Symbolic computation on the bright soliton solutions for the generalized coupled nonlinear...

Optics Communications 285 (2012) 3567–3577

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Optics Communications

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Symbolic computation on the bright soliton solutions for the generalized couplednonlinear Schrödinger equations with cubic–quintic nonlinearity

Pan Wang, Bo Tian ⁎State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, ChinaSchool of Science, P. O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China

⁎ Corresponding author.E-mail address: [email protected] (B. Tian).

0030-4018/$ – see front matter © 2012 Elsevier B.V. Alldoi:10.1016/j.optcom.2012.04.023

a b s t r a c t
a r t i c l e i n f o
Article history:Received 21 December 2011Received in revised form 12 April 2012Accepted 17 April 2012Available online 1 May 2012

Keywords:Generalized coupled nonlinear Schrödingerequations with cubic–quintic nonlinearitySoliton solutionsHirota methodSymbolic computation

Under investigation in this paper are the generalized coupled nonlinear Schrödinger equations with cubic–quintic nonlinearity which describe the effects of the quintic nonlinearity on the ultrashort optical solitonpulse propagation in the non-Kerr media. Via the dependent variable transformation and Hirota method, the bi-linear form is derived. Based on the bilinear form obtained, the one-, two- and three-soliton solutions are pres-ented in the form of exponential polynomials with the help of symbolic computation. Propagation andinteractions of solitons are investigated analytically and graphically. Evolution of one soliton is discussed withthe analysis of such physical quantities as the soliton amplitude, width, velocity, initial phase and energy. Inter-actions of the solitons appear in the forms of the repulsion or attraction alternately and propagation in parallel.Inelastic and head-on interactions of the solitons are also showed. Finally, via the asymptotic analysis, conditionsof the elastic and inelastic interactions are obtained.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Optical solitons have a promising potential to become the carriers intelecommunication due to their capability of propagating the long dis-tances without attenuation [1–4]. Hence, attention is being paid theo-retically and experimentally to analyze the dynamics of opticalsolitons [5,6]. Such investigations are helpful for realizing the opticalsoliton applications, particularly in the soliton-based optical communi-cation systems [5] and nonlinear optical switches [6]. The waveguideswhich are used in such optical systems are usually of the Kerr type[6]. Consequently, the dynamics of light pulses are described by thenonlinear Schrödinger (NLS) family of equations with cubic nonlinearterms [7]. However, as the intensity of the incident light field becomesstronger, non-Kerr nonlinearity effect comes into play [2]. The influ-ences of the non-Kerr nonlinearity on the NLS soliton propagation aredescribed by the NLS family of equations with higher-degree nonlinearterms [8–15].

The cubic–quintic NLS equation appears in several physical areassuch as the nonlinear optics [16], nuclear physics [17] and Bose–Einstein condensation [18]. In the nonlinear optics, it describes thepropagation of pulses in the double-doped optical fibers [16]. Periodicalvariation of the nonlinearity can be achieved by varying the type of dop-ants along the fibers [19]. Moreover, the quintic NLS equation has theapplications in water wave theory as well as other physical systemswhere the time and space scales are greater than those described bythe cubic NLS equation [20–22].

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In this paper, we will investigate the generalized coupled NLS equa-tions with cubic–quintic nonlinearity, i.e., the quintic generalization ofthe coupled cubic NLS equations [23,24] which describe the effects ofquintic nonlinearity on the ultrashort optical pulse propagation innon-Kerr media [25,26],

iq1z þ q1tt þ 2 q1j j2 þ q2j j2� �

q1 þ ρ1 q1j j2 þ ρ2 q2j j2� �2

q1

−2i ρ1 q1j j2 þ ρ2 q2j j2� �

q1h i

tþ 2i ρ1q

�1q1t þ ρ2q

�2q2tð Þq1 ¼ 0;

ð1aÞ

iq2z þ q2tt þ 2 q1j j2 þ q2j j2� �

q2 þ ρ1 q1j j2 þ ρ2 q2j j2� �2

q2

−2i ρ1 q1j j2 þ ρ1 q2j j2� �

q2h i

tþ 2i ρ1q

�1q1t þ ρ2q

�2q2tð Þq2 ¼ 0;

ð1bÞ

where the components q1 and q2 of the electromagnetic fields propa-gate along the coordinate z in the two cores of an optical waveguide,t is the local time, ρ1 and ρ2 are the free parameters and * denotes thecomplex conjugate. When q1=q and q2=0 (or q1=0 and q2=q),Eqs. (1a) and (1b) will reduce to the integrable Kundu–Eckhaus equa-tion [23,24,27]. Eqs. (1a) and (1b) have been studied in the followingaspects: conserved quantities and Lax pair have been presented [23];relationship between the Manakov model and Eqs. (1a) and (1b) hasbeen established [23]; Darboux transformation has been constructedand the corresponding soliton solutions have been obtained [26]; Eqs.(1a) and (1b) without the quintic nonlinearity have been investigated[28]. Ref. [23] has showed that Eqs. (1a) and (1b) are integrable. InRef. [29], the non-integrable coupled cubic–quintic NLS equationshave been studied based on the calculation of the interaction term inthe full Hamiltonian of the system. We can find that both Eqs. (1a)

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3568 P. Wang, B. Tian / Optics Communications 285 (2012) 3567–3577

and (1b) and the non-integrable coupled cubic–quintic NLS equationshave the cubic–quintic nonlinearity terms. They are the 1+1 dimen-sional integrable and multi-dimensional non-integrable equations,respectively.

However, to our knowledge, multi-soliton solutions in the exponen-tial polynomial form and interaction properties of solitons for Eqs. (1a)and (1b) have not been uncovered as yet. In Section 2, via the depen-dent variable transformation and Hirota method, the bilinear formwill be derived. Based on the bilinear form obtained, the one-, two-and three-soliton solutions for Eqs. (1a) and (1b) will be presentedwith symbolic computation. Propagation and interactions of solitonswill be investigated analytically and graphically in Section 3. Section 4will be our conclusions.

2. Bilinear form and soliton solutions for Eqs. (1a) and (1b)

2.1. Bilinear form

Introducing the dependent variable transformation

q1 ¼ gfexp i∫ ρ1

gg�

f 2þ ρ2

hh�

f 2

� �dt

� �;

q2 ¼ hfexp i∫ ρ1

gg�

f 2þ ρ2

hh�

f 2

� �dt

� �;

ð2Þ

where g(z, t) and h(z, t) are both the complex differentiable functions,while f(z, t) is a real one, we derive the bilinear form for Eqs. (1a) and(1b) as follows:

iDz þ D2t −λ

� �g·fð Þ ¼ 0; ð3aÞ

iDz þ D2t −λ

� �h·fð Þ ¼ 0; ð3bÞ

D2t −λ

� �f ·f þ 2 gj j2 þ hj j2

� �¼ 0; ð3cÞ

with λ as a real constant. Hereby, Dz and Dt are the bilinear derivativeoperators [30,31] defined by

Dmz D

nt ν·υð Þ ¼ ∂

∂z−∂∂z′

� �m ∂∂t−

∂∂t′

� �n

ν z; tð Þυ z′; t′ð Þ z′¼z;t′¼t ;��

where m and n are the nonnegative integers.

2.2. Bright solitons

Eqs. (1a) and (1b) can be solved by the following expansions forg(z, t), h(z, t) and f(z, t):

g z; tð Þ ¼ εg1 z; tð Þ þ ε3g3 z; tð Þ þ ε5g5 z; tð Þ þ ⋯; ð4aÞ

h z; tð Þ ¼ εh1 z; tð Þ þ ε3h3 z; tð Þ þ ε5h5 z; tð Þ þ ⋯; ð4bÞ

f z; tð Þ ¼ 1þ ε2f 2 z; tð Þ þ ε4f 4 z; tð Þ þ ε6f 6 z; tð Þ þ ⋯; ð4cÞ

where ε is a formal expansion parameter, the coefficients fi(z, t)'s(i=2, 4, 6,…), gj(z, t)'s (j=1, 3, 5,…) and hj(z, t)'s (j=1, 3, 5,…)are the differentiable functions to be determined. The recursion rela-tion for fn(z, t)'s, gn(z, t)'s and hn(z, t)'s (n=1, 2,…) can be derived bysubstituting Expressions (4a), (4b) and (4c) into Bilinear Form (3a,b,c) and equating the coefficients of the same powers of ε. It should benoticed that gn(z, t) and hn(z, t) have only the odd items of ε, while fn(z,t) has only the even items of ε. In the physical context, in general, thebright soliton solutions can be obtained when gn(z, t) and hn(z, t) haveonly the odd items, while fn(z, t) has only the even items of ε; thedark soliton solutions can be derived when gn(z, t), hn(z, t) and fn(z, t)

all have the odd and even items. In the mathematical context, in orderto obtain the bright soliton solutions, we will find that the coefficientsof even terms for both gn(z, t) and hn(z, t) equal to zero and the onesof odd terms for fn(z, t) equal to zero after some calculations.

Without loss of generality, we set ε=1. Truncating Expressions (4a),(4b) and (4c) with gm=0, hm=0 (m=3, 5,…) and fn=0 (n=4, 6,…),and substituting them into Bilinear Form (3a,b,c), we derive

g ¼ g1 ¼ βeξ; h ¼ h1 ¼ γeξ;λ ¼ 0;

f ¼ 1þ f 2 ¼ 1þ ββ� þ γγ�

wþw�ð Þ2 eξþξ�; ξ ¼ iw2zþwt þ θ;

where w, θ, β and γ are the complex constants. Thus, the bright one-soliton solutions for Eqs. (1a) and (1b) can be expressed as

q1 ¼ β2sech 2−1 ξþ ξ� þ 2Δ

h i� exp −Δþ iIm ξð Þ þ i∫ ρ1ββ

� þ ρ2γγ�ð Þexp ξþ ξ�ð Þ

1þ exp 2Δþ ξþ ξ�ð Þ dt� �

;

ð5aÞ

q2 ¼ γ2sech 2−1 ξþ ξ� þ 2Δ

h i� exp −Δþ iIm ξð Þ þ i∫ ρ1ββ

� þ ρ2γγ�ð Þexp ξþ ξ�ð Þ

1þ exp 2Δþ ξþ ξ�ð Þ dt� �

;ð5bÞ

where

e2Δ ¼ ββ� þ γγ�

wþw�ð Þ2 ;

with Im(ξ) denoting the imaginary part of ξ and Re(ξ) denoting the realpart of ξ.

For obtaining the bright two-soliton solutions, we take

g ¼ g1 þ g3; h ¼ h1 þ h3; f ¼ 1þ f 2 þ f 4; ð6Þ

where λ=0, g1=β1eξ1+β2e

ξ2, h1=γ1eξ1+γ2e

ξ2 and ξj= iwj2z+wjt+

θj, βj's, γj's,wj's and θj's (j=1, 2) are the complex constants. SubstitutingExpression (6) into Bilinear Form (3a,b,c), we obtain the bright two-soliton solutions for Eqs. (1a) and (1b) as follows:

q1 ¼ g1 þ g31þ f 2 þ f 4

exp i∫ρ1 g1 þ g3ð Þ g�1 þ g�3ð Þ þ ρ2 h1 þ h3ð Þ h�1 þ h�3ð Þ1þ f 2 þ f 4ð Þ2 dt

" #;

ð7aÞ

q2 ¼ h1 þ h31þ f 2 þ f 4

exp i∫ρ1 g1 þ g3ð Þ g�1 þ g�3ð Þ þ ρ2 h1 þ h3ð Þ h�1 þ h�3ð Þ1þ f 2 þ f 4ð Þ2 dt

" #;

ð7bÞ

where

g3 ¼ ι1eξ1þξ2þξ�1 þ ι2e

ξ1þξ2þξ�2 ;h3 ¼ ς1eξ1þξ2þξ�1 þ ς2e

ξ1þξ2þξ�2 ;

f 2 ¼ m11eξ1þξ�1 þm12e

ξ1þξ�2 þm21eξ2þξ�1 þm22e

ξ2þξ�2 ; f 4 ¼ τeξ1þξ2þξ�1þξ�2 ;

m11 ¼ β1β�1 þ γ1γ

�1

w1 þw�1

2 ; m12 ¼ β1β�2 þ γ1γ

�2

w1 þw�2

2 ; m21 ¼ β2β�1 þ γ2γ

�1

w2 þw�1

2 ;

m22 ¼ β2β�2 þ γ2γ

�2

w2 þw�2

2 ;

ι1 ¼ w1−w2ð Þ −β2γ1γ�1 w2 þw�

1ð Þ þ β1β2β�1 w1−w2ð Þ þ β1γ2γ

�1 w1 þw�

1ð Þ½ �w1 þw�

1

2 w2 þw�1

2 ;

ι2 ¼ w1−w2ð Þ −β2γ1γ�2 w2 þw�

2ð Þ þ β1β2β�2 w1−w2ð Þ þ β1γ2γ

�2 w1 þw�

2ð Þ½ �w1 þw�

2

2 w2 þw�2

2 ;

-7

7

z

-3

5

t

0

1

q1

-7

7

z

-3

5

t

0

1.4

q2

(a)

(b)

Figs. 1. Two solitons via Solutions (7a,b). Parameters are ρ1=3, ρ2=−2, w1=1,w2=0.5, β1=1, β2=−1, γ1=2, γ2=−2, θ1=0 and θ2=0.

-7

7

z

-4

6

t

0

1

q1

-7

7

z

-4

6

t

0

1

q2

(a)

(b)

Figs. 2. Two solitons via Solutions (7a,b). Parameters are ρ1=3, ρ2=−2, w1=1,w2=0.5, β1=2, β2=−1, γ1=2, γ2=−2, θ1=0 and θ2=0.

3569P. Wang, B. Tian / Optics Communications 285 (2012) 3567–3577

ς1 ¼ w1−w2ð Þ β2γ1β�1 w1 þw�

1ð Þ−β1γ2β�1 w�

1 þw2ð Þ−γ1γ2γ�1 w2−w1ð Þ½ �

w1 þw�1

2 w2 þw�1

2 ;

ς2 ¼ w1−w2ð Þ β2γ1β�2 w1 þw�

2ð Þ−β1γ2β�2 w�

2 þw2ð Þ−γ1γ2γ�2 w2−w1ð Þ½ �

w1 þw�2

2 w2 þw�2

2 ;

τ ¼ w1−w2ð Þ w�1−w�

2ð Þ β1τ1 þ γ1τ2ð Þw1 þw�

1

2 w2 þw�1

2 w1 þw�2

2 w2 þw�2

2 ;τ1 ¼ β2β

�1β

�2 w1−w2ð Þ w�

1−w�2

þ γ2 −β�

2γ�1 w1 þw�

1

w2 þw�2

þ β�1 γ�

2 w2 þw�1

w1 þw�

2 � �

;

τ2 ¼ γ2γ�1γ

�2 w1−w2ð Þ w�

1−w�2

þ β2 β�

2γ�1 w2 þw�

1

w1 þw�2

−β�

1γ�2 w1 þw�

1

w2 þw�2

� �:

With symbolic computation [32], the three-soliton solutions canbe obtained as follows:

q1 ¼ g1 þ g3 þ g51þ f 2 þ f 4 þ f 6

expni∫hρ1 g1 þ g3 þ g5ð Þ g�1 þ g�3 þ g�5

þρ2 h1 þ h3 þ h5ð Þ� h�1 þ h�3 þ h�5

i1þ f 2 þ f 4 þ f 6ð Þ−2dt

o;

ð8aÞ

q2 ¼ h1 þ h3 þ h51þ f 2 þ f 4 þ f 6

expni∫hρ1 g1 þ g3 þ g5ð Þ g�1 þ g�3 þ g�5

þρ2 h1 þ h3 þ h5ð Þ � h�1 þ h�3 þ h�5

i1þ f 2 þ f 4 þ f 6ð Þ−2dt

o;

ð8bÞ

where

g1 ¼ β1eξ1 þ β2e

ξ2 þ β3eξ3 ;h1 ¼ γ1e

ξ1 þ γ2eξ2 þ γ3e

ξ3 ;

f 6 ¼ h6eξ1þξ2þξ3þξ�1þξ�2þξ�3 ;λ ¼ 0; ξj ¼ iw2

j zþwjt þ θj j ¼ 1;2;3ð Þ;

g3 ¼ n11eξ1þξ2þξ�1 þ n12e

ξ1þξ2þξ�2 þ n13eξ1þξ2þξ�3 þ n21e

ξ1þξ3þξ�1 þ n22eξ1þξ3þξ�2

þ n23eξ1þξ3þξ�3 þ n31e

ξ2þξ3þξ�1 þ n32eξ2þξ3þξ�2 þ n33e

ξ2þξ3þξ�3 ;

h3 ¼ s11eξ1þξ2þξ�1 þ s12e

ξ1þξ2þξ�2 þ s13eξ1þξ2þξ�3 þ s21e

ξ1þξ3þξ�1 þ s22eξ1þξ3þξ�2

þs23eξ1þξ3þξ�3 þ s31e

ξ2þξ3þξ�1 þ s32eξ2þξ3þξ�2 þ s33e

ξ2þξ3þξ�3 ;

g5 ¼ r1eξ1þξ2þξ3þξ�1þξ�2 þ r2e

ξ1þξ2þξ3þξ�1þξ�3 þ r3eξ1þξ2þξ3þξ�2þξ�3 ;

h5 ¼ r4eξ1þξ2þξ3þξ�1þξ�2 þ r5e

ξ1þξ2þξ3þξ�1þξ�3 þ r6eξ1þξ2þξ3þξ�2þξ�3 ;

f 2 ¼ m11eξ1þξ�1 þm12e


ξ2þξ�1 þm22eξ2þξ�2

þm23eξ2þξ�3 þm31e


ξ3þξ�3 ;

f 4 ¼ h12eξ1þξ2þξ�1þξ�2 þ h13e

ξ1þξ2þξ�1þξ�3 þ h11eξ1þξ2þξ�2þξ�3 þ h21e

ξ1þξ3þξ�1þξ�2

þ h22eξ1þξ3þξ�2þξ�3 þ h23e

ξ1þξ3þξ�1þξ�3 þ h31eξ2þξ3þξ�1þξ�3

þ h32eξ2þξ3þξ�1þξ�2 þ h33e

ξ2þξ3þξ�2þξ�3 ;

where wj's, βj's, γj's and θj's (j=1,2,3) are all the complex constants.The coefficient expressions can be seen in Appendix A.

3. Analysis and discussions

Based on Solutions (5a,b), (7a,b) and (8a,b), we can reveal and un-derstand the propagation and interaction mechanisms of the solitonsfor Eqs. (1a) and (1b). More on the solitonic interactions can be seenin Refs. [33,34]. The amplitude, width, velocity, initial phase and ener-gy are, respectively, given as

A1 ¼ β2eΔ

; A2 ¼ γ2eΔ

; W1 ¼ W2 ¼ 2wþw� ;

V1 ¼ V2 ¼ −i w–w� ; P1 ¼ P2 ¼ ln

ββ� þ γγ�

wþw�ð Þ2� �

= wþw� ;

E1 ¼ −ββ� wþw�ð Þ3eiw�2x

ββ� þ γγ�ð Þ wþw�ð Þ2eiw�2x þ ββ� þ γγ�ð Þeiw2xþ wþw�ð Þtþθþθ�h i ;

-2

2

z

-3

2

t

0

1

q1

-2

2

z

-3

2

t

0

2

q2

(a)

(b)

Figs. 3. Two solitons via Solutions (7a,b). Parameters are ρ1=2, ρ2=−1,w1=1,w2=−2, β1=−1, β2=1, γ1=2, γ2=3, θ1=2 and θ2=0.

-8 8t

1

q1

z 3

z 3

-8 8t

1

q2

z 3

z 3

(a)

(b)

Figs. 5. Two solitons via Solutions (7a,b). Parameters are the same with Fig. 4.


E2 ¼ −γγ� wþw�ð Þ3eiw�2x

ββ� þ γγ�ð Þ wþw�ð Þ2eiw�2x þ ββ� þ γγ�ð Þeiw2xþ wþw�ð Þtþθþθ�h i :

Interactions of the two solitons via Solutions (7a,b) are showed inFigs. 1–6, and the situations of three solitons via Solutions (8a,b) aregiven in Figs. 7–14. In Fig. 1(a), the main effect of the two solitons isthe repulsion or attraction alternately, and there are small bumps inthe interaction regions. The similar situation occurs in Fig. 1(b). In

-4

4

z

-5

5

t

0

1

q1

-4

4

z

-4

4

z

-5

5

t

0

1.5

q2

-4

4

z

(a)

(b)

Figs. 4. Two solitons via Solutions (7a,b). Parameters are ρ1=1, ρ2=−2, w1=1− i,w2=1, β1=−1, β2=1, γ1=2, γ2=3, θ1=0 and θ2=0.

Figs. 1, the interactions at the both sides of the raised areas are symmet-ric about the z-axis, since the relation β1/β2=γ1/γ2 is satisfied. Whenthe parameters meet the relation β1/β2≠γ1/γ2, the interactions at theboth sides of the raised areas are asymmetric about the z-axis inFigs. 2. The asymmetry can also be seen in Figs. 2.

From Figs. 3, we can see that the amplitudes take the different sit-uations in modes q1 and q2 as the parameters satisfy β1/β2≠γ1/γ2.Another feature in Figs. 3 is that the solitons propagate in parallel, in-dependently of each other, as we change the parameters θ1 or θ2. This

-2

2

z-3

3

t

0

1

q1

-2

2

z-3

3

t

0

1.7

q2

(a)

(b)

Figs. 6. Two solitons via Solutions (7a,b). Parameters are ρ1=1, ρ2=−2, w1=1− i,w2=−1+ i, β1=−1, β2=1, γ1=2, γ2=3, θ1=0 and θ2=0.

-7 7z

-3

5

t

(a)

-7 7z

-3

5

t

(b)

Figs. 7. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1, w1=2,w2=0.5, w3=1, β1=1, β2=1, β3=1, γ1=1, γ2=1, γ3=1, θ1=0, θ2=0 and θ3=0.

-7 7z

-3

5

t

-7 7z

-3

5

t

(a)

(b)

Figs. 8. Three solitons via Solutions (8a,b). Parameters are the same with Fig. 11 exceptfor w1=−2.


is conducive to the multi-channel optical fiber transmission. Interac-tions for the solitons in Figs. 4 are inelastic, i.e., if a soliton getssuppressed and the other one gets enhanced after the interaction inmode q1, they will be enhanced and suppressed after the interactionin mode q2, respectively. The changes of the soliton amplitudes afterthe interactions can be seen in Figs. 5.

With the choice of parameters, the head-on interactions of the twosolitons are showed in Figs. 6. There are the slight concave and convexin the modes of q1 and q2, respectively. Figs. 6 give the interactions ofthe two solitons under the parameters β1/β2b0 and γ1/γ2>0. Whenthe parametersβ1/β2>0 and γ1/γ2>0, orβ1/β2b0 and γ1/γ2b0, the in-teractions present the similar properties.

Figs. 7 give the interactions of the three solitons as parametersw1,w2 and w3 are all positive real numbers and β1/β2/β3=γ1/γ2/γ3. Asymmetrical structure can be observed in Figs. 8, as we only changethe sign of w1 by comparing Figs. 7 and 8. When the parametersmeet the relation β1/β2/β3≠γ1/γ2/γ3, the asymmetrical structure ofthe interactions among the three solitons appears in Figs. 9.

Interactions among a stationary soliton and the other two ones areshowed in Figs. 10. From Figs. 11, we can see that there are threebumps at the interaction regions. Figs. 12 give the head-on interactionsamong the two parallel solitons and another one. There are the slightconcave and convex at the interaction regions. Interactions among thetwo solitons with the bounded states and another one are showed in

Figs. 13. We can also find that the soliton velocities will be the samewhen the parameters wj's have the same imaginary part. Two solitonswith the same velocity will periodically attract and repel, and formthe bounded states in Figs. 13. Figs. 14 present the inelastic interactionsamong the three solitons, i.e., the two solitons get suppressed andanother gets enhanced in mode q1, while they will also be suppressedand enhanced after the interaction inmode q2, respectively. The changeof the soliton amplitudes at the different times can be seen in Figs. 15.

In order to check whether the interaction between two solitons iselastic, we make the asymptotic analysis to investigate Two SolitonSolutions (7a,b) as follows:

Before the interaction (z→−∞):

q1−1 →β1½ �2eΔ1

sech12

ξ1 þ ξ�1 þ 2Δ1 � �

; ξ1 þ ξ�1∼0; ξ2 þ ξ�2→−∞

; ð9aÞ

q1−2 →γ1j j2eΔ1

sech12

ξ1 þ ξ�1 þ 2Δ1 � �

; ξ1 þ ξ�1∼0; ξ2 þ ξ�2→−∞

; ð9bÞ

q2−1 →ι1j j

2m11eΔ2

sech12

ξ1 þ ξ�1 þ 2Δ2 � �

; ξ2 þ ξ�2∼0; ξ1 þ ξ�1→þ ∞

;

ð9cÞ

-7 7z

-7 7z

-3

5t

-3

5

t(a)

(b)

Figs. 9. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1, w1=−2,w2=0.5, w3=1, β1=1, β2=1, β3=3, γ1=1, γ2=2, γ3=1, θ1=0, θ2=0 and θ3=0.

-3

3

z

-4

7

t

0

3.4

q1

-3

3

z

-4

7

t

0

3.4

q2

(a)

(b)

Figs. 10. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1, w1=2,w2=1+ i, w3=1− i, β1=1, β2=1, β3=1, γ1=1, γ2=1, γ3=1, θ1=0, θ2=0 andθ3=0.


q2−2 →ς1j j

2m11eΔ2

sech12

ξ1 þ ξ�1 þ 2Δ2 � �

; ξ2 þ ξ�2∼0; ξ1 þ ξ�1→þ ∞

;

ð9dÞ

where e2Δ1=m11 and e2Δ2=τ/m11, with qj1−'s and qj

2−'s (j=1, 2)denoting the asymptotic expressions for the two solitons of modesq1 and q2 before the interaction, respectively.

After the interaction (z→+∞):

q1þ1 →ι2j j

2m22eΔ3

sech12

ξ1 þ ξ�1 þ 2Δ3 � �

; ξ1 þ ξ�1∼0; ξ2 þ ξ�2→þ ∞

;

ð10aÞ

q1þ2 →ς2j j

2m22eΔ3

sech12

ξ1 þ ξ�1 þ 2Δ3 � �

; ξ1 þ ξ�1∼0; ξ2 þ ξ�2→þ ∞

;

ð10bÞ

q2þ1 →β2j j2eΔ4

sech12

ξ1 þ ξ�1 þ 2Δ4 � �

; ξ2 þ ξ�2∼0; ξ1 þ ξ�1→−∞

;

ð10cÞ

q2þ2 →γ2j j2eΔ4

sech12

ξ1 þ ξ�1 þ 2Δ4 � �

; ξ2 þ ξ�2∼0; ξ1 þ ξ�1→−∞

;

ð10dÞ

where e2Δ3=τ/m22 and e2Δ4=m22, with qj1+'s and qj

2+'s (j=1, 2)denoting the asymptotic expressions for the two solitons of modesq1 and q2 after the interaction, respectively. When the parameters ac-cord with the following conditions:

β1j jeΔ1

¼ ι2j jm22e

Δ3and

ι1j jm11e

Δ2¼ β2j j

eΔ4; ð11aÞ

γ1j jeΔ1

¼ ς2j jm22e

Δ3and

ς1j jm11e

Δ2¼ γ2j j

eΔ4; ð11bÞ

the interactions of the two solitons for modes q1 and q2 are both elastic.When |β1|e−Δ1≠m22

−1|ι2|e−Δ3 or m11−1|ι1|e−Δ2≠ |β2|e−Δ4, the interac-

tion of the two solitons for mode q1 is inelastic. When |γ1|e−Δ1≠m22

−1|ς2|e−Δ3 or m11−1|ς1|e−Δ2≠ |γ2|e−Δ4, the interaction of the two

solitons for mode q2 is inelastic.

4. Conclusions

In this paper, with symbolic computation, the generalized couplednonlinear Schrödinger equations [i.e., Eqs. (1a) and (1b)] with cubic–quintic nonlinearity have been studied, which describe the effects ofquintic nonlinearity on the ultrashort optical pulse propagation innon-Kerr media. Via Dependent Variable Transformation (2) andHirota method, Bilinear Form (3a,b,c) for Eqs. (1a) and (1b) havebeen obtained. The bright one-, two- and three-soliton solutions[i.e., Solutions (5a,b), (7a,b) and (8a,b)] have been derived.

Propagation and interactions of solitons have been investigatedanalytically and graphically. According to Solutions (5a,b), the

-3 3z

-3 3z

-4

7t

-4

7

t

(a)

(b)

Figs. 11. The contour plots of Figs. 14.

-3

2

z

-2

7

t

0

2.4

q1

-3

2

z

-2

7

t

0

2.4

q2

(a)

(b)

Figs. 12. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1, w1=1.1+ i,w2=1− i, w3=1+ i, β1=1, β2=1, β3=1, γ1=1, γ2=1, γ3=1, θ1=0, θ2=0 andθ3=0.

-3

3

z

-6

7

t

0

2.4

q1

-3

3

z

-6

7

t

0

2.4

q2

(a)

(b)

Figs. 13. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1,w1=−1+ i,w2=2− i, w3=1.1− i, β1=1, β2=1, β3=1, γ1=1, γ2=1, γ3=1, θ1=0, θ2=0 andθ3=0.


propagation of one soliton has been discussed with the analysis ofsuch physical quantities as the soliton amplitude, width, velocity, ini-tial phase and energy. Interactions of the two solitons via Solutions(7a,b) have been shown in Figs. 1–6, while the situations of thethree solitons through Solutions (8a,b) have been shown inFigs. 7–14. In Figs. 1, the main effect of the two solitons is the repul-sion or attraction alternately, and there are bumps in the interactionregions. The interactions at the both sides of the raised areas areasymmetric about the z-axis in Figs. 2. Figs. 3 show that two solitonspropagate in parallel, independently of each other. Inelastic interac-tions of two solitons are presented in Figs. 4, i.e., if a soliton getssuppressed and the other one gets enhanced after the interaction inmode q1, they will be enhanced and suppressed after the interactionin mode q2. The change of the soliton amplitudes can be seen inFigs. 5. Figs. 6 give the head-on interactions of the two solitons. Thesymmetrical and asymmetrical interaction structures of the three sol-itons are showed in Figs. 7–9, respectively. Interactions among astationary soliton and the other two solitons are showed in Figs. 10and 11, and the situations of the two parallel solitons and anothersoliton are given in Figs. 12. Figs. 13 reveal the evolution of thetwo solitons with the bounded states interacting with another soli-ton. From the velocity expressions, we can find that the soliton ve-locities will be the same when the parameters wj's have the sameimaginary part. Those solitons with the same velocity will perio-dically attract and repel, and form the bounded states in Figs. 13. In-elastic interactions of the three solitons are presented in Figs. 14

and 15. Finally, via the asymptotic analysis, Conditions (11a) and(11b) of the elastic interactions for modes q1 and q2 are obtained.When the parameters do not meet Conditions (11a) and (11b), in-elastic interactions will occur.

-2

2

z

-6

5

t

0

2

q1

-2

2

z

-6

5

t

0

2

q2

(a)

(b)

Figs. 14. Three solitons via Solutions (8a,b). Parameters are ρ1=1, ρ2=1,w1=−1+ i,w2=−1−2i,w3=−1.1− i, β1=1, β2=2, β3=1, γ1=1, γ2=1, γ3=1, θ1=0, θ2=0and θ3=0.

-10 10

-10 10

t

1 z 2

z 2

t

1

q1

q2

z 2z 2

(a)

(b)

Figs. 15. Three solitons via Solutions (8a,b). Parameters are the same with Fig. 14.


In the framework of the present paper, other coupled nonlinearmodels with the cubic–quintic nonlinearity can be correspondinglyinvestigated. For instance, the N-coupled nonlinear Schrödingerequations describing the simultaneous propagation of N fields in op-tical fibers are as follows:

iqjz þ qjtt þ 2XNm¼1

qmj j2 !

qj þXNm¼1

ρm qmj j2 !2

qj−2iXNm¼1

ρm qmj j2 !

qj

" #t

þ2iXNm¼1

ρmq�mqmt

!qj ¼ 0; j ¼ 1;2;…N;m ¼ 1;2;…Nð Þ:

ð12Þ

Through the dependent variable transformation qj ¼gjfexp i∫

XNm¼1

"ρmgmg

�m

f 2

!dt#, we can derive the bilinear form for Eq. (12) in the follow-

ing form:

iDz þ D2t −λ

� �gj·f� �

¼ 0; ð13aÞ

D2t −λ

� �f ·f þ 2

XNm¼1

gmj j2 !

¼ 0: ð13bÞ

Results on Eq. (12) will be published elsewhere.

Acknowledgments

This work has been supported by the National Natural ScienceFoundation of China under Grant no. 60772023, by the FundamentalResearch Funds for the Central Universities of China under GrantNo. 2011BUPTYB02, and by the Specialized Research Fund for theDoctoral Program of Higher Education (No. 200800130006), ChineseMinistry of Education.

Appendix A

n1j ¼w1−w2ð Þ −β2γ1γ

�j w2 þw�

j

� �þ β1β2β

�j w1−w2ð Þ þ β1γ2γ

�j w1 þw�

j

� �h iw1 þw�

j

� �2w2 þw�

j

� �2j ¼ 1;2;3ð Þ;


�j w3 þw�

j

� �þ β1β3β


�j w1 þw�

j

� �h iw1 þw�

j

� �2w3 þw�

j

� �2j ¼ 1;2; 3ð Þ;


�j w3 þw�

j

� �þ β2β3β


�j w2 þw�

j

� �h iw2 þw�

j

� �2w3 þw�

j

� �2j ¼ 1;2; 3ð Þ;

s1j ¼w1−w2ð Þ β2γ1β

�j w1 þw�

j

� �−γ2β1β

�j w2 þw�

j

� �−γ2γ1γ

�j w2−w1ð Þ

h iw1 þw�

j

� �2w2 þw�

j

� �2j ¼ 1;2; 3ð Þ;


�j w1 þw�

j

� �−γ3β1β

�j w3 þw�

j

� �−γ3γ1γ

�j w3−w1ð Þ

h iw1 þw�

j

� �2w3 þw�

j

� �2j ¼ 1;2; 3ð Þ;


�j w2 þw�

j

� �−γ3β2β

�j w3 þw�

j

� �−γ3γ2γ

�j w3−w2ð Þ

h iw2 þw�

j

� �2w3 þw�

j

� �2j ¼ 1;2;3ð Þ;


r1 ¼n

w1−w2ð Þ w�1−w�

2ð Þ w1−w3ð Þ w2−w3ð Þn

w1−w2ð Þβ1β2r11

þ β1γ2r12 þ γ1

hw1−w2ð Þ w�

1−w�2ð Þ w3 þw�

1ð Þ w3 þw�2ð Þβ3γ2γ

�1γ

�2

þ β2r13ioo.h

w1 þw�1ð Þ2 w2 þw�

1ð Þ2 w1 þw�2ð Þ2

� w2 þw�2ð Þ2 w3 þw�

1ð Þ2 w3 þw�2ð Þ2i;

r11 ¼ w�1−w�

2ð Þ w1−w3ð Þ w2−w3ð Þβ3β�1β

�2

þγ3

h− w1 þw�

1ð Þ w2 þw�1ð Þ w3 þw�

2ð Þβ�2γ

�1

þβ�1γ

�2 w1 þw�


1ð Þi;

r12 ¼ w1 þw�1ð Þ w�


2ð Þ w2−w3ð Þγ3γ�1γ

�2

þ w1−w3ð Þβ3

hw1 þw�

1ð Þ w2 þw�2ð Þ

� w3 þw�1ð Þβ�

2γ�1−β�

1γ�2 w2 þw�


2ð Þi;

r13 ¼ − w2 þw�1ð Þ w�


2ð Þ w1−w3ð Þγ3γ�1γ

�2

− w2−w3ð Þβ3

hw2 þw�


� w3 þw�1ð Þβ�

2γ�1−β�

1γ�2 w1 þw�


2ð Þi;

rj ¼n

w1−w2ð Þ w�j−1−w�

3

� �w1−w3ð Þ w2−w3ð Þ

nw1−w2ð Þβ1β2rj1 þ β1γ2rj2

þγ1 w1−w2ð Þ w�j−1−w�

3

� �w3 þw�

j−1

� �w3−w�

3ð Þβ3γ2γ�j−1γ

�3 þ β2rj3

h ioo� w1 þw�

2ð Þ2 � w2 þw�2ð Þ2 w1 þw�

3ð Þ2 w2 þw�3ð Þ2 w3 þw�


h i× j ¼ 2;3ð Þ;

rj1 ¼ w�j−1−w�

3

� �w1−w3ð Þ w2−w3ð Þβ3β

�j−1β

�3

þγ3

h−β�

3γ�j−1 w1 þw�

j−1

� �w2 þw�

j−1

� �w3 þw�

3ð Þ

þβ�j−1γ

�3 w1 þw�


j−1

� �ij ¼ 2;3ð Þ;

rj2 ¼ w1 þw�j−1

� �w�

j−1−w�3

� �w1 þw�

3ð Þ w2−w3ð Þγ3γ�j−1γ

�3

þ w1−w3ð Þβ3

hw1 þw�

j−1

� �w2 þw�

3ð Þ w3 þw�j−1

� �β�3γ

�j−1

−β�j−1γ

�3 w2 þw�

j−1

� �w1 þw�

3ð Þ w3 þw�3ð Þi

j ¼ 2;3ð Þ;

rj3 ¼ − w2 þw�j−1

� �w�

j−1−w�3

� �w2 þw�

3ð Þ w1−w3ð Þγ3γ�j−1γ

�3

− w2−w3ð Þβ3

hw2 þw�

j−1

� �w1 þw�


� �β�3γ

�j−1

−β�j−1γ

�3 w1 þw�

j−1

� �w2 þw�

3ð Þ w3 þw�3ð Þi

j ¼ 2;3ð Þ;

rj ¼n


j−2

� �w1−w3ð Þ w2−w3ð Þ

× β2rj1 þ γ2 β1rj2 þ γ1 w1−w2ð Þrj3h in oo

�hw1 þw�


j−2

� �2w2 þw�

j−2

� �2× w3 þw�

1ð Þ2 w3 þw�j−2

� �2ij ¼ 4;5ð Þ;

rj1 ¼ w1 þw�1ð Þ w�

1−w�j−2

� �w1 þw�

j−2

� �w2−w3ð Þβ3β

�1β

�j−2γ1

þγ3

nw1−w2ð Þ w�

1−w�j−2

� �w3 þw�


� �β1β

�1β

�j−2

−γ1 w1−w3ð Þhβ�j−2γ

�1 w1 þw�

j−2

� �w2 þw�


� �−β�

1γ�j−2 w1 þw�


� �w3 þw�

1ð Þio

j ¼ 4;5ð Þ;

rj2 ¼ − w�1 þw2ð Þ w�

1−w�j−2

� �w2 þw�

j−2

� �w1−w3ð Þβ3β

�1β

�j−2

þ w2−w3ð Þγ3

hw1 þw�


� �w3 þw�

j−2

� �β�j−2γ

�1

−β�1γ

�j−2 w2 þw�


� �w3 þw�

1ð Þi

j ¼ 4;5ð Þ;

rj3 ¼ w�1−w�

j−2

� �w1−w3ð Þ w2−w3ð Þγ3γ

�1γ

�j−2

þβ3

hw1 þw�

j−2

� �w2 þw�

j−2

� �w3 þw�

1ð Þβ�2γ

�1

−β�1γ

�j−2 w1 þw�


j−2

� �ij ¼ 4;5ð Þ;

r6 ¼n


3ð Þ w1−w3ð Þ w2−w3ð Þnβ2r61

þγ2

hβ1r62 þ γ1 w1−w2ð Þr63

ioo�hw1 þw�



× w2 þw�3ð Þ2 w3 þw�

3ð Þ2i;

r61 ¼ w1 þw�2ð Þ w�

2−w�3ð Þ w1−w�

3ð Þ w2−w3ð Þβ3β�2β

�3γ1

þγ3

nw1−w2ð Þ w�


2ð Þβ�2β

�3

−γ1 w1−w3ð Þhβ�3γ

�2 w1 þw�


3ð Þ

−β�2γ

�3 w1 þw�


2ð Þio

;

r62 ¼ − w�2 þw2ð Þ w�


3ð Þ w1−w3ð Þβ3β�2β

�3

þ w2−w3ð Þγ3

hw1 þw�


3ð Þβ�3γ

�2

−β�2γ

�3 w2 þw�


2ð Þi;

r63 ¼ w�2−w�

3ð Þ w1−w3ð Þ w2−w3ð Þγ3γ�2γ

�3

þβ3

hw1 þw�


2ð Þβ�3γ

�2

−β�2γ

�3 w1 þw�


3ð Þi;

mij ¼βiβ

�j þ γiγ

�j

wi þw�j

� �2 i ¼ 1;2;3; j ¼ 1;2;3ð Þ;

h1j ¼n


j

� �nβ1

nw1−w2ð Þ w�

1−w�j

� �β2β

�1β

�j

þγ2 −β�j γ

�1 w1 þw�

1ð Þ w2 þw�j

� �þ w2 þw�

1ð Þ w1 þw�j

� �β�1γ

�j

h ioþγ1

nw1−w2ð Þ w�

1−w�j

� �γ2γ

�1γ

�j þ β2

hw2 þw�

1ð Þ w1 þw�j

� �β�j γ

�1

− w1 þw�1ð Þ w2 þw�

j

� �β�1γ

�j

iooo.hw1 þw�

1ð Þ2

� w2 þw�1ð Þ2 w1 þw�

j

� �2w2 þw�

j

� �2ij ¼ 2;3ð Þ;

h11 ¼n


3ð Þnβ1

nw1−w2ð Þ w�

2−w�3ð Þβ2β

�2β

�3

þγ2

h−β�

3γ�2 w1−w�

2ð Þ w2 þw�3ð Þ þ w2 þw�

2ð Þ w1 þw�3ð Þβ�

2γ�3

ioþγ1

nw1−w2ð Þ w�

2−w�3ð Þγ2γ

�2γ

�3 þ β2

hw2 þw�


3γ�2


3ð Þβ�2γ

�3

iooo� w1 þw�



h i;


h21 ¼n


2ð Þnβ1

nw1−w3ð Þ w�

1−w�2ð Þβ3β

�1β

�2

þγ3

h−β�

2γ�1 w1 þw�



1γ�2

ioþγ1

nw1−w3ð Þ w�

1−w�2ð Þγ3γ

�1γ

�2 þ β3

hw1 þw�


2γ�1


2ð Þβ�1γ

�2

iooo� w1 þw�



h i;

h22 ¼n


3ð Þnβ1

nw1−w3ð Þ w�

2−w�3ð Þβ3β

�2β

�3

þγ3

h−β�

3γ�2 w1 þw�



2γ�3

ioþγ1

nw1−w3ð Þ w�

2−w�3ð Þγ3γ

�2γ

�3 þ β3

hw1 þw�


3γ�2


3ð Þβ�2γ

�3

iooo� w1 þw�



h i;

h23 ¼n


3ð Þnβ1

nw1−w3ð Þ w�

1−w�3ð Þβ3β

�1β

�3

þγ3

h−β�

3γ�1 w1 þw�



1γ�3

ioþγ1

nw1−w3ð Þ w�

1−w�3ð Þγ3γ

�1γ

�3 þ β3

hw1 þw�


3γ�1


3ð Þβ�1γ

�3

iooo� w1 þw�



h i;

h31 ¼n


3ð Þnβ2

nw2−w3ð Þ w�

1−w�3ð Þβ3β

�1β

�3

þγ3

h−β�

3γ�1 w2 þw�



1γ�3

ioþγ2

nw2−w3ð Þ w�

1−w�3ð Þγ3γ

�1γ

�3 þ β3

hw2 þw�


3γ�1


3ð Þβ�1γ

�3

iooo� w2 þw�



h i;

h32 ¼n


2ð Þnβ2

nw2−w3ð Þ w�

1−w�2ð Þβ3β

�1β

�2

þγ3

h−β�

2γ�1 w2 þw�



1γ�2

ioþγ2

nw2−w3ð Þ w�

1−w�2ð Þγ3γ

�1γ

�2 þ β3

hw2 þw�


2γ�1


2ð Þβ�1γ

�2

iooo� w2 þw�



h i;

h33 ¼n


3ð Þnβ2

nw2−w3ð Þ w�

2−w�3ð Þβ3β

�2β

�3

þγ3

h−β�

3γ�2 w2 þw�



2γ�3

ioþγ2

nw2−w3ð Þ w�

2−w�3ð Þγ3γ

�2γ

�3 þ β3

hw2 þw�


3γ�2


3ð Þβ�2γ

�3

iooo� w2 þw�



h i;

h6 ¼ f w1−w2ð Þ w�1−w�

2ð Þ w1−w3ð Þ w�1−w�

3ð Þ w2−w3ð Þ w�2−w�

3ð Þfβ1 β2h61 w1−w2ð Þ þ γ2 w2−w3ð Þγ3h62 þ w1−w3ð Þβ3h63½ �f gþγ1fγ2 w1−w2ð Þ½ w�

1−w�2ð Þ w1−w3ð Þ

� w2−w3ð Þ w�1−w�

3ð Þ w�2−w�

3ð Þγ3γ�1γ

�2γ

�3 þ β3h64�

þβ2½−γ3h65 w1−w3ð Þ þ β3h66

� w2−w3ð Þ�ggg=½ w1 þw�1ð Þ2 w1 þw�


1ð Þ2

× w2 þw�2ð Þ2 w2 þw�


2ð Þ2 w3 þw�3ð Þ2�

;

h61 ¼ w�1−w�

2ð Þ w1−w3ð Þ w2−w3ð Þ w�1−w�

3ð Þ w�2−w�

3ð Þβ3β�1β

�2β

�3

þγ3

n− w1 þw�


1ð Þ w�1−w�

3ð Þ

× w3 þw�3ð Þβ�

1β�3γ

�2 þ β�

2 w3 þw�2ð Þhw1 þw�


� w�2−w�


3γ�1 þ w�



× w2 þw�3ð Þβ�

1γ�3

io;

h62 ¼ w1 þw�1ð Þ w�



3ð Þβ�3γ

�1γ

�2

þγ�3 w1 þw�

3ð Þh−β�

2γ�1 w1 þw�


2ð Þ w�1−w�

3ð Þ

þβ�1γ

�2 w2 þw�


1ð Þ w�2−w�

3ð Þi;

h63 ¼ w2 þw�1ð Þ w�



2ð Þβ�1β

�3γ

�2

−β�2 w2 þw�

2ð Þhβ�3γ

�1 w1 þw�


3ð Þ w�2−w�

3ð Þ

þβ�1γ

�3 w2 þw�

1ð Þ w�1−w�


3ð Þi;

h64 ¼ w�1−w�



3ð Þβ�3γ

�1γ

�2

þγ�3 w3 þw�

3ð Þh−β�

2γ�1 w1 þw�


1ð Þ w�1−w�

3ð Þ

þβ�1γ

�2 w1 þw�


2ð Þ w�2−w�

3ð Þi;

h65 ¼ w2 þw�1ð Þ w�



3ð Þβ�3γ

�1γ

�2

þγ�3 w2 þw�

3ð Þh−β�

2γ�1 w2 þw�


2ð Þ w�1−w�

3ð Þ

þβ�1γ

�2 w1 þw�


1ð Þ w�2−w�

3ð Þi;

h66 ¼ − w1 þw�1ð Þ w2 þw�


3ð Þ w�1−w�

3ð Þβ�3β

�1γ

�2

þβ�2 w1 þw�

2ð Þhβ�3γ

�1 w2 þw�

1ð Þ w3 þw�1ð Þ w�


3ð Þ

þβ�1γ

�3 w1 þw�


3ð Þ w�1−w�

2ð Þi:

References

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Symbolic computation on the bright soliton solutions for the generalized coupled nonlinear...

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