 70ψt3(ψ∗)2 +14ψ∗(5ψttψttt+3ψtψtttt),
K8[ψ(x, t)]=ψtttttttt+14∗ψ3[40|ψt|2(ψ∗)2+20ψtt(ψ∗)3+ 2ψtttt∗ψ∗ +3(ψtt∗)2+4ψt∗ψttt∗] +ψ2[28ψ∗(14ψttψtt∗+ 11ψtttψt∗+6ψtψttt∗)+238ψtt(ψt∗)2+336|ψt|2ψtt∗+ 560(ψt)2(ψ∗)3+98ψtttt(ψ∗)2+2ψtttttt∗]+ 2ψ{21ψt2[9(ψt∗)2+14ψ∗ψtt∗] +ψt[728ψttψt∗ψ∗+ 238ψttt(ψ∗)2+6ψttttt∗]+34|ψttt|2+ 36ψttttψtt∗+ 22ψttψtttt∗+20ψtttttψt∗+161ψtt2(ψ∗)2+8ψttttttψ∗}+ 182ψtt|ψtt|2 +308ψttψtttψt∗+252ψtψtttψtt∗+196ψtψttψttt∗+168ψtψttttψt∗+42ψt2ψtttt∗+ 14ψ∗(30ψt3ψt∗+4ψtttttψt+5ψttt2+8ψttψtttt)+ 490ψt2ψtt(ψ∗)2+140ψ4ψ∗[(ψt∗)2+ψ∗ψtt∗]+ 7ψ|ψ|8.
The expressions for a_j, b_j, and c_j are given as follows:
a0=− iÅ8{35|ψ|8+21ψtt2(ψ∗)2−21|ψt|4+ 14ψ∗ψtt∗+ 70ψ3ψ∗((ψt∗)2+ψ∗ψtt∗)−ψtttψttt∗+ψtt∗ψtttt+7ψ2[2ψ∗ψtttt∗+ 3(ψtt∗)2+4ψt∗ψttt∗+10ψt(ψ∗)2ψt∗+10ψtt(ψ∗)3]−ψtttttψt∗+ ψt(28ψ∗ψt∗ψtt+28(ψ∗)2ψttt−ψttttt∗)+ψ∗ψtttttt+ ψ[70(ψ∗)3(ψt)2 +14(ψt∗)2ψtt +28ψtψt∗ψtt∗+ 14ψt∗(4ψttψtt∗+ψt∗ψttt+ψtψttt∗) +14(ψ∗)2ψtttt+ψtttttt∗} + Å7{−30ψ3(ψ∗)2ψt∗+20(ψ∗)2ψtψtt+ψtt∗ψttt+ 10ψ2[3(ψ∗)3ψt−2ψt∗ψtt∗−ψ∗ψttt∗] −ψttψttt∗−ψt∗ψtttt+ ψtψtttt∗+ψ∗(10ψt2ψt∗+ψttttt)−ψ[10ψ∗ψt∗ψtt+10ψt((ψt∗)2− ψ∗ψtt∗)]−10(ψ∗)2ψttt+ψttttt∗}−iÅ6{10|ψ|6 +5(ψ∗)2ψtt2+ ψttψtt∗+5ψ2[(ψt∗)2 +2ψ∗ψtt∗]−ψt∗ψttt−ψtψttt∗+ψt∗ψtttt+ ψ[10(ψ∗)2ψtt+ψtttt∗]}+Å5{−6ψ2ψ∗ψt∗−ψt∗ψtt+ψtψtt∗+ ψ∗ψttt+ψ[6(ψ∗)2ψt−ψttt∗]}−iÅ4[3ψ2(ψ∗)2−ψtψt∗+ψ∗ψtt+ ψψtt∗]+Å3(ψ∗ψt−ψψt∗)−iÅ2ψψ∗,
a1=2Å8{30ψttt(ψ∗)2ψt∗−20(ψ∗)2ψtψtt−ψtt∗ψttt+ 10ψ2(−3(ψ∗)3ψt+2ψt∗ψtt∗+ψ∗ψttt∗) +ψttψttt∗ψt∗ψtttt− ψtψtttt∗−ψ∗(10ψt2ψt∗+ψttttt)+ψ[10ψ∗ψt∗ψtt+10ψt((ψt∗)2− ψ∗ψtt∗) −10(ψ∗)2ψttt+ψttttt∗]}−2iÅ7{10|ψ|6+ 5(ψ∗)2(ψt)2+ ψttψtt∗+5ψ2((ψt∗)2+2ψ∗ψtt∗) −ψt∗ψttt−ψtψttt∗+ψ∗ψtttt+ ψ(10(ψ∗)2ψtt+ψtttt∗)}+2Å6{6ψ2ψ∗ψt∗+ψt∗ψtt−ψtψtt∗− ψt∗ψttt+ψ(−6(ψt∗)2ψt+ψttt∗)}−2iÅ5{3|ψ|4− ψtψt∗+ψ∗ψtt+ ψψtt∗}+2Å4{ψ∗ψt +ψψt∗}−2iÅ3ψψ∗,
a2=4iÅ8{10|ψ|6+ 5(ψ∗)2ψt2+ψttψtt∗+5ψ2((ψt∗)2+2ψ∗ψtt∗)− ψt∗ψttt−ψtψttt∗ +ψ∗ψtttt+ψ(10(ψ∗)2ψtt+ψtttt∗)}+ 4Å7{6ψ2ψ∗ψt∗ +ψt∗ψtt−ψtψtt∗−ψ∗ψttt + ψ(−6(ψ∗)2ψt+ψttt∗)} +4iÅ6{3|ψ|4−ψtψt∗+ψ∗ψtt+ψψtt∗}+ 4Å5{ψtψ∗−ψψt∗} +4iÅ4ψψ∗+2iÅ2,
a3=− 8Å8{6ψ2ψ∗ψt∗+ψt∗ψtt−ψtψtt∗−ψ∗ψttt+ ψ(−6(ψ∗)2ψt+ψttt∗)}+8iÅ7{3ψ2(ψ∗)2− ψtψt∗+ψttψ∗+ψψtt∗}−8Å6{−ψtψ∗+ψψt∗}+ 8iÅ5ψψ∗+4iÅ3,
a4=− 16iÅ8{3|ψ|4− ψtψt∗+ψ∗ψtt+ψψtt∗}+16Å7{ψtψ∗−ψψt∗}− 16iÅ6ψψ∗−8iÅ4,
a5=32Å8{−ψtψ∗+ψψt∗}−32iÅ7ψψ∗−16iÅ5,
a6=64iÅ8ψψ∗+32iÅ6,
a7=64iÅ7,
a8=−128iÅ8,
b0=Å8{140|ψ|6ψt+70(ψ∗)2ψt3+70ψtt2ψt∗+112ψtψttψtt∗+ 98ψtψtttψt∗+70ψ2(ψt(ψt∗)2+2ψ∗ψtt∗)+ψ∗(2ψttψt∗+ψ∗ψttt)+ 28ψt2ψttt∗+14ψ∗(5ψttψttt+3ψtψtttt)+14ψ(20(ψ∗)2ψtψtt+ 3ψtt∗ψttt+ψttψttt∗+2ψt∗ψtttt+ψtψtttt∗+ψ∗(20ψt2ψt∗+ψttttt))+ ψttttttt}+iÅ7{20ψ4(ψ∗)3+20ψ∗ψtt2+20ψt2ψtt∗+ 10ψ3(5ψt∗ψtt+3ψtttψ∗) +2ψ(35(ψ∗)2ψt2+11ψttψtt∗+ 9ψ∗ψttt +4ψtψttt∗+6ψttttψ∗)+2ψ2(30ψ∗ψtψt∗+ 25(ψ∗)2ψtt+ψtttt∗)+ψtttttt}+Å6{30|ψ|4ψt+ 10ψt2ψt∗+ 20ψ∗ψtψtt +10ψ(ψt∗ψtt+ψtψtt∗+ψ∗ψttt)+ψttttt}+ iÅ5{6ψ|ψ|4+6ψ∗ψt2+4ψ(ψtψt∗+2ψ∗ψtt)+ 2ψ2ψtt∗+ψtttt}+Å4{6ψψ∗ψt+ψttt}+iÅ3{2ψ2ψ∗+ψtt}+ Å2ψt,
b1=− 2iÅ8{20ψ|ψ|6+ 20ψ∗ψtt2+20ψt2ψtt∗+10ψ3((ψt∗)2+ 2ψ∗ψtt∗)+10ψt(5ψt∗ψtt+3ψt∗ψttt)+2ψ(35(ψ∗ψt)2+ 11ψttψtt∗+9ψt∗ψttt+4ψtψttt∗+6ψ∗ψtttt)+2ψ2(30ψ∗ψtψt∗+ 25(ψ∗)2ψtt+ψtttt∗)+ψtttttt}+2Å7{30|ψ|4ψt+ 10ψt2ψt∗+ 20ψ∗ψtψtt +10ψ(ψt∗ψtt+ψtψtt∗+ψ∗ψttt)+ψttttt}− 2iÅ6{6ψ|ψ|4+6ψ∗ψt2+4ψ(ψtψt∗+2ψ∗ψtt)+ 2ψ2ψtt∗+ψtttt}+2Å5{6ψψ∗ψt+ψttt}−2iÅ4{2ψ2ψ∗+ ψtt}+2Å3ψt−2iÅ2ψ ,
b2=− 4Å8{30|ψ|4ψt+10ψt2ψt∗+20ψ∗ψtψtt+ 10ψ(ψt∗ψtt+ψtψtt∗+ψ∗ψttt)+ψttttt}−4iÅ7{6ψ|ψ|4+ 6ψ∗ψt2+4ψ(ψtψt∗+2ψ∗ψtt)+2ψ2ψtt∗+ψtttt}− 4Å6{6|ψ|2ψt+ψttt}−4iÅ5{2ψ|ψ|2 +ψtt}− 4Å4ψt−4iÅ3ψ,
b3=8iÅ8{6ψ|ψ|4+ 6ψ∗ψt2+4ψ(ψtψt∗+2ψ∗ψtt)+ 2ψ2ψtt∗+ψtttt}−8Å7{6|ψ|2ψt+ψttt}+ 8iÅ6{2ψ|ψ|2+ψtt} −8Å5ψt+8iÅ4ψ,
b4=16Å8{6|ψ|2ψt+ ψttt}+16iÅ7{2ψ|ψ|2+ ψtt}+16Å6ψt+ 16iÅ5ψ,
b5=− 32iÅ8{2ψ|ψ|2+ ψtt}+32Å7ψt−32iÅ6ψ,
b6=− 64Å8ψt−64iÅ7ψ,
b7=128iÅ8ψ,
b8=0,
cj=bj∗.

References

[1] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).10.1063/1.1654836Search in Google Scholar

[2] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).10.1103/PhysRevLett.45.1095Search in Google Scholar

[3] Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).10.1016/S0370-1573(97)00073-2Search in Google Scholar

[4] M. P. Barnett, J. F. Capitani, J. Von Zur Gathen, and J. Gerhard, Int. J. Quantum Chem. 100, 80 (2004).10.1002/qua.20097Search in Google Scholar

[5] J. R. DeOliveira and M. A. DeMoura, Phys. Rev. E 57, 4751 (1998).10.1103/PhysRevE.57.4751Search in Google Scholar

[6] Y. Kodama and A. Hasegawa, Solitons in Optical Communications, Oxford University Press, Oxford 1995.10.1093/oso/9780198565079.001.0001Search in Google Scholar

[7] E. Bourkoff, W. Zhao, R. I. Joseph and D. N. Christodoulides, Opt. Lett. 12, 272 (1987).10.1364/OL.12.000272Search in Google Scholar

[8] F. D. Zong, C. Q. Dai, and J. F. Zhang, Commun. Theor. Phys. 45, 721 (2006).10.1088/0253-6102/45/4/029Search in Google Scholar

[9] J. R. de Oliveria and M. A. de Moura, Phys. Rev. E 57, 4751 (1998).10.1103/PhysRevE.57.4751Search in Google Scholar

[10] A. Chowdury, D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Phys. Rev. E 90, 032922 (2014).10.1103/PhysRevE.90.032922Search in Google Scholar PubMed

[11] F. DeMartini, Phys. Rev. 164, 312 (1967).10.1103/PhysRev.164.312Search in Google Scholar

[12] G. Yang and Y. R. Shen, Opt. Lett. 9, 510 (1984).10.1364/OL.9.000510Search in Google Scholar

[13] D. Anderson and M. Lisak, Phys. Rev. A 27, 1393 (1983).10.1103/PhysRevA.27.1393Search in Google Scholar

[14] Q. Zhou, S. Liu, Nonlinear Dyn. 81, 733 (2015).10.1007/s11071-015-2023-3Search in Google Scholar

[15] A. Ankiewicz, D. J. Kedziora, A. Chowdury, U. Bandelow, and N. Akhmediev, Phys. Rev. E 93, 012206 (2016).10.1103/PhysRevE.93.012206Search in Google Scholar

[16] V. E. Zakharov, J. Appl. Mech. Tech. Phys. 14, 805 (1973).Search in Google Scholar

[17] R. Hirota, J. Math. Phys. 14, 805 (1973).10.1063/1.1666399Search in Google Scholar

[18] A, Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, Phys. Rev. E 81, 046602 (2010).10.1103/PhysRevE.81.046602Search in Google Scholar

[19] K. Porsezian, M. Daniel, and M. Lakshmanan, J. Math. Phys. 33, 1807 (1992).10.1063/1.529658Search in Google Scholar

[20] M. Daniel, L. Kavitha, and R. Amuda, Phys. Rev. B 59, 13774 (1999).10.1103/PhysRevB.59.13774Search in Google Scholar

[21] B. Yang, W. G. Zhang, H. Q. Zhang, and S. B. Pei, Phys. Scr. 88, 065004 (2013).10.1088/0031-8949/88/06/065004Search in Google Scholar

[22] M. Lakshmanan, K. Porsezian, and M. Daniel, Phys. Lett. A 133, 483 (1988).10.1016/0375-9601(88)90520-8Search in Google Scholar

[23] A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, and M. K. Oberthaler, Phys. Rev. Lett. 101, 130401 (2008).10.1103/PhysRevLett.101.130401Search in Google Scholar PubMed

[24] C. Becker, S. Stellmer, P. Solitan-Panahi, S. DöRscher, M. Baumert, et al., Nature Phys. 4, 496 (2008).10.1038/nphys962Search in Google Scholar

[25] D. E. Pelinovsky, V. V. Afanasjev, and Y. S. Kivshar, Phys. Rev. E 53, 1940 (1996).10.1103/PhysRevE.53.1940Search in Google Scholar

[26] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scatering, Cambridge University Press 1992.10.1017/CBO9780511623998Search in Google Scholar

[27] A. S. Fokas, Stud. Appl. Math. 77, 253 (1987).10.1002/sapm1987773253Search in Google Scholar

[28] Q. Zhou and M. Mirzazadeh, Z. Naturforsch. A 71, 807 (2016).10.1515/zna-2016-0201Search in Google Scholar

[29] W. Hereman, Int. J. Quantum Chem. 106, 278 (2006).10.1002/qua.20727Search in Google Scholar

[30] M. Hisakado and M. Wadati, J. Phys. Soc. Japan 64, 408 (1995).10.1143/JPSJ.64.408Search in Google Scholar

Received: 2016-8-22

Accepted: 2016-9-8

Published Online: 2016-10-10

Published in Print: 2016-12-1

Articles in the same Issue

https://doi.org/10.1515/zna-2016-0310

Keywords for this article

Higher-Order Dispersive Effects and Nonlinearity; Nonlinear Schrödinger Equation; Optical Fibre; Oscillations; Solitons