A Generalized Sub-Equation Expansion Method and Some AnalyticalSolutions to the Inhomogeneous Higher-Order Nonlinear SchrodingerEquation

Biao Lia,c, Yong Chena,b, and Yu-Qi Lia,c

a Nonlinear Science Center, Ningbo University, Ningbo 315211, Chinab Institute of Theoretical Computing, East China Normal University, Shanghai 200062, Chinac Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080,

China

Reprint requests to Dr. B. L.; E-mail: biaolee2000@yahoo.com.cn

Z. Naturforsch. 63a, 763 – 777 (2008); received May 5, 2008

On the basis of symbolic computation a generalized sub-equation expansion method is presentedfor constructing some exact analytical solutions of nonlinear partial differential equations. To illus-trate the validity of the method, we investigate the exact analytical solutions of the inhomogeneoushigh-order nonlinear Schrodinger equation (IHNLSE) including not only the group velocity disper-sion, self-phase-modulation, but also various high-order effects, such as the third-order dispersion,self-steepening and self-frequency shift. As a result, a broad class of exact analytical solutions of theIHNLSE are obtained. From our results, many previous solutions of some nonlinear Schrodinger-typeequations can be recovered by means of suitable selections of the arbitrary functions and arbitraryconstants. With the aid of computer simulation, the abundant structure of bright and dark solitarywave solutions, combined-type solitary wave solutions, dispersion-managed solitary wave solutions,Jacobi elliptic function solutions and Weierstrass elliptic function solutions are shown by some fig-ures.

Key words: Inhomogeneous High-Order NLS Equation; Elliptic Function Solutions; Solitary WaveSolutions.

PACS numbers: 05.45.Yv, 42.65.Tg, 42.81.Dp

1. Introduction

The search for a new mathematical algorithm to dis-cover exact solutions of nonlinear partial differentialequations (NLPDEs) is an important and essential taskin nonlinear science. Many methods have been devel-oped by mathematicians and physicists to find spe-cial solutions of NLPDEs, such as the inverse scatter-ing transformation [1] and Darboux transformation [2],the formal variable separation approach [3], the Ja-cobi elliptic function method [4], the tanh method [5 –8], various extended tanh methods [9 – 15], andthe generalized projective Riccati equations method[16, 17].

Recently, Fan [18] developed a new algebraicmethod which further exceeds the applicability of thetanh method in obtaining a series of exact solutions ofnonlinear equations. More recently, Yan [19], Chen etal. [20] and Yomba [21] have further developed thismethod and obtained some new and more general so-lutions for some NLPDEs.

0932–0784 / 08 / 1200–0763 $ 06.00 c© 2008 Verlag der Zeitschrift fur Naturforschung, Tubingen · http://znaturforsch.com

In this paper, firstly, on the basis of the work donein [5 – 21], we will propose a generalized sub-equationmethod by a more general ansatz, so that it can beused to obtain more types and general formal solu-tions, which include a series of nontravelling wave andcoefficient function solutions, such as bright-like soli-tary wave solutions, dark-like solitary wave solutions,W-shaped solitary wave solutions, combined brightand dark solitary wave solutions, single and combinednondegenerate Jacobi elliptic function solutions andWeierstrass elliptic doubly periodic solutions for someNLPDEs. Secondly, to illustrate the validity of the pro-posed method, we will investigate some exact analyti-cal solutions of the following inhomogeneous higher-order nonlinear Schrodinger equation (IHNLSE) [22],which governs the envelope wave equation for fem-tosecond optical pulse propagation in inhomogeneousfibers:

qz = i(α1(z)qtt + α2(z)|q|2q)+ α3(z)qttt

+ α4(z)(|q|2q)t + α5(z)q(|q|2)t +Γ (z)q,(1)

764 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

where q(z, t) represents the complex envelope of theelectrical field, z is the normalized propagation dis-tance, t is the normalized retarded time, and α1(z),α2(z), α3(z), α4(z), and α5(z) are the distributed pa-rameters, which are functions of the propagation dis-tance z, related to the group velocity dispersion (GVD),self-phase-modulation (SPM), third-order dispersion(TOD), self-steepening, and the delayed nonlinear re-sponse effect, respectively. Γ (z) denotes the ampli-fication or absorption coefficient. The study of theIHNLSE (1) is of great interest due to its wide rangeof applications. Its use is not only restricted to opti-cal pulse propagation in inhomogeneous fiber media,but also to the core of dispersion-managed solitons andcombined managed solitons. In [23], exact multisoli-ton solutions of (1) are presented by employing theDarboux transformation based on the Lax pair. An ex-act dark soliton solution of the IHNLSE (1) has beenobtained in [24]. When Γ (z) = 0 in the IHNLSE (1),two exact analytical solutions, that describe the modu-lation instability and the soliton propagation on a con-tinuous wave background, are obtained by using theDarboux transformation [25]. Using a direct assump-tion method, Yang et al. [26] presented three exactcombined solitary wave solutions for the IHNLSE (1),analyzed the features of the solutions and numeri-cally discussed the stabilities of these solutions un-der slight violations of the parameter conditions andfinite initial perturbations for some exact analytical so-lutions. Besides the results mentioned above, in re-cent years, many authors have analyzed the nonlin-ear Schrodinger-type equation from different points ofview and have obtained some interesting and signifi-cant results, such as in [27 – 38].

The present paper is organized as follows: In Sec-tion 2, the generalized sub-equation expansion methodis presented. In Section 3, the proposed method is ap-plied to investigate the IHNLSE (1) and a series ofexact analytical solutions of the IHNLSE (1) are ob-tained. Furthermore some figures are given with com-puter simulations to show the properties of the kinkprofile solitary wave solutions, bright-like and dark-like solitary wave solutions, Jacobi elliptic function so-lutions and Weierstrass elliptic function solutions. Fi-nally, some conclusions are given briefly.

2. The Generalized Sub-Equation ExpansionMethod

Now we establish the generalized sub-equation ex-pansion method as follows:

Given a NLPDE with, say, two variables {z, t}:

E(u,ut ,uz,uzt ,utt ,uzz, · · · ) = 0. (2)

Step 1. We assume that the solutions of (2) are asfollows:

u(z, t) =A0 + ∑M

i=1 φ i−1(ξ ) [Aiφ(ξ )+ Biφ ′(ξ )]a0 + ∑N

j=1 φ j−1(ξ ) [a jφ(ξ )+ b jφ ′(ξ )], (3)

where A0 = A0(z, t), Ai = Ai(z, t), Bi = Bi(z, t) (i =1, · · · ,M), a0 = a0(z, t), a j = a j(z, t), b j = b j(z, t) ( j =1, · · · ,N), ξ = ξ (z, t) are all differentiable functionsof {z, t}, and the new variable φ = φ(ξ ) satisfies

φ ′2(ξ ) =(

dφ(ξ )dξ

)2

= h0 + h1φ(ξ )+ h2φ2(ξ )

+ h3φ3(ξ )+ h4φ4(ξ ),(4)

where hi (i = 0,1,2,3,4) are constants.The parameters M and N can be determined by bal-

ancing the highest-order derivative term and the non-linear terms in (2). M,N are usually positive integers; ifnot, some proper transformation u(z, t)→ um(z, t) maybe used in order to satisfy this requirement.

Step 2. Substituting (3) along with (4) into (2), ex-tracting the numerator of the obtained system, separat-ing the system into a real and imaginary part, then set-ting the coefficients of φ i(ξ )(φ ′(ξ )) j (i = 0,1, · · · ; j =0,1) to zero, we obtain a set of over-determined PDEs[or ordinary differential equations (ODEs)] with regardto the differential functions A0, Ai, Bi (i = 1,2, · · · ,M),a0, a j, b j ( j = 1,2, · · · ,N), ξ and the varying differen-tial function coefficients included in (2).

Step 3. Solving the overdetermined PDEs (orODEs) by a symbolic computation system (Maple), wewould end up with the explicit expressions for A0, Ai,Bi (i = 1, · · · ,M), a0, a j, b j ( j = 1, · · · ,N), ξ and thefunctions coefficients or the constraints among them.

Step 4. By using the results obtained in the abovesteps and the various solutions of (4), a rich set of so-lutions for (2) can be expected.

By considering the different values of h0, h1, h2, h3and h4, we can derive that (4) admits a series of fun-damental solutions. For simplicity, we only list somehyperbolic function solutions, Jacobi elliptic functionsolutions, and Weierstrass elliptic doubly periodic so-lutions as follows.

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 765

Case 1. If h0 = h1 = 0, two solutions of (4) arederived:

φ1I (ξ ) = −4h2C0sech(

√h2ξ )

{2C0h3sech(

√h2ξ )

− (∆1 + 1) tanh(√

h2ξ )+ (∆1 −1)}−1

,(5)

φ2I (ξ )=−4h2C0sech(

√h2ξ )

{2C0h3sech(

√h2ξ )

+ (∆2+C20) tanh(

√h2ξ )+(∆2−C2

0)}−1

,(6)

where h2 > 0, C0 = exp(√

h2C1) is an arbitrary con-stant, and ∆1 = C0

2(4h2h4−h32), ∆2 = (4h2h4−h3

2).If further choosing the parameters h2, h3, h4, ∆1, ∆2,

and C0 in (5) and (6), we can derive the sech-type so-lutions

φ3I (ξ ) = −h2

h3sech2

(√h2

2ξ)

,

h0 = h1 = h4 = 0, ∆1 = −1 (or ∆2 = −C20),

(7)

φ4I (ξ ) =

√−h2

h4sech

(√h2ξ)

, h0 = h1 = h3= 0,

h4 < 0, ∆1 = −1(or ∆2 = −C20). (8)

Case 2. If h1 = h3 = 0, the Jacobi elliptic functionsolutions for (4) can be derived:

φ1II(ξ ) =

√−h2m2

h4(2m2 −1)cn

(√h2

2m2 −1ξ ,m

),

h4 < 0, h2 > 0, h0 =h2

2m2(1−m2)h4(2m2 −1)2 ,

(9)

φ2II(ξ ) =

√−h2

h4(2−m2)dn

(√h2

2−m2 ξ ,m

),

h4 < 0, h2 > 0, h0 =h2

2(1−m2)h4(2−m2)2 ,

(10)

φ3II(ξ ) =

√−h2m2

h4(m2 + 1)sn

(√− h2

m2 + 1ξ ,m

),

h4 > 0, h2 < 0, h0 =h2

2m2

h4(m2 + 1)2 ,

(11)

where 0 ≤ m ≤ 1 is a modulus.If m → 1 and m → 0, the Jacobi functions degener-

ate to the hyperbolic functions and the trigonometricfunctions, respectively, i. e.

m → 1,

snξ → tanhξ , cnξ → sechξ , dnξ → sechξ ;(12)

m→ 0, snξ → sinξ , cnξ → cosξ , dnξ → 1. (13)

Therefore, when m → 1, the solutions (9) and (10) de-generate to the solution (8), and the solution (11) de-generates to the following solitary wave solution:

φ4II(ξ ) =

√− h2

2h4tanh

(√−h2

2ξ

),

h0 =h2

2

4h4, h1 = h3 = 0, h2 < 0, h4 > 0.

(14)

Case 3. If h2 = h4 = 0, (4) has the following Weier-strass elliptic doubly periodic type solution:

φIII(ξ ) =℘(√

h3

2ξ ,g2,g3

),

where h3 > 0, g2 = −4h1

h3, g3 = −4

h0

h3.

(15)

Remark 1. 1) The ansatz (3) presented in this paperis more general than the ansatz by some authors [18 –21]. (i) When a0 = 1, ai = bi = 0 (i = 1,2, · · · ,N),Ai,Bi (i = 0,1,2, · · · ,M) are all constants and ξ =ξ (z, t) is only a linear function of {z, t}, our ansatz inthe paper is changed into the ansatz in [18 – 20]. By thisansatz, we only obtain the travelling wave solutions ofgiven PDEs. (ii) When Bi = 0 (i = 1,2, · · · ,M), a0 = 1,and a j = b j = 0 ( j = 1,2, · · · ,N), Ai (i = 0, · · · ,M) andξ are undetermined functions of {z, t}, the ansatz ischanged into the ansatz in [21].

2) Theoretically speaking, one could introduceterms up to 6th, to 8th etc. order or even consider termsof infinite order which might be summed up to giverise to a functionally completely new ansatz. Due tocomplexity of the computation, the PDEs (or ODEs)systems derived may not be solved in most cases.

3) In order that the PDEs (or ODEs) derived instep 2 can be solved by symbolic computation, we usu-ally choose some special forms of Ai, Bi, a j, b j and ξ .

3. Exact Analytical Solutions of the Inhomo-geneous Nonlinear Schrodinger Equation

We now investigate the IHNLSE (1) with the gen-eralized sub-equation expansion method of Section 2.According to the method, we assume that the solutionsof the IHNLSE (1) are of the special forms

q(z, t) =A0(z)+ A1(z)φ(ξ )+ B1(z)φ ′(ξ )

1 + a1(z)φ(ξ )+ b1(z)φ ′(ξ )· exp

{i[t2λ2(z)+ tλ1(z)+ λ0(z)]

},

(16)

766 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

0

5

10

15

20

z–10

–5

0

5

10

t

0

1

2

3

4

| |11

2q

b)

0

5

10

15

20

z

–10

–50

510

t

0

50

100

150

112| |q

c)

–4

–2

0

2

4

0

5

10

15

20

z

0

1

2

3

4

5

6

ω

|F |

1

d)

–4

–2

0

2

4

0

5

10

15

20

z

0

5

10

15

20

25

30

ω

|F |

1

Fig. 1. Evolution plots of the solution (20) with the parameters: α1(z) = α3(z) = sin(z), h2 = 2, h3 = −1, c0 = 1, c1 = 1,c2 = 0.1, c4 = 0, µ = 0.1. (a) Γ (z) = 0.01; (b) Γ (z) = −0.4sin(2z)/(1 − 0.4sin2 z); (c, d) spatial Fourier spectra of thesolution (20) corresponding to Fig. 1a and Fig. 1b, respectively.

where

ξ = ξ (z, t) = p(z)t + η(z), (17)

and A0(z), A1(z), B1(z), a1(z), b1(z), λ0(z), λ1(z),λ2(z), p(z), and η(z) are real functions of z to be de-termined, and φ(ξ ) satisfies (4).

Substituting (16) and (17) along with (4) into theIHNLSE (1), removing the exponential term, collect-ing the coefficients of the polynomial of φ(ξ ), φ ′(ξ )and t of the resulting system’s numerator, then sep-

arating each coefficient into the real part and imagi-nary part and setting each part to zero, we obtain anODE system with respect to the differentiable func-tions α1(z), α2(z), α3(z), α4(z), α5(z), A0(z), A1(z),B1(z), a1(z), b1(z), λ0(z), λ1(z), λ2(z), p(z), and η(z).For simplicity, we omit the ODE system in this paper.

Solving the ODE system with the symbolic compu-tation system Maple, we obtain a rich set of solutionsof the system. For simplicity, we omit the too complexand the trivial solutions. Then together with the solu-tions of the ODE system and (5) – (17), the following

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 767

10 families of solutions to the IHNLSE (1) can be de-rived.

Family 1.

q1(z, t) =a1(z)φI(ξ )1 + µφI(ξ )

exp(i(c2t + λ0(z))), (18)

where

a1(z) = c0 exp[∫

Γ (z)dz], h3 = 2µh2, h0 = h1 = 0,

ξ = c1(t +∫

(c12h2−3c2

2)α3(z)−2c2α1(z)dz)+c4,

α5(z) = −32

(−2µ2h2 + 2h4)c12α3(z)

(a1(z))2 − 32

α4(z),

α2(z) = − 1(a1(z))2

[(−2µ2h2 + 2h4)c1

2α1(z)

+ (−6c2µ2h2 + 6c2h4)c12α3(z)

]−α4(z)c2,

λ0(z) =∫

(c12h2 − c2

2)α1(z)

+ (3c12c2h2 − c2

3)α3(z)dz+ c3. (19)

Here µ , h2, h4, c0, c1, c2, c3, c4, c5 are arbitrary con-stants, and α1(z), α3(z), α4(z), Γ (z) are arbitrary func-tions of z.

If furthermore h4 = 0, from φ3I and (18) the follow-

ing solution of the IHNLSE (1) can be derived:

q11(z, t) =a1(z)h2

−h3 cosh(√

h22 ξ

)2+ µh2

· exp(i(c2t + λ0(z))).

(20)

In this situation, the solution (20) describes abright-like solitary wave with the amplitude and thewidth determined by a1(z) = c0 exp[

∫Γ (z)dz] and√

h2c1/2, respectively. We note that the time shift[χ(z) = −

∫(c1

2h2 −3c22)α3(z)−2c2α1(z)dz] and the

group velocity [V (z) = dχ/dz = 2c2α1(z)− (c12h2 −

3c22)α3(z)] of the solitary wave are dependent on z,

which leads to a change of the center position of thesolitary wave along the propagation direction of thefiber, and means that we may design a fiber system tocontrol the time shift and the velocity of the solitarywave.

In order to understand the evolution of these so-lutions expressed by (20), let us consider a solitonmanagement system, where the system parameters aretrigonometric and hyperbolic functions. For simplicity,

we only consider some examples for each solution ob-tained in Family 1 – 10 under some special parameters.

Figure 1a presents the evolution plots of the solu-tion (20) with Γ (z) = 0.01 > 0, which correspondsto a dispersion increasing fiber. From it we can seethat the intensity of the solitary wave increases whenΓ (z) > 0, and the time shift and the group velocity ofthe solitary wave are changing while the solitary wavekeeps its shape in propagating along the fiber. Fig-ure 1b presents the evolution plots of the solution (20)with Γ (z) = −0.4sin(2z)/(1− 0.4sin2 z), which cor-responds to a fiber, whose dispersion is periodicallychanging. From it we can see that the intensity of thesolitary wave changes periodically, and the time shiftand the group velocity of the solitary wave are alsochanging while the solitary wave keeps its shape inpropagating along the fiber. Especially when Γ (z)→ 0,we can derive that the amplitude of q11(z, t) keeps in-variable, unlike in Fig. 1, the amplitude of q11(z, t) ei-ther increases exponentially or changes periodically.Figures 1c and 1d are the spatial Fourier spectra ofthe solution q11(z, t) corresponding to Figs. 1a and 1b,respectively. From the Fourier spectra of the solutionq11(z, t), we can derive that the solution q11(z, t) isdominant for low frequencies.

Family 2.

q2(z, t) = c0 exp[∫

Γ (z)dz](c1 + c2φI + c7φ ′

I(ξ ))

· exp(i(c4t + c5)),(21)

where

ξ = c3t + c6, α1(z) = α3(z) = 0,

α2(z) = −c4α4(z), α5(z) = −32

α4(z),(22)

and h0 = h1 = 0, c7 = 0 (or c7 = 1), and h2, h3, h4, c0,c1, c2, c3, c4, c5, c6 are arbitrary constants, and α4(z),Γ (z) are arbitrary functions of z.

From φ3I and (21), one obtains a solution of the

IHNLSE (1) as follows:

q21(z, t) = c0 exp[∫

Γ (z)dz]

·[

c1 −h2

h3

(c2 −

√h2 tanh

(√h2

2ξ))

sech2(√

h2

2ξ)]

· exp(i(c4t + c5)), (23)

where the corresponding parameters are determinedby (22) and c7 = 1.

768 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

0

5

10

15

20

z

–10

–5

0

5

10

t

0

100

200

300

400

500

212| |q

b)

0

5

10

15

20

z

–10

–5

0

5

10

t

0

0.2

0.4

0.6

0.8

1

212| |q

c)

0

5

10

15

20

z

–10

–5

0

5

10

t

0

50

100

150

200

212| |q

d)

0

5

10

15

20

z

–10

–5

0

5

10

t

0

1

2

3

4

212| |q

Fig. 2. Evolution plots of the solution (23) with the parameters: Γ (z) = −0.01, h2 = 1, c0 = 1, c3 = 2, c6 = 1. (a) h3 = 0.4,c1 = 1, c2 = 10; (b) h3 = 10, c1 = 1, c2 = 10; (c) h3 = 0.4, c1 = 10, c2 = 10; (d) h3 = 0.4, c1 = 1, c2 = 0.01.

Because the solution q21(z,t) includes both func-tion tanh(ξ ) and function sech(ξ ), the solution maypresent different shapes. If tanh(ξ ) [sech(ξ )] is domi-nant, the solution describes the dark solitary wave (thebright solitary wave). If both effects are nearly equal,the solution presents a combined bright and dark soli-tary wave. As shown in Fig. 2, under different parame-ters {h2,h3,c1,c2}, the shapes of the intensity (23) takea bright-like solitary wave, dark-like solitary wave,W-shaped solitary wave and a combined bright anddark solitary wave, respectively. It is worth noting thatthe solitary velocity does not change and there is onlya shape-presenting time shift in the propagation due tothe constant c6. In particular, as shown in Fig. 2d, abright solitary wave and a dark solitary wave are foundto combine under some conditions and propagate si-multaneously in an inhomogeneous fiber.

Family 3.

q3(z, t) = a0(z)(

1∓√

2h4

h2φI

)· exp(i(c3t + λ0(z))),

(24)

where

a0(z) = c0 exp[∫

Γ (z)dz],

h0 = h1 = 0, h3 = ∓2h2h4√h2h4

,

ξ = c1t− 12

c1

∫4α1(z)c3 +(c1

2h2+6c32)α3(z)dz+c7,

α2(z) = − 12(a0(z))2

[2α4(z)(a0(z))2c3 + α1(z)c1

2h2

+3α3(z)c12c3h2(a0(z))2],

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 769

a)

0

10

20

30

40

z

–40

–20

0

20

40

t

0

1

2

3

4

312| |q

b)

0

10

20

30

40

z

–40

–20

0

20

40

t

0

1

2

3

4

312| |q

Fig. 3. Evolution plots of the solution (26) with the parameters: α1(z) = α3(z) = cos(z), h2 = h4 =C0 = 1, c0 =−2, c1 = 0.4,c3 = 0.2, c7 = 0. (a) Γ (z) = 0.001; (b) Γ (z) = 0.01sin(z).

α5(z) = −34

c12α3(z)h2 + 2(a0(z))2α4(z)

(a0(z))2 ,

λ0(z) =12

∫(−2c3

2 − c12h2)α1(z)

+ (−2c33 −3c1

2c3h2)α3(z)dz+ c5, (25)

and c0, c1, c3, c5, c7, h2, h4 are arbitrary constants, andα1(z), α4(z), α3(z), Γ (z) are arbitrary functions of z.

From h3 = ∓2h2h4/√

h2h4 we obtain ∆1 = ∆2 = 0in (5) and (6). Thus from (5) and (24) one can obtain asolution for the IHNLSE (1) as follows:

q31(z, t) = a0(z){

1∓√

2h4/h2[−4h2C0sech(

√h2ξ )

]·[2C0h3sech(

√h2ξ )− tanh(

√h2ξ )−1]−1}

· exp(i(c3t + λ0(z))), (26)

where the corresponding parameters are determinedby (25).

Due to the existence of both function tanh(ξ ) andfunction sech(ξ ), the solution q3(z,t) may present dif-ferent shapes. From Fig. 3 one can see that the intensityof (26) presents mainly kink profile while the solitarywave propagates along the fiber.

Family 4.

q4(z, t) =a1(z)(c1 + φI(ξ ))

1 + µφI(ξ )· exp(i(λ2(z)t2 + c3λ2(z)t + λ0(z))),

(27)

where

ξ = c2λ2(z)(

t +12

c3

)+ c5, h0 = h1 = 0,

α1(z) = −14

ddz λ2(z)(λ2(z))2 , α2(z) =

12

( ddz λ2(z)

)c2

2h4

(a1(z))2(1 + µc1)2 ,

h3 = 4c1h4

1 + µc1, h2 = 4

c12h4

(1 + µc1)2 ,

Γ (z) =12

2λ2(z) ddz a1(z)−a1(z) d

dz λ2(z)a1(z)λ2(z)

,

λ0(z) =14

(c2

3 +2c2

1c22h4

(1 + µc1)2

)λ2(z)+ c4, (28)

and α3(z) = α4(z) = α5(z) = 0, c1, c2, c3, c4, c5, h4,µ are arbitrary constants, and λ2(z), a1(z) are arbitraryfunctions of z.

From (28), we can derive that h23 −4h2h4 = 0. Thus

from (8), (27), and (28) a solution of the IHNLSE (1)can be derived:

q41(z, t) =

√−2h4α1(z)

α2(z)c2λ2(z)1 + µc1

[4h2 −2c1h3

−c1C0(sinh(√

h2ξ )+ cosh(√

h2ξ ))][

4µh2 −2h3

−C0(sinh(√

h2ξ )+ cosh(√

h2ξ ))]−1

· exp(i(λ2(z)t2 + c3λ2(z)t + λ0(z))), (29)

770 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

0

5

10

15

20

z–40

–20

0

20

40

t

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

412| |q

b)

0

1

2

3

4

5

z

–40–20

0

20

40

t

0

0.02

0.04

0.06

0.08

412| |q

Fig. 4. Evolution plots of the solution (29) with the parameters: (a) α1(z) = sin(z), α2(z) = −sin(z), h2 = 1, h3 = 2, c0 = 6,C0 = 6, c1 = 0.1, c2 = 1, c3 = 1, c5 = 0, µ = −2; (b) α1(z) = 1−0.8sin(4z)2, α2 = −1, h2 = 1, h3 = 2, c0 = 5, C0 = 10,c1 = c2 = c3 = 1, c5 = 0, µ = −2.

where α1(z) and α2(z) are arbitrary functions of z, andthe corresponding parameters are constrained by

λ2(z) =1∫

4α1(z)dz+ c0,

Γ (z) =α ′

1(z)α2(z)−α1(z)α ′2(z)

2α1(z)α2(z)−2α1(z)λ2(z),

ξ = c2λ2(z)(

t +12

c3

)+ c5,

h3 = 4c1h4

1 + µc1, h2 = 4

c12h4

(1 + µc1)2 ,

λ0(z) =14

(c2

3 +2c2

1c22h4

(1 + µc1)2

)λ2(z)+ c4.

(30)

Figure 4 presents the evolution plots of the solu-tion (29) under some special parameters. From Fig. 4aone can see that, if the main control functions areα1(z) = sin(z), α2(z) = −sin(z), the intensity of thesolitary wave presents the property of the kink profilesolitary wave, and the time shift is changing while thesolitary wave keeps its shape in propagating along thefiber. As shown in Fig. 4b, when the main control func-tions are α1(z) = 1−0.8sin(4z)2, α2 = −1, the inten-sity of the solution (29) deceases periodically while thekink profile solitary wave propagates along the fiber.

Family 5.

q51(z, t) =

√−2α1(z)h4

α2(z)c1λ2(z)

√−h2m2

h4(2m2 −1)

· cn

(√h2

2m2 −1ξ ,m

)

· exp(i(λ2(z)t2 + c2λ2(z)t + λ0(z))),

(31)

q52(z, t) =

√−2α1(z)h4

α2(z)c1λ2(z)

√−h2

h4(2−m2)

·dn

(√h2

2−m2 ξ ,m

)

· exp(i(λ2(z)t2 + c2λ2(z)t + λ0(z))),

(32)

q53(z, t) =

√−2α1(z)h4

α2(z)c1λ2(z)

√−h2m2

h4(m2 + 1)

· sn

(√− h2

m2 + 1ξ ,m

)

· exp(i(λ2(z)t2 + c2λ2(z)t + λ0(z))),

(33)

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 771

a)

0

2

4

6

8

10

12

14

16

z

–8–6

–4–2

02

46

8

t

0

1

2

3

512| |q

b)

0

2

4

6

8

10

12

14

16

z

–8–6

–4–2

02

46

8

t

0

2

4

512| |q

c)

0

5

10

15

20

z

–40

–20

0

20

40

t

0

0.02

0.04

0.06

0.08

512| |q

d)

0

5

10

15

20

z

–40

–20

0

20

40

t

0

0.02

0.04

0.06

512| |q

Fig. 5. Evolution plots of the solution (31) with the parameters: α1(z) = 1− 0.8sin(z)2, c0 = 100, c1 = 40, c2 = c4 = 0.(a, b) α2(z) = 1, h2 = 10, h4 = −1, m = 1; (c, d) α2(z) = −1, h2 = −0.5, h4 = 1, m = 0.9.

where

ξ = c1λ2(z)(

t +12

c2

)+ c4, h1 = h3 = 0,

λ0(z) =14(c2

2 − c12h2)λ2(z)+ c3,

λ2(z) =1∫

4α1(z)dz+ c0,

Γ (z) =α ′

1(z)α2(z)−α1(z)α ′2(z)

2α1(z)α2(z)−2α1(z)λ2(z),

(34)

and h0, h2, h4 are constrained by the conditions inφ1

II(ξ ), φ2II(ξ ) and φ3

II(ξ ), respectively, c0, c1, c2, c3,c4 are arbitrary constants, α3(z) = α4(z) = α5(z) = 0,α1(z), and α2(z) are arbitrary functions of z.

From (12), we can derive that the Jacobi ellipticfunction solutions q51(z, t), q52(z, t) and q53(z, t) re-duce to hyperbolic function solution. At the same time,when m = 1, we can conclude that the width, the groupvelocity and the height of the solitary wave solutionq51(z, t) are determined by λ2(z) = 1∫

4α1(z)dz+c0, c2 and

772 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

0

5

10

15

20

z–10

–5

0

5

10

t

0

0.1

0.2

0.3

0.4

0.5

0.6

62| |q

b)

0

5

10

15

20

z

–6–4

–20

24

6

t

00.20.40.6

62| |q

Fig. 6. Evolution plots of the solution (35) of the form φ1II(ξ ) with the parameters: Γ (z) = 0.01, α1(z) = sin(z), α3(z) = sin(z),

h2 = 0.4, h4 = −1, c0 = 1, c1 = 2, c2 = 0.1, c3 = c4 = 0. (a) m = 1; (b) m = 0.9.

√−α1(z)α2(z)

λ2(z), respectively. In essentially, the widthand the height are controlled by α1(z) and α2(z), thevelocity is determined by c2. Similarly, we can ob-tain the control functions and control parameters ofthe solitary wave solution q52(z,t) and q53(z, t) un-der m = 1.

The time-space evolution of the dispersion Ja-cobi elliptic periodic wave solution (31) is shownin Figure 5. From Figs. 5a and 5c, the intensitiesof the dispersion-managed bright solitary wave anddispersion-managed dark solitary wave are decreas-ing periodically while the widths of them are increas-ing in propagating along the fiber. Figures 5b and 5dpresent the periodic properties of the Jacobi ellipticfunctions cn(ξ ,m) and sn(ξ ,m).

Family 6.

q6(z, t) = a1(z)φ iII exp(i(c2t +λ0(z))), i = 1,2,3, (35)

where

a1(z) = c0 exp[∫

Γ (z)dz],

ξ = c1

(t+∫−2c2α1(z)+(c2

1h2−3c22)α3(z)dz

)+c4,

λ0(z) =∫

(−c22 + c1

2h2)α1(z)

+ (−c23 + 3c1

2c2h2)α3(z)dz+ c3,

α2(z) = − 1(a1(z))2

[α4(z)(a1(z))2c2 + 2α1(z)c1

2h4

+ 6c12h4α3(z)c2

],

α5(z) = −32

2α3(z)c12h4 + α4(z)(a1(z))2

(a1(z))2 ,

h1 = h3 = 0, (36)

and h0, h2, h4 are constrained by the conditions inφ1

II(ξ ), φ2II(ξ ) and φ3

II(ξ ), respectively, c0, c1, c2, c3, c4are arbitrary constants, and α1(z), α3(z), α4(z), Γ (z)are arbitrary functions of z.

As shown in Fig. 6, when taking φ1II(ξ ) in the solu-

tion (35), the intensity of the bright-like solitary wave(m = 1) and the Jacobi elliptic periodic wave (0 < m <1) are increasing exponentially due to Γ (z) = 0.01 > 0.If α1(z) and α3(z) take trigonometric periodic func-tions, the time shift and the group velocity of the soli-tary wave and Jacobi elliptic wave are changing whilethe waves keeps their shape in propagating along thefiber.

Family 7.

q7(z, t) = c0 exp[∫

Γ (z)dz]

·(c3 + c2φ i

II(ξ )+ c1(φ iII(ξ ))′

)· exp(i(c4t + c5)), i = 1,2,3,

(37)

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 773

a)

0

2

4

6

8

10

z

–10

–5

0

5

10

t

1

1.5

2

2.5

3

3.5

4

72| |q

b)

0

2

4

6

8

10

z–10

–5

0

5

10

t

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

72| |q

c)

0

2

4

6

8

10

z

–10

–5

0

5

10

t

0

1

2

3

4

5

72| |q

d)

0

2

4

6

8

10

z

–10

–5

0

5

10

t

0

10

20

30

40

72| |q

Fig. 7. Evolution plots of the solution (37) with the parameters: Γ (z) = −0.01, h2 = 1, h4 = −1, c0 = c3 = c4 = c5 = m = 1,c6 = 0. (a) c1 = −0.2, c2 = 1; (b) c1 = 0.1, c2 = −0.4; (c) c1 = −2, c2 = 0.4; (d) c1 = 10, c2 = 0.4.

where

ξ = c3t + c6, α1(z) = α3(z) = 0,

α2(z) = −c4α4(z), α5(z) = −32

α4(z),(38)

and h1 = h3 = 0, h0, h2, h4 are constrained by theconditions in φ1

II(ξ ), φ2II(ξ ) and φ3

II(ξ ), respectively,c3 = 0 (or c3 = 1), c0, c1, c2, c4, c5, c6 are arbi-trary constants, and α4(z), Γ (z) are arbitrary functionsof z.

As shown in Fig. 7, when taking φ1II(ξ ) in (35), the

shapes of the intensity (35) are determined by c1 and c2under h2 = 1, h4 = −1, c3 = m = 1. A bright-like soli-

tary wave and dark-like solitary wave are shown inFigs. 7a and 7b. In particular, as shown in Figs. 7cand 7d, a bright solitary wave, a dark solitary wave, andtwo bright solitary waves are found to be combined un-der some conditions; they propagate simultaneously inan inhomogeneous fiber. It is worth noting that the soli-tary velocity does not change and there is only a con-stant time shift in propagation due to constant c6. Atthe same time, it is necessary to point out that we onlyconsider some special cases under m = 1. If 0 < m < 1,the evolution of the solutions is similar to Figs. 5and 6. For simplicity, we do not simulate them in thispaper.

774 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

05

1015

20

z–5

0

5

10

t

40

60

80

100

120

82| |q

b)

0

5

10

15

20

z

–6–4

–20

24

6

t

0

0.5

1

82| |q

Fig. 8. Evolution plots of the solution (39) of the form φ1II(ξ ) with the parameters: Γ (z) = 0.008, α3(z) = cos(z), h2 = 4,

h0 = 2, c0 = c2 = c3 = m = 1, c4 = c5 = 0, µ = 3. (a) c1 = 10, h4 = −0.1; (b) c1 = −0.4, h4 = −1.

Family 8.

q8(z, t) =a1(z)(c1 + φ i

II(ξ ))1 + µφ i

II(ξ )exp(i(c3t + λ0(z))),

i = 1,2,3,

(39)

where

ξ =c2t− 1µ(c12µ2−1)

[(2c2

3µ4h0c1 −2c23c1h4

− 3c2c32µ3c1

2 −4c23c1

2h4µ

+ 4c23µ3h0 + 3c2c3

2µ)∫

α3(z)dz]+ c4,

α1(z) = −3α3(z)c3, α2(z) = −α4(z)c3,

λ0(z) = 2c33∫

α3(z)dz+ c5,

α5(z) = −32

α4(z)+ 3c2

2α3(z)(µ4h0 −h4)(a1(z))2 ,

h2 = −2µ3h0 + c1h4

(c1µ + 1)µ, a1(z) = c0exp

[∫Γ (z)dz

],

h1 = h3 = 0, (40)

and h0, h2, h4 are also constrained by the conditions inφ1

II(ξ ), φ1II(ξ ) and φ1

II(ξ ), respectively, µ �= 0, c0, c1,c2, c3, c4, c5 are arbitrary constants, and α4(z), Γ (z)are arbitrary functions of z.

As shown in Fig. 8, the intensity of a dark-like soli-tary wave and a W-shaped solitary wave is increasing

exponentially due to Γ (z) = 0.01 > 0. At the sametime, due to the periodicity of the trigonometric peri-odic functions α3(z), the time shift and the group ve-locity of the solitary wave and W-shaped solitary waveare changing while the waves keep their shape in prop-agating along the fiber.

Family 9.

q9(z, t) =

√−10h0α1(z)

α2(z)µ2c1λ2(z)

℘(√

h32 ξ , 2

µ2 , 12µ3

)

1+µ℘(√

h32 ξ , 2

µ2 , 12µ3

)· exp(i(λ2(z)t2 + c2λ2(z)t + λ0(z))), (41)

where

ξ = c1λ2(z)t +12

c1c2λ2(z)+ c3, h3 = −8µ3h0,

h1 = 4µh0, h2 = h4 = 0, λ2(z) =1∫

4α1(z)dz+ c0,

Γ (z) =α ′

1(z)α2(z)−α1(z)α ′2(z)

2α1(z)α2(z)−2α1(z)λ2(z),

λ0(z) =32

λ2(z)c21µ2h0 +

14

c22λ2(z)+ c4,

α3(z) = α4(z) = α5(z) = 0, (42)

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 775

a)

0

5

10

15

20

z

–60

–40

–20

0

20

40

60

t

0

0.01

0.02

92| |q

b)

0

5

10

15

20

z

–20

–10

0

10

20

t

0.02

0.04

0.06

0.08

92| |q

Fig. 9. Evolution plots of the solution (41) with the parameters: (a) α1(z) = 1−0.8sin(z)2, α2(z) = 10, h0 = −1/8, c0 = 20,c1 = 8, c2 = 2, c3 = 0, µ = 1; (b) α1(z) = 0.1, α2(z) = 1, h0 = −1/8, c0 = 10, c1 = 8, c2 = −0.40, c3 = 0, µ = 1.

and h0, c0, c1, c2, c3, c4 are arbitrary constants, andα1(z), α2(z) are arbitrary functions of z.

From Fig. 9 one can see that the evolution of the so-lution (41) is similar to that shown in Figs. 5b and 5dif the main control functions α1(z) and α2(z) are of thesame (or similar) forms. As shown in Fig. 9, under thesame (or similar) control functions, the evolution of theWeierstrass elliptic doubly periodic type solution (41)is similar to that of the Jacobi elliptic function solu-tion (31).

Family 10.

q10(z, t) =

c0 exp[∫

Γ (z)dz] ℘

(√h32 ξ , 2

µ2 , 12µ3

)1 + µ℘

(√h32 ξ , 2

µ2 , 12µ3

)· exp(i(c2t + λ0(z))),

(43)

where

ξ =c1

(t−∫

(6c21µ2h0+3c2

2)α3(z)+2c2α1(z)dz)

+c3,

h3 = −8µ3h0, h1 = 4µh0, h2 = h4 = 0,

λ0(z) =∫

(−6c21µ2h0 − c2

2)α1(z)

+ (−18µ2c21c2h0 − c3

2)α3(z)dz+ c4,

α2(z) = − 1a2

1(z)

[10µ4h0α1(z)c2

1 + α4(z)a21(z)c2

+ 30α3(z)c21µ4c2h0

],

α5(z) = −32

(a21(z)α4(z)+ 10α3(z)c2

1µ4h0)a2

1(z),

a1(z) = c0 exp[∫

Γ (z)dz]. (44)

Here h0, c0, c1, c2, c3, c4 are arbitrary constants, andα1(z), α2(z), α3(z) are arbitrary functions of z.

From Fig. 10 one can see that the evolution of thesolution (43) is similar to that shown in Fig. 6 if themain control functions α1(z) and α2(z) are of the same(or similar) forms. Under the same (or similar) controlfunctions, the evolution of the Weierstrass elliptic dou-bly periodic type solution (41) is similar to that of theJacobi elliptic function solution (35).

Remark 2. 1) The solutions obtained in the presentpaper include many previous results by many authors,such as: a) if c2 = 0, m = 1, the Theorems 1 and 2in [27] can be reproduced by our results (31) – (33);b) if m = 1, the solutions (51) and (52) in [17] canbe also reproduced by (31) – (33); c) if setting m = 1and α3(z) = α4(z) = α5(z) = 0 in Family 6, the solu-tions (48) and (49) in [17] can be recovered. But to ourknowledge, the other solutions have not been reportedearlier.

776 B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation

a)

0

5

10

15

20

z

–40

–20

0

20

40

t

0.2

0.4

0.6

0.8

110

2| |q

b)

0

10

20

30

40

z

–40

–20

0

20

40

t

0.2

0.4

0.6

0.8

110

2| |q

Fig. 10. Evolution plots of the solution (43) with the parameters: h0 = −1, c0 = 1, c1 = 0.1, c2 = 1, c3 = c4 = 0, µ = 1.(a) Γ (z) = 0.001, α1(z) = α3(z) = cos(z); (b) Γ (z) = 0.001, α1(z) = α3(z) = 1/(4z+4).

2) The solution presenting the property of the brightsolitary wave or dark solitary wave is mainly de-termined by the function form of the solution. Ifthe solution is only related to the function tanh(ξ )or sech(ξ ), the solution describes only the dark soli-tary wave or bright solitary wave, such as q11(z, t),q5(z, t) and q6(z,t), each solution presents one form.But if the solution combines function tanh(ξ ) withfunction sech(ξ ), such as q21(z,t), q31(z,t), q41(z, t),q7(z, t) and q8(z,t), the solution may present variousshapes, which are determined by whose effect is dom-inant. If tanh(ξ ) [sech(ξ )] is dominant, the solutiondescribes the dark solitary wave (the bright solitarywave). If both effects are nearly equal, the solutionpresents a combined bright and dark solitary wave.

3) Because the solutions obtained contain some ar-bitrary functions and arbitrary constants, the solutionspresent abundant structures. As shown in the figures,we found W-shaped solitary waves, combined-typesolitary waves, dispersion-managed solitary waves, pe-riodic propagating Jacobi elliptic function waves andWeierstrass elliptic function waves. For simplicity, toeach solution, we only considered some special param-eter conditions. If taking different parameters in eachsolution, abundant structures can be expected.

4. Summary and Discussion

In this paper, on the basis of symbolic computa-tion, we presented a generalized sub-equation expan-

sion method to construct some exact analytical solu-tions of nonlinear partial differential equations. Mak-ing use of the method, we investigated several exactanalytical solutions of the inhomogeneous high-ordernonlinear Schrodinger equation including not only thegroup velocity dispersion, self-phase-modulation, butalso various high-order effects, such as the third-orderdispersion, self-steepening and self-frequency shift.As a result, a broad class of exact analytical solu-tions of the IHNLSE was obtained, which includebright and dark solitary wave solutions, combined-type solitary wave solutions, dispersion-managed soli-tary wave solutions, Jacobi elliptic function solu-tions and Weierstrass elliptic function solutions. Asshown in the figures produced by computer simu-lation, these solutions possess abundant structures.From our results, many previous solutions of somenonlinear Schrodinger-type equations can be recov-ered by means of some suitable choices of the arbi-trary functions and arbitrary constants. The methoddeveloped provides a systematic way to generatevarious exact analytical solutions of the IHNLSEwith varying coefficients and can be used also forother PDEs. The development of a new mathemati-cal algorithm to discover analytical solutions, espe-cially solitary wave solutions, in nonlinear dispersionsystems with spatial parameter variations is impor-tant and might have significant impact on future re-search.

B. Li et al. · The Inhomogeneous Higher-Order Nonlinear Schrodinger Equation 777

Acknowledgements

The work was supported by NSF of China un-der Grant Nos. 10747141 and 10735030, Zhejiang

Provincial NSF of China under Grant No. 605408,Ningbo NSF under Grant Nos. 2007A610049 and2006A610093, and K. C. Wong Magna Fund ofNingbo University .

[1] M. J. Ablowitz and P. A. Clarkson, Soliton, Nonlin-ear Evolution Equations and Inverse Scattering, Cam-bridge University Press, New York 1991.

[2] C. H. Gu, H. S. Hu, and Z. X. Zhou, Darboux Trans-formation in Soliton Theory and its Geometric Appli-cations, Shanghai Scientific and Technical Publishers,Shanghai 1999.

[3] S. Y. Lou, Phys. Lett. A 277, 94 (2000); S. Y. Lou andH. Y. Ruan, J. Phys. A 34, 305 (2001); X. Y. Tang, S. Y.Lou, and Y. Zhang, Phys. Rev. E 66, 046601 (2002).

[4] S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, Phys. Lett.A 289, 69 (2001).

[5] S. Y. Lou, G. X. Huang, and H. Y. Ruan, J. Phys. A 24,L587 (1991).

[6] Z. B. Li and Y. P. Liu, Comput. Phys. Commun. 148,256 (2002).

[7] W. Malfliet and W. Hereman, Phys. Scr. 54, 563 (1996).[8] E. J. Parkes and B. R. Duffy, Comput. Phys. Commun.

98, 288 (1996).[9] E. Fan, Phys. Lett. A 277, 212 (2000); 294, 26 (2002).

[10] Z. Y. Yan, Phys. Lett. A 292, 100 (2001); Z. Y. Yan andH. Q. Zhang, Phys. Lett. A 285, 355 (2001).

[11] Z. S. Lu and H. Q. Zhang, Chaos, Solitons and Fractals17, 669 (2003).

[12] B. Li, Y. Chen, and H. Q. Zhang, J. Phys. A 35, 8253(2002).

[13] E. Yomba, Chaos, Solitons and Fractals 20, 1135(2004); 22, 321 (2004).

[14] Y. T. Gao and B. Tian, Comput. Phys. Commun. 133,158 (2001); B. Tian and Y. T. Gao, Comput. Math.Appl. 45, 731 (2003).

[15] B. Li, Y. Chen, H. N. Xuan, and H. Q. Zhang, Chaos,Solitons and Fractals 17, 885 (2003).

[16] B. Li and Y. Chen, Chaos, Solitons and Fractals 21, 241(2004); B. Li and H. Q. Zhang, Int. J. Mod. Phys. C 15,741 (2004).

[17] B. Li, Z. Naturforsch. 59a, 919 (2004).[18] E. Fan, Phys. Lett. A 243, 243 (2002); J. Phys. A 36,

7009 (2003); E. Fan and Y. C. Hon, Chaos, Solitons andFractals 15, 559 (2003); E. Fan and H. H. Dai, Comput.Phys. Commun. 153, 17 (2003).

[19] Z. Y. Yan, Chaos, Solitons and Fractals 21, 1013(2004).

[20] Y. Chen, Q. Wang, and Y. Z. Lang, Z. Naturforsch. 60a,127 (2005).

[21] E. Yomba, Phys. Lett. A 336, 463 (2005); Chaos, Soli-tons and Fractals 27, 187 (2006).

[22] E. Papaioannou, D. J. Frantzeskakis, and K. Hizanidis,IEEE J. Quantum Electron. 32, 145 (1996).

[23] R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou, Phys. Rev. E70, 066603 (2004).

[24] R. C. Yang, R. Y. Hao, L. Li, Z. H. Li, and G. S. Zhou,Opt. Commun. 242, 285 (2004).

[25] Z. Y. Xu, L. Li, Z. H. Li, G. S. Zhou, and K. Nakkeeran,Phys. Rev. E 68, 046605 (2003).

[26] R. C. Yang, L. Li, R. Y. Hao, Z. H. Li, and G. S. Zhou,Phys. Rev. E 71, 036616 (2005).

[27] V. N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85,4502 (2000); V. N. Serkin, T. L. Belyaeva, I. V. Alexan-drov, and G. Melo Melchor, Proceedings of SPIE 2001,Vol. 4271, p. 280.

[28] K. Porsezian and K. Nakkeeran, Phys. Rev. Lett. 76,3955 (1996).

[29] J. Kim, Q. H. Park, and H. J. Shin, Phys. Rev. E 58,6746 (1998).

[30] M. Gedalin, T. C. Scott, and Y. B. Band, Phys. Rev.Lett. 78, 448 (1997).

[31] S. L. Palacios, A. Guinea, J. M. Fernandez-Diaz, andR. D. Crespo, Phys. Rev. E 60, R45 (1999).

[32] Z. H. Li, L. Li, H. P. Tian, and G. S. Zhou, Phys. Rev.Lett. 84, 4096 (2000); Z. H. Li, L. Li, H. P. Tian, G. S.Zhou, and K. H. Spatschek, Phys. Rev. Lett. 89, 263901(2002).

[33] W. P. Hong, Opt. Commun. 194, 217 (2001).[34] L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H.

Spatschek, Phys. Rev. E 66, 04661 (2002); L. Li, Z. H.Li, S. Q. Li, and G. S. Zhou, Opt. Commun. 234, 169(2004).

[35] B. Li, Int. J. Mod. Phys. C 16, 1225 (2005); B. Li andY. Chen, Z. Naturforsch. 60a, 768 (2005); Z. Natur-forsch. 61a, 509 (2006).

[36] M. N. Vinoj, V. C. Kuriakose, and K. Porsezian, Chaos,Solitons and Fractals 12, 2569 (2001).

[37] K. Nakkeeran, J. Phys. A 34, 5111 (2001); Phys. Rev.E 62, 1313 (2000).

[38] R. Ganapathy and V. C. Kuriakose, Chaos, Solitons andFractals 15, 99 (2003).