A Note on Exact Traveling Wave Solutions of the ...

Commun. Theor. Phys. 57 (2012) 764–770

Vol. 57, No. 5, May 15, 2012

A Note on Exact Traveling Wave Solutions of the Perturbed Nonlinear Schr¨ odinger’s ∗ Equation with Kerr Law Nonlinearity

Ü2), o#²)

ZHANG Zai-Yun ( and LI Xin-Ping (

†

[), YU De-Min ({¬), ZHANG Ying-Hui (ÜN),

GAN Xiang-Yang (

School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

(Received October 20, 2011; revised manuscript received December 29, 2011)

Abstract In this paper, we investigate nonlinear the perturbed nonlinear Schr¨odinger’s equation (NLSE) with Kerr law nonlinearity given in [Z.Y. Zhang, et al., Appl. Math. Comput. 216 (2010) 3064] and obtain exact traveling solutions by using infinite series method (ISM), Cosine-function method (CFM). We show that the solutions by using ISM and CFM are equal. Finally, we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM). PACS numbers: 02.30.Jr, 04.20.Jb

Key words: exact solutions, NLSE with Kerr law nonlinearity, infinite series method (ISM), Cosine-function method (CFM), Jacobi elliptic function expansion method (JEFEM)

1 Introduction In the recent decades, the investigation of the traveling wave solutions of nonlinear partial differential equations (NLPDEs) plays an important role in the study of nonlinear physical phenomena and many direct methods have been developed to construct traveling wave solutions to the NLPDEs, such as the trigonometric function series method,[1−2] the modified mapping method and the extended mapping method,[3] the modified trigonometric function series method,[4] the dynamical system approach,[5−6] the exp-function method,[7−8] the multiple exp-function method,[9] the transformed rational function method,[10] the symmetry algebra method (consisting of Lie point symmetries),[11] the Wronskian technique,[12] the modified (G′ /G)-expansion method[13] and so on. In this paper, we will consider the perturbed NLSE with Kerr law nonlinearity[3] iut + uxx + α|u|2 u + i[γ1 uxxx + γ2 |u|2 ux + γ3 (|u|2 )x u] = 0,

(1)

where γ1 is the third order dispersion, γ2 is the nonlinear dispersion, while γ3 is also a version of nonlinear dispersion. More details are presented in Refs. [14] and [15]. Equation (1) describes the propagation of optical solitons in nonlinear optical fibers that exhibits a Kerr law nonlinearity. Recently, there are lots of contributions about Eq. (1) (see for instance[16−20] ). These papers have been concerned with finding various types of solutions, including fronts (kinks), bright solitary waves, and dark solitary waves in various media, such as power law (or dual-power law), parabolic law, Kerr law. Equation (1) has important application in various fields, such as semiconductor

materials, optical fiber communications, plasma physics, fluid and solid mechanics. More details are presented in [4, 14, 21–22] and the references therein. In the absence of γ1 , γ2 , γ3 (i.e. γ1 = γ2 = γ3 = 0), Eq. (1) reduces to iut + uxx + α|u|2 u = 0.

(2)

It is well known that the NLSE (2) admits the bright soliton solution:[23] r 2 2 2 sech (k(x − 2µt)) e i[µx−(µ −k )t] , u(x, t) = k α where α and k are arbitrary real constants, for the selffocusing case α > 0, and the dark soliton solution:[24] r 2 2 2 u(x, t) = k − tanh(k(x − 2µt)) e i[µx−(µ +2k )t] , α where α and k are arbitrary real constants, for the defocusing case α < 0. In this paper, we employ the ISM and CFM to investigate Eq. (1) and we obtain some explicit expressions of solutions for Eq. (1). We find that the solutions by using ISM and CFM are equal. Finally, we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM). Remark 1 It is worth mentioning that Zhang et al.[3−6,13] considered the NLSE with Kerr law nonlinearity and obtained some new exact traveling wave solutions of Eq. (1). In Ref. [3], by using the modified mapping method and the extended mapping method, Zhang et al. derived some new exact solutions of Eq. (1), which are the linear combination of two different Jacobi elliptic functions and investigated the solutions in the limit cases. In Ref. [4], by

∗ Supported by the Research Foundation of Education Bureau of Hunan Province under Grant No. 11C0628 and Foundation of Hunan Institute of Science and Technology under Grant No. 2011Y49 † Corresponding author, E-mail: [email protected] c 2011 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

Communications in Theoretical Physics

No. 5

using the modified trigonometric function series method, Zhang et al. studied some new exact traveling wave solutions. In Ref. [5], by using qualitative theory of dynamical systems, Zhang et al. obtained the traveling wave solutions in terms of bright and dark optical solitons and the cnoidal waves. The authors found that NLSE with Kerr law nonlinearity has only three types of bounded traveling wave solutions, namely, bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we pointed out the region which these periodic wave solutions lie in. We showed the relation between the bounded traveling wave solution and the energy level h. We observed that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution. In Ref. [6], by using the dynamical system approach, Zhang et al. investigated the dynamic behavior of traveling wave solutions to the Eq. (1). Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained. In Ref. [13], by using the modified (G′ /G)-expansion method, Miao and Zhang obtained the traveling wave solutions of Eq. (1), which are expressed by the hyperbolic functions, trigonometric functions and rational functions.

2 ISM and Exact Traveling Wave Solutions of Eq. (1) In this section, we investigate the perturbed NLSE with Kerr law nonlinearity Eq. (1) given in Ref. [3] and obtain new exact traveling solutions by using ISM. To facilitate further on our analysis, we assume that Eq. (1) has traveling wave solutions in the form[3] u(x, t) = φ(ξ) exp(i(Kx − Ωt)),

ξ = k(x − ct),

(3)

where c is the propagation speed of a wave. Substituting (3) into Eq. (1) yields i(γ1 k 3 φ′′′ − 3γ1 K 2 kφ′ + γ2 kφ2 φ′ + 2γ3 kφ2 φ′ − ckφ′ + 2Kkφ′ )

+ (Ωφ + k 2 φ′′ − K 2 φ + αφ3

+ 3γ1 Kk 2 φ′′ + γ1 K 3 φ − γ2 Kφ3 ) = 0,

where γi (i = 1, 2, 3), α, k are positive constants and the prime meaning differentiation with respect to ξ. By virtue of Ref. [3], pp. 3065, we have ′′

3

Aφ (ξ) + Bφ(ξ) + Cφ (ξ) = 0, 2

2

(4)

where A = γ1 k , B = 2K − c − 3γ1 K , C = (1/3)γ2 + (2/3)γ3 . For convenience, we assume that k = 1, A = 1 − c2 , B = −1, C = 2(1 − c)/(1 + c) (Chosen appropriate constants K, γ1 , γ2 , γ3 ). Infinite Series Method

765

Consider the nonlinear partial differential equation: F (u, ut , ux , uxx , . . .) = 0,

(5)

where u = u(x, t) is the solution of Eq. (5). Owing to transformations u(x, t) = f (ξ),

ξ = x + λt,

(6)

where λ is constant, we obtain ∂ ∂ ∂ ∂ (·) = λ (·), (·) = (·), ∂t ∂ξ ∂x ∂ξ ∂ ∂ (·) = 2 (·), . . . (7) 2 ∂x ∂ξ From Eqs. (5) and (7), we get the nonlinear ordinary differential equation (ODE) ∂f (ξ) ∂ 2 f (ξ) G f (ξ), , , . . . = 0. ∂ξ ∂ξ 2 Next, we apply the approach of Hereman et al.[25] and then suppose the solution in the form ∞ X f (ξ) = an g n (ξ), (8) n=1

where g(ξ) is a solution of linear terms and the expansion coefficients an (n = 1, 2, . . .) are to be determined. To deal with the nonlinear terms and perturbed term, we need to apply the extension of Cauchy’s Product Rule for multiple series. Lemma 1 (Extension of Cauchy’s product rule): If I X (i) F = a(i) i = 1, 2, . . . , I, n1 , n1 =1

represents I infinite convergent series, then I Y

F (i) =

∞ n−1 X X

...

n=1 r=I−1

i=1

k−1 X m−1 X

(1) (2)

(I)

al am−l . . . an−r .

m=2 l=1

Substituting (5) into ODE yields recursion relation, which gives the values of the coefficients. We assume Eq. (4) has the solution in the form ξ . g(ξ) = exp √ 1 − c2 Therefore, we look for the solution of Eq. (4) in the form ∞ nξ X φ(ξ) = an exp √ . (9) 1 − c2 n=1 Substituting (9) into (4) and by using Lemma 1, we obtain the recursion relation as follows a1 is arbitrary, (n2 − 1)an +

a2 = 0,

n−1 m−1 2(1 − c) X X al am−l an−m = 0, 1 + c m=2 l=1

n ≥ 3.

(10)

It follows form (10) that a2d = 0, a2d+1 =

−

2(1 − c) d a2d+1 1 , 1+c 23d

d = 1, 2, 3, . . .

(11)


766

Substituting (11) into (9) yields ∞ (2d + 1)ξ X 2(1 − c) d a2d+1 1 φ(ξ) = − exp √ 3d 1+c 2 1 − c2 d=0 √ a1 exp(ξ/ 1 − c2 ) √ . = 1 + ((1 − c)/(1 + c))(a21 /4) exp(2ξ/ 1 − c2 ) p If we choose a1 = ±2 (1 + c)/(1 − c), then r 1+c ξ sech √ φ(ξ) = ± . (12) 1−c 1 − c2 Hence we get the exact solutions of Eq. (3) r x − ct 1 + c |u(x, t)| = (13) sech √ , 1−c 1 − c2 for c2 < 1, see Fig. 1, and r 1 + c x − ct sech √ |u(x, t)| = (14) , 1−c c2 − 1 for c2 > 1, see Fig. 2, where |u| is the norm of u.

where ν, β, and γ are unknown parameters, which will be determined. Then we conclude φ′ (ξ) = − νβγ cosβ−1 (γξ) sin(γξ),

φ′′ (ξ) = − νβγ 2 cosβ (γξ)

+ νβγ 2 (β − 1) cosβ−2 (γξ) sin2 (γξ)

= − νβγ 2 cosβ (γξ) + νβγ 2 (β − 1) × cosβ−2 (γξ)(1 − cos2 (γξ))

= − νβ 2 γ 2 cosβ (γξ)

+ νβγ 2 (β − 1) cosβ−2 (γξ).

(1 − c2 )[−νβ 2 γ 2 cosβ (γξ) + νβγ 2 (β − 1) cosβ−2 (γξ)] 1−c 3 ν cos3β (γξ) = 0. − ν cosβ (γξ) + 2 1+c By equating the exponents and the coefficients of each pair of the cosine function we have the following system of algebraic equations:

u↼x֒t↽

2.0

(1 − c2 )νβγ 2 (β − 1) + 2

1.5

3β = β − 2.

0.5 1 1 0 t 0 −1

0 x

Fig. 1 (Color online) The graphics of solution (13), taking c = 1/2.

u↼x֒t↽

150 100 50 0 1.0 1 0.5 t

0 −1

0 x

Fig. 2 (Color online) The graphics of solution (14), taking c = 2.

3 CFM and Exact Traveling Wave Solutions of Eq. (1) In this section, the cosine function method[26−29] is applied to Eq. (1). We assume that Eq. (4) has traveling wave solutions in the form φ(ξ) = ν cosβ (γξ), (15)

(16)

Substituting (15) and (16) into Eq. (4) reduces

ν − νβ 2 γ 2 (1 − c2 ) = 0,

1.0

Vol. 57

1−c 3 ν = 0, 1+c (17)

It follows from (17) that r 1+c 1 ν=± . (18) , β = −1, γ = ± √ 2 1−c c −1 Combining (15) with (18), we obtain the exact solution to Eq. (4) as follows: r 1+c ξ φ(ξ) = ± , c2 > 1. sech √ 1−c 1 − c2 Then, the exact solution to the Eq. (1) can be written as r x − ct 1 + c |u(x, t)| = (19) sech √ 2 , 1−c c −1 for c2 > 1. Remark 2 In (14) and (19), if we choose a1 = p ±2 (1 + c)/(1 − c), c2 > 1, then the solutions are obtained of Eq. (1) by using the infinite series and cosine function methods are equal. Remark 3 In our contribution, we consider the exact traveling wave solutions of 1D-NLSE with Kerr law nonlinearity. It is worth mentioning that Chen et al.[26] proposed a multi-symplectic splitting (MSS) method to solve 2DNLSE using the idea of splitting numerical methods and the multi-symplectic methods. It is further shown that the method constructed in this way preserve the global symplecticity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method. In the absence of the perturbed term, Eq. (1) becomes Eq. (2). In Ref. [27], Wang et al. considered Eq. (2) and investigated the rogue waves with a controllable center in terms of rational-like functions by using a direct method. Moreover, the authors showed that the position


No. 5

of these solutions can be controlled by choosing different center parameters and this may describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, Bose– Einstein condensates respectively. More details are presented in Ref. [27]. In 3D-NLSE case, Lai[28] considered the (2+1)-dimensional generalized NLSE including linear and nonlinear gain (loss) with variable coefficients and obtained the exact chirped soliton-like and quasi-periodic wave solutions. In 4D-NLSE case, Xu et al.[29] have generalized the (1+1)D Jacobi elliptic function F-expansion and applied the improved method for solving exact solutions of a generalized (3+1)D NLSE. Exact solutions to the (3+1)D NLSE has been obtained whose distributed dispersion/diffraction, nonlin-earity, and gain or loss are allowed to be constant or variable as a function of one spatial or temporal dimension. More details are presented in Ref. [27] and the references therein.

4 JEFEM and Exact Traveling Wave Solutions of Eq. (1) In this section, we investigate the perturbed NLSE with Kerr law nonlinearity Eq. (1) given in Ref. [3] and we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM),[30] including the solutions of sn, cn, dn, cs type.

That is, (B + Ca20 )a0 = 0, 3Ca0 a21 = 0,

Taking m = 1, we have r |u2 (x, t)| =

see Fig. 4 as follows:

We observe that dφ = a1 cnξdnξ, dξ d2φ = −(1 + m2 )a1 snξ + 2m2 sn3 ξ, dξ 2 φ3 (ξ) = a30 + 3a20 a1 snξ + 3a0 a21 sn2 ξ + a31 sn3 ξ,

+ 3Ca0 a21 sn2 ξ + (Ca21 + 2m2 A)a1 sn3 ξ = 0.

as m → 1.

2A − tanh C

r

−

B (x − ct) , 2

(29)

u1↼x֒t↽

0.4 0.3 0.2 0.1 0 1.0 1 0.5 t

0 −1

0 x

Fig. 3 (Color online) The graphics of solution (28), taking m = 1/4, A = −1, B = −1, C = 2, c = 1.

(21) 1.0

(22) (23) (24)

where m (0 < m < 1) is the modulus of the Jacobi elliptic functions, see Ref. [3]. It follows from (3) and (20)–(24) that (B + Ca20 )a0 + (B + 3Ca20 − (1 + m2 )A)a1 snξ

(25)

0.5

0.8 u2↼x֒t↽

φ(ξ) = a0 + a1 snξ.

(Ca21 + 2m2 A)a1 = 0.

snξ → tanh ξ,

j=0

where aj (j = 1, 2, . . . , n) are constants and the positive integer n can be determined by considering the homogeneous balance the highest order derivatives and highest order nonlinear appearing in ODE (3). So, we get n = 1. Equation (20) reduces to

(B + 3Ca20 − (1 + m2 )A)a1 = 0,

Solving the system (25) by Mathematica gives r 2A a0 = 0, a1 = ±m − . (26) C From (21) and (26), we obtain r r 2A B φ(ξ) = ±m − sn − (x − ct), (27) C 1 + m2 where AC < 0 and B < 0. By (3) and (27), we get the traveling wave solutions of Eq. (1) as follows: r r 2A B (x − ct) (28) |u1 (x, t)| = m − sn − , C 1 + m2 see Fig. 3 as follows: We notice that the Jacobi elliptic functions degenerate to the following functions (see Ref. [3], pp. 3064)

4.1 Jacobi Elliptic sn Function Expansion Method and Exact Traveling Wave Solutions of Eq. (1) To facilitate further on our analysis, we assume that Eq. (3) has n X φ(ξ) = aj snj (ξ), (20)

767

0.6 0.4 0.2 0 0 −1

0.2 0.4 t 0.6

0.8

1.0 1

0 x

Fig. 4 (Color online) The graphics of solution (29), taking A = −1, B = −1, C = 2, c = 1.


768

4.2 Jacobi Elliptic cn Function Expansion Method and Exact Traveling Wave Solutions of Eq. (1)

see Fig. 6 as follows:

We assume that Eq. (3) has

0.4

aj cnj (ξ),

(30)

j=0

where aj (j = 1, 2, . . . , n) are constants and the positive integer n can be determined by considering the homogeneous balance the highest order derivatives and highest order nonlinear appearing in ODE (3). So, we get n = 1. Equation (30) reduces to φ(ξ) = a0 + a1 cnξ.

+

3Ca0 a21 = 0,

0.8

(Ca21 − 2m2 A)a1 = 0.

0 x

0.4 0.2 1.0

0 −1

0 x

Fig. 6 (Color online) The graphics of solution (39), taking A = 1, B = −1, C = 2, c = 1.

(35)

as m → 1.

√ 2A | sech −B(x − ct)|. C

0.6

1

Solving the system (35) by Mathematica gives r 2A a0 = 0, a1 = ±m . (36) C From (31) and (36), we obtain r r 2A B φ(ξ) = ±m cn − (x − ct), (37) C 2m2 − 1 where AC > 0. By (3) and (37), we get the traveling wave solutions of Eq. (1) as follows: r r 2A B |u3 (x, t)| = m (x − ct) (38) cn − , C 2m2 − 1 see Fig. 5 as follows: We notice that the Jacobi elliptic functions degenerate to the following functions (see Ref. [3], pp. 3064)

|u4 (x, t)| =

0 −1

0.5 t

3

(B + 3Ca20 + (2m2 − 1)A)a1 = 0,

Taking m = 1, we have r

t

(34)

− 2m A)a1 cn ξ = 0.

cnξ → sech ξ,

1 0.5

1.0

That is, (B + Ca20 )a0 = 0,

0 1.0

(33)

(B + Ca20 )a0 + (B + 3Ca20 + (2m2 − 1)A)a1 cnξ +

0.1

(32)

where m (0 < m < 1) is the modulus of the Jacobi elliptic functions, see Ref. [3]. Substituting (30)–(34) into (3) yields 2

0.2

Fig. 5 (Color online) The graphics of solution (38), taking m = 1/2, A = 1, B = 1, C = 2, c = 1.

φ3 (ξ) = a30 + 3a20 a1 cnξ + 3a0 a21 cn2 ξ + a31 cn3 ξ,

(Ca21

0.3

(31)

We observe that dφ = −a1 snξdnξ, dξ d2φ = (2m2 − 1)a1 cnξ − 2m2 a1 cn3 ξ, dξ 2

3Ca0 a21 cn2 ξ

u3↼x֒t↽

n X

0.5

u4↼x֒t↽

φ(ξ) =

Vol. 57

(39)

4.3 Jacobi Elliptic dn Function Expansion Method and Exact Traveling Wave Solutions of Eq. (1) We assume that Eq. (3) has n X φ(ξ) = aj dnj (ξ).

(40)

j=0

Similarly, we get n = 1. Equation (40) reduces to φ(ξ) = a0 + a1 dnξ. We observe that dφ = −m2 a1 snξcnξ, dξ d2φ = (2 − m2 )a1 dnξ − 2a1 dn3 ξ, dξ 2 φ3 (ξ) = a30 + 3a20 a1 dnξ + 3a0 a21 dn2 ξ + a31 dn3 ξ,

(41)

(42) (43) (44)

where m (0 < m < 1) is the modulus of the Jacobi elliptic functions, see Ref. [3]. Substituting (40)–(44) into (3) yields (B + Ca20 )a0 + (B + 3Ca20 + (2 − m2 )A)a1 dnξ + 3Ca0 a21 dn2 ξ + (Ca21 − 2A)a1 dn3 ξ = 0.


No. 5

That is, (B + Ca20 )a0 3Ca0 a21 = 0,

= 0, (Ca21

(B +

3Ca20

2

+ (2 − m )A)a1 = 0,

− 2A)a1 = 0.

(45)

Solving the system (45) by Mathematica gives r 2A . (46) a0 = 0, a1 = ± C From (41) and (46), we obtain r r 2A B φ(ξ) = ± dn − (x − ct), (47) C 2 − m2 where AC > 0 and B < 0. By (3) and (47), we get the traveling wave solutions of Eq. (1) as follows: r r 2A B (x − ct) (48) |u5 (x, t)| = dn − , C 2m2 − 1 see Fig. 7 as follows: We notice that the Jacobi elliptic functions degenerate to the following functions (see Ref. [3], pp. 3064) dnξ → sech ξ,

as m → 1.

Taking m = 1, we have r

√ 2A |u6 (x, t)| = | sech −B(x − ct)|. C In fact, u6 (x, t) is u4 (x, t).

(49)

We observe that dφ = −a1 (1 + cs2 ξ)dnξ, dξ d2φ = (2 − m2 )a1 csξ + 2a1 cs3 ξ, dξ 2

(52) (53)

φ3 (ξ) = a30 + 3a20 a1 csξ + 3a0 a21 cs2 ξ + a31 cs3 ξ,

(54)

where m (0 < m < 1) is the modulus of the Jacobi elliptic functions, see Ref. [3]. Substituting (50)–(54) into (3) yields (B + Ca20 )a0 + (B + 3Ca20 + (2 − m2 )A)a1 csξ + 3Ca0 a21 cs2 ξ + (Ca21 + 2A)a1 cs3 ξ = 0.

That is, (B + Ca20 )a0 = 0, 3Ca0 a21 = 0,

(B + 3Ca20 + (2 − m2 )A)a1 = 0,

(Ca21 + 2A)a1 = 0.

(55)

Solving the system (55) by Mathematica gives r 2A a0 = 0, a1 = ± . C From (51) and (56), we obtain r r 2A B φ(ξ) = ± − cs − (x − ct), C 2 − m2 where AC < 0 and B < 0.

(56)

(57)

Τ1015 15 u7↼x֒t↽

0.50 u5↼x֒t↽

769

0.45 0.40

10 5 0 1.0 1

0.35 1.0

0.5 t

1 0.5 t

0 x

0 −1

Fig. 8 (Color online) The graphics of solution (58), taking m = 1/2, A = 1, B = −1, C = −2, c = 1.

Fig. 7 (Color online) The graphics of solution (48), taking m = 1/2, A = 1, B = 1, C = 2, c = 1.

4.4 Jacobi Elliptic cs Function Expansion Method and Exact Traveling Wave Solutions of Eq. (1) We assume that Eq. (3) has φ(ξ) =

n X

aj dnj (ξ).

(50)

j=0

Similarly, we get n = 1. Equation (50) reduces to φ(ξ) = a0 + a1 csξ,

csξ =

cnξ . snξ

0 −1

0 x

(51)

By (3) and (57), we get the traveling wave solutions of Eq. (1) as follows: r r 2A B |u7 (x, t)| = − cs − (x − ct) (58) , C 2 − m2 see Fig. 8 as follows: We notice that the Jacobi elliptic functions degenerate to the following functions (see Ref. [31], pp. 1992, Appendix C) cs ξ → csch ξ, Taking m = 1, we have r |u8 (x, t)| =

−

as m → 1.

√ 2A | csch −B(x − ct)|, C

(59)


770

see Fig. 9 as follows: Τ1016

u8↼x֒t↽

4 3 2 1 0 1.0 1 0.5 t

0 −1

0 x

Fig. 9 (Color online) The graphics of solution (59), taking m = 1/4, A = 1, B = −1, C = 2, c = 1.

5 Conclusion and Discussion In this paper, to find the traveling wave solutions of nonlinear partial differential equations (NPDEs), u(x, t) =

References [1] Z.Y. Zhang, Turk. J. Phys. 32 (2008) 235. [2] W.X. Ma and B. Fuchssteiner, In. J. Nonlinear Mechanics 31 (1996) 329. [3] Z.Y. Zhang, Z.H. Liu, X.J. Miao, and Y.Z. Chen, Appl. Math. Comput. 216 (2010) 3064. [4] Z.Y. Zhang, Y.X. Li, Z.H. Liu, and X.J. Miao, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 3097. [5] Z.Y. Zhang, Z.H. Liu, X.J. Miao, and Y.Z. Chen, Phys. Lett. A 375 (2011) 1275. [6] Z.Y. Zhang, X.Y. Gan, and D.M. Yu, Zeitschrift f¨ ur Naturforschung 66a (2011) 721. [7] A. Yıldırım and Z. Pınar, Comput. Math. Appl. 60 (2010) 1873. [8] F. Khani, M.T. Darvishi, A. Farmany, and L. Kavitha, ANZIAM J. 52 (2010) 110. [9] W.X. Ma, T.W. Huang, and Y. Zhang, Phys. Scr. 82 (2010) 065003. [10] W.X. Ma and J.H. Lee, Choas, Solitons and Fractals 42 (2009) 1356. [11] W.X. Ma and M. Chen, Appl. Math. Comput. 215 (2009) 2835. [12] W.X. Ma and Y. You, Trans. Amer. Math. Soc. 357 (2005) 1753. [13] X.J. Miao and Z.Y. Zhang, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 4259. [14] A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, CRC Press, Boca Raton, FL, USA (2007). [15] A. Biswas, Optical Fiber Technology 9 (2003) 224.

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u(ξ), ξ = k(x − ct), where k and c are constants. So, we obtain the following ordinary differential equation (ODE): Aφ′′ (ξ) + Bφ(ξ) + Cφ3 (ξ) = 0. Indeed, the above equation is the well known the Duffing equation. It is well known that the Duffing equation is the equation governing the oscillations of a mass attached to the end of a spring whose tension (or compression). We can see Ref. [32]. Then, we obtain new exact traveling solutions by using ISM and CFM. Under some parameter conditions, we find that the solutions by using ISM and CFM are equal. Finally, we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM).

Acknowledgement The authors would like to present our sincere thanks to the referee for their valuable and helpful comments and suggestions. I would like to express my gratitude to Dr. Ying-Hui Zhang, De-Ming Yu, Xiang-Yang Gan, and XinPing Li for their useful discussions concerning this paper.

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