Applied Mathematics and Computation 217 (2011) 5967–5971

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

New types of exact solutions for the fourth-order dispersive cubic–quinticnonlinear Schrödinger equation

Gui-Qiong XuDepartment of Information Management, Shanghai University, Shanghai 200444, PR China

a r t i c l e i n f o

Keywords:The nonlinear Schrödinger equationSoliton solutionPeriodic wave solution

0096-3003/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.amc.2010.12.008

E-mail address: xugq@staff.shu.edu.cn

a b s t r a c t

In this study, we use two direct algebraic methods to solve a fourth-order dispersive cubic–quintic nonlinear Schrödinger equation, which is used to describe the propagation of opti-cal pulse in a medium exhibiting a parabolic nonlinearity law. By using complex envelopeansatz method, we first obtain a new dark soliton and bright soliton, which may approachnonzero when the time variable approaches infinity. Then a series of analytical exact solu-tions are constructed by means of F-expansion method. These solutions include solitarywave solutions of the bell shape, solitary wave solutions of the kink shape, and periodicwave solutions of Jacobian elliptic function.

� 2010 Elsevier Inc. All rights reserved.

1. Introduction

The nonlinear Schrödinger (NLS) equation has been widely recognized as a ubiquitous mathematical model for describingthe evolution of a slowly varying wave packet in a general nonlinear wave system, thus it plays an important role in variousbranches of physics such as nonlinear optics [1,2], water waves [3], plasma physics, quantum mechanics, superconductivityand Bose–Einstein condensate theory. In optics, the propagation of a picosecond optical pulse in a monomode optical fiber(not including optical fiber loss) is described by the classic NLS equation. For water waves, the NLS equation describes theevolution of the envelope of modulated nonlinear wave groups.

Extending the NLS equation to the generalized NLS equations received much attention [4–8], due to these models beingmore realistic and many important applications of different types of generalized NLS equations. All these physical phenom-ena can be better understood with the help of exact solutions. Recently, much effort has been spent on the construction ofexact solutions of generalized NLS equation with different forms [4–23].

In this paper, we consider the higher order dispersive NLS equation with both fourth-order dispersion effects and a quin-tic nonlinearity [19]

iqt þ aqxx � bqxxxx þ cðjqj2 þ djqj4Þq ¼ 0; ð1Þ

which describes the propagation of optical pulse in a medium that exhibits a parabolic nonlinearity. In Eq. (1), qðx; tÞ is theslowly varying envelope of the electromagnetic field, x represents the distance along the direction of propagation (the lon-gitudinal coordinate), and t represents the retarded time (in the group velocity frame). The parameters a, b, c and d are realconstants.

Eq. (1) has been studied for its importance in diverse physical areas such as nonlinear optics [20], in particular the prop-agation of solitonic in a bosonic gases and the coalescence of droplets in the first-order phase transition in Bose–Einstein

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5968 G.-Q. Xu / Applied Mathematics and Computation 217 (2011) 5967–5971

condensates. A considerable amount of research work has been devoted to the study of Eq. (1) with special parametricchoices. For example, (i) b ¼ 0 [21,22], (ii) d ¼ 0 [23] and so on.

Very recently, Biswas and Milovic [24] studied the exact solutions of Eq. (1) when the parameters a, b, c and d are nonzero.As a result, the authors obtained the optical soliton solution of Eq. (1) as follows,

qðx; tÞ ¼ Akþ coshðBðx� mtÞÞ e

ið�jxþxtþhÞ; ð2Þ

where A;B;x; k and m were given by Eqs. (16)–(21) of Ref. [24]. However, the ‘‘solution’’ (2) does not satisfy Eq. (1), which ispartly because of several errors in Eqs. (12) and (13) given in Ref. [24].

The natural and important problem is that whether Eq. (1) possesses optical soliton in the form (2) and more types ofexact solutions. In this paper, we will revisit exact solutions of Eq. (1) by using two direct algebraic methods.

2. Optical solitons by using complex envelope ansatz method

Motivated by the work of Li et al. [7], the solutions of Eq. (1) are supposed as,

qðx; tÞ ¼ Eðx; tÞ eiðkx�xtþhÞ; ð3Þ

where Eðx; tÞ is now considered as a complex envelope function, and k;x; h are real constants. Substituting Eq. (3) into Eq. (1)and removing the exponential term, we can rewrite Eq. (1) as

iEt þ 2ikðaþ 2bk2ÞEx þ ðaþ 6bk2ÞExx � 4ibkExxx � bExxxx þ ðx� ak2 � bk4ÞEþ cjEj2Eþ cdjEj4E ¼ 0: ð4Þ

Firstly, we now take the complex envelope ansatz function Eðx; tÞ as

Eðx; tÞ ¼ ibþ k tanhðnÞ; n ¼ pxþ st; ð5Þ

where b; k; p; s are real constants. Substituting Eq. (5) into Eq. (4) and setting the coefficients of tanhðnÞj ðj ¼ 0; . . . ;5Þ to zero,one obtains the following algebraic equations:

kðdck4 � 24p4bÞ ¼ 0; kðk3bcdþ 24bp3kÞ ¼ 0;

kðs� 2kb3cd� kbc þ 2kpaþ 32bp3kþ 4bpk3Þ ¼ 0;

kðk2c þ 2k2b2cdþ 40p4bþ 2p2aþ 12k2p2bÞ ¼ 0;

kð2p2aþ k2a� b4cd� b2c þ k4bþ 16p4b�xþ 12k2p2bÞ ¼ 0;

bbk4 � bx� b3c � 2apkk� b5cd� sk� 4bpkk3 � 8bp3kkþ bak2 ¼ 0:

Solving it we obtain one set of nontrivial solution,

s ¼ 8bpkðk2 þ p2Þ; b ¼ � kkp; x ¼ 2p2aþ 3k2aþ 37k4bþ 52k2p2bþ 16p4b;

c ¼ �2p2ð30bk2 þ 20bp2 þ aÞk2 ; d ¼ � 12bp2

k2ð30bk2 þ 20bp2 þ aÞ:

ð6Þ

From (3), (5) and (6), the optical solitary wave of Eq. (1) can be obtained,

q1ðx; tÞ ¼ � ikkpþ k tanhðpxþ 8bpkðk2 þ p2ÞtÞ

� �ei kx�ð2p2aþ3k2aþ37k4bþ52k2p2bþ16p4bÞtþhð Þ;

where c; d satisfy the parametric constraints given by (6). Thus the intensity jq1ðz; tÞj2 reads,

Q1 � jq1ðx; tÞj2 ¼ k2k2

p2 þ k2tanh2ðpxþ 8bpkðk2 þ p2ÞtÞ;

which is an optical dark soliton (the intensity profile contains a dip in a uniform background) as shown in Fig. 1(a). Since theearly work of Zakharov and Shabat [25], dark optical solitons have been an active topic of research both theoretically andexperimentally. This is because of possible application of dark solitons for long-distance communications, making use ofits stability under the influence of material losses.

Secondly, the solutions of Eq. (4) also can be supposed as

Eðx; tÞ ¼ bþ ik tanhðnÞ; n ¼ pxþ st; ð7Þ

where b; k; p; s are real constants. Performing the similar computations as before, we can obtain another exact solution,

q2ðx; tÞ ¼kkpþ ik tanhðpxþ 8bpkðk2 þ p2ÞtÞ

� �ei kx�ð2p2aþ3k2aþ37k4bþ52k2p2bþ16p4bÞtþhð Þ;

(a)

–10

–5

0

5

10

x

–2

–1

0

1

2

t

1.5

2 Q1

(b)

–10–5

05

10x–1

01

2

t

0.2

0.4

0.6

0.8

1

1.2

Q2

Fig. 1. The intensity of a dark soliton given by jq1ðx; tÞj2 (see (a)) and the intensity of a bright soliton given by jq2ðx; tÞj

2 (see (b)), with a ¼ �15:075; b ¼0:25; c ¼ 2; d ¼ 3; k ¼ p ¼ 1; k ¼ 1:1.

G.-Q. Xu / Applied Mathematics and Computation 217 (2011) 5967–5971 5969

where c; d satisfy the parametric constraints given by (6). Thus we have

Q 2 � jq2ðx; tÞj2 ¼ k2k2

p2 � k2tanh2ðpxþ 8bpkðk2 þ p2ÞtÞ;

which is an optical bright soliton as shown in Fig. 1(b).It is easily seen that the intensities jq1ðz; tÞj

2 and jq2ðx; tÞj2 approach nonzero when the time variable approaches infinity.

In order to derive twin-hole dark (W-shaped) solitons, we may suppose

Eðx; tÞ ¼ ibþ k tanhðnÞ þ iqsechðnÞ; n ¼ pxþ st:

Substituting it to Eq. (4) leads to an over-determined algebraic system with respect to b; k;q; p; s; k; k. However, by solving theobtained algebraic system, one cannot derive nontrivial solutions for q – 0.

3. A series of exact solutions by using F-expansion method

We suppose that the solution of (1) is of the form

qðx; tÞ ¼ PðsÞ eig; s ¼ Bðx� mtÞ; g ¼ ð�jxþxt þ hÞ; ð8Þ

where PðsÞ is a real function, and B; m;j;x are real constants to be determined. Substituting Eq. (8) to Eq. (1) and separatingthe real and imaginary parts, one may obtain the following equations,

� Bðmþ 2ajþ 4bj3ÞP0 þ 4bjB3P000 ¼ 0; ð9Þðxþ aj2 þ bj4ÞP � cP3 � cdP5 � B2ðaþ 6bj2ÞP00 þ bB4P

0000 ¼ 0: ð10Þ

The linear ordinary differential equation (9) has no solitary wave solutions, thus we have to take j ¼ m ¼ 0. In this case Eq.(9) is satisfied identically, and Eq. (10) becomes,

xP � cP3 � cdP5 � aB2P00 þ bB4P0000 ¼ 0: ð11Þ

Eq. (11) can be solved by using the F-expansion method [26–29]. According to the F-expansion method, we suppose,

PðsÞ ¼Xn

i¼0

AiFiðsÞ; An – 0; ð12Þ

where Ai ði ¼ 0; . . . ;nÞ are real constants to be determined, the integer n is determined by balancing the linear highest orderterm and nonlinear term. And FðsÞ in (12) satisfies,

dFðsÞds

� �2

¼ h0 þ h2FðsÞ2 þ h4FðsÞ4; ð13Þ

where h0;h2;h4 are real constants. By balancing the linear highest order derivative term P0000

with nonlinear term P5 in Eq.(11), we find n ¼ 1. Thus Eq. (12) becomes,

PðsÞ ¼ A0 þ A1FðsÞ: ð14Þ

Substituting Eq. (14) into Eq. (11) along with Eq. (13), collecting all terms with the same power of FjðsÞ ðj ¼ 0; . . . ;5Þ, andequating the coefficients of these terms yields a set of algebraic equations with respect to A0;A1;B;x; a; b; c; d;h0; h2;h4:

5970 G.-Q. Xu / Applied Mathematics and Computation 217 (2011) 5967–5971

dcA41A0 ¼ 0; xA0 � cA3

0 � dcA50 ¼ 0;

10dcA21A3

0 þ 3cA21A0 ¼ 0; 24bA1B4h2

4 � dcA51 ¼ 0;

20bA1B4h2h4 � 10dcA31A2

0 � cA31 � 2A1B2ah4 ¼ 0;

xA1 � 3cA1A20 � A1B2ah2 þ 12bA1B4h4h0 þ bA1B4h2

2 � 5dcA1A40 ¼ 0:

Solving the above algebraic equations, we have a set of nontrivial solution,

A0 ¼ 0; A1 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12bB2h4

dð10bB2h2 � aÞ

vuut ;

x ¼ B2ðah2 � 12bB2h4h0 � bB2h22Þ; c ¼ dð10bB2h2 � aÞ2

6b:

ð15Þ

Special analytical solutions to Eq. (13) exists for certain choices of the constants h0;h2 and h4. Whenh0 ¼ 1;h2 ¼ �ð1þm2Þ;h4 ¼ m2, Eq. (13) has the solution FðsÞ ¼ snðs;mÞ. From Eqs. (8) and (14), Eq. (1) has the Jacobianelliptic sine function solution,

q3ðx; tÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 12B2bm2

dð10B2bþ 10B2bm2 þ aÞ

vuut snðBx;mÞeið�B2ðB2bþB2bm4þ14B2bm2þam2þaÞtþhÞ;

where B is determined by dð10bB2 þ 10m2bB2 þ aÞ2 � 6bc ¼ 0.When h0 ¼ 1�m2; h2 ¼ 2m2 � 1; h4 ¼ �m2, Eq. (13) has the solution FðsÞ ¼ cnðs;mÞ. Inserting it into (14) and using the

transformation (8), Eq. (1) has the Jacobian elliptic cosine function solution,

q4ðx; tÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12B2bm2

dð10B2b� 20B2bm2 þ aÞ

vuut cnðBx;mÞeiðB2ð16B2bm2�16B2bm4�B2b�aþ2am2ÞtþhÞ;

where B is determined by dðaþ 10bB2 � 20bB2m2Þ2 � 6bc ¼ 0.Some solitary wave solutions can be obtained if the modulus m approaches to 1. For example, when m! 1, the solution

q3ðx; tÞ degenerates to the kink type envelope wave solution,

q5ðx; tÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 12bB2

dð20B2bþ aÞ

stanhðBxÞeið�2B2ð8B2bþaÞtþhÞ;

where B is determined by dðaþ 20bB2Þ2 � 6bc ¼ 0.When m! 1, the solution q4ðx; tÞ degenerates to the bell type envelope wave solution,

q6ðx; tÞ ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12B2b

dða� 10B2bÞ

ssechðBxÞeiðB2ða�B2bÞtþhÞ;

where B is determined by dða� 10bB2Þ2 � 6bc ¼ 0.As pointed out in Ref. [26], Eq. (13) has many other Jacobian elliptic function solutions in terms of

dnðnÞ;nsðnÞ;ndðnÞ;ncðnÞ; scðnÞ; csðnÞ; sdðnÞ;dsðnÞ; cdðnÞ;dcðnÞ as well as the corresponding solitary wave and trigonometricfunction solutions. For simplicity, such types of solutions to Eq. (1) are not listed here.

With the aid of Maple, we have checked the solutions qjðx; tÞ ðj ¼ 1; . . . ;6Þ by putting them back into Eq. (1).

Acknowledgments

I would like to express my sincere thanks to the referees for their helpful suggestions. This work was supported by theNational Natural Science Foundation of China (No. 10801037).

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