Travelling Wave Solutions for Nonlinear Schrödinger Equation with a Higher-Order Dispersive Term

Abstract

A nonlinear Schrödinger equation with a higher-order dispersive term describing the propagation of ultrashort femtosecond pulses in optical fibres is considered and is transformed into a second-order nonlinear ordinary differential equation. We investigate the exact travelling wave solutions of the nonlinear Schrödinger equation using three methods, namely, the auxiliary equation method, the first integral method, and the direct integral method. As a result, Jacobi elliptic function solution, hyperbolic function solution, trigonometric function solution, and rational solution with parameters are obtained successfully. When the parameters are taken as special values, the two known solitary wave solution and periodic wave solution are derived from the solutions obtained. The aim of the paper is to compare the efficiency of the three methods.

1. Introduction

We are concerned with the following nonlinear Schrödinger equation with a higher-order dispersive term [1]: 𝑖𝜓_𝑥+𝛼₁𝜓_𝑡𝑡+𝛼₂𝜓𝜓²−𝑖𝛼₃𝜓_𝑡𝑡𝑡−𝑖𝛼₄𝜓𝜓²_𝑡−𝑖𝛼₅𝜓²_𝑡𝜓=0, (1) where 𝜓=𝜓(𝑥,𝑡) is slowly varying envelop of the electric field, the subscripts 𝑥 and 𝑡 are the spatial and temporal partial derivative in retard time coordinates, and 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄, and 𝛼₅ are real parameters related to the group velocity, self-phase modulation, third-order dispersion, self-steepening, and self-frequency shift arising from stimulated Raman scattering, respectively. Equation (1) can be applied to describe the propagation of ultrashort femtosecond pulses in optical fibres.

The nonlinear Schrödinger (NLS) equations occur in many branches of physics such as nonlinear optics, quantum mechanics, condensed matter physics, and plasma physics [2, 3]. There have been various Schrödinger-type equations in the study of nonlinear phenomena. All these phenomena can be better understood with the help of exact analytical solutions. Recently, a considerable amount of research work has been devoted to searching for exact solution of NLS equation with a variety of nonlinearities. For example, Li and Wang [4] applied the (𝐺^′/𝐺)-expansion method to study the higher-order nonlinear Schrödinger equation (1) and obtained travelling wave solutions under constraint condition. Ma and Lee [5] proposed the transformed rational function method for solving the (3+1)-dimensional Jimbo-Miwa equation and obtained exact solutions. Wu and Dai [6] investigated quintic nonlinear derivative Schrödinger equation and obtained soliton solutions including chirped bright, dark, and kink soliton solutions with the aid of a special auxiliary equation. Ma [7] introduced a class of bilinear differential operators which describe generalized bilinear differential equations and discussed their links with the Bell polynomial. More details are presented in [8, 9].

Exact solutions (especially travelling wave solution) of nonlinear evolution equation (NLEE) play an important role in the study of nonlinear physical phenomenon [2, 10]. During the past decades, various analytical and numerical methods for constructing the travelling wave solutions to NLEE have been proposed, such as Hirota’s bilinear method [11, 12], the Bäcklund transformation method [13], the sine-cosine method [14], the tanh-function method [15, 16], the Exp-function method [17, 18], the Jacobi elliptic expansion method [19], the first integral method [20], and the Riccati equation method [21, 22]. The reader is referred to [1, 23, 24] and the references therein.

In this paper, our main purpose is to investigate the travelling wave solutions for nonlinear Schrödinger equation with a higher-order dispersive term (1). The rest of the paper is organized as follows. In Section 2, we describe these methods for solving nonlinear evolution equations including the auxiliary equation method, the first integral method, and the direct integral method. In Section 3, we apply these methods to (1) and obtain enveloped travelling wave solutions. Finally, the concluding remarks are presented in Section 4.

2. Description of Methods

2.1. The Auxiliary Equation Method

For a given nonlinear partial differential equation with a physical field 𝑢(𝑥,𝑡) in two independent variables 𝑥,𝑡, 𝐻𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑡𝑥,𝑢_𝑥𝑥,…=0. (2)

Step 1. Making use of the travelling wave transformation, 𝑢(𝑥,𝑡)=𝑢(𝜉),𝜉=𝑘(𝑥−𝑣𝑡). (3) Then, (2) is reduced to the following ordinary differential equation (ODE): 𝑄𝑢,−𝑘𝑣𝑢^′,𝑘𝑢^′,𝑘²𝑣²𝑢^′′,−𝑘²𝑣𝑢^′′,𝑘²𝑢^′′,…=0. (4)

Step 2. We introduce a new ansatz: 𝑢(𝜉)=𝑎₀+^𝑁_𝑖=1𝑎_𝑖𝜙^𝑖(𝜉)+𝑏_𝑖𝜙^𝑖−1(𝜉)𝑅+𝜙²(𝜉)^1/2+𝑐_𝑖𝜙^−𝑖(𝜉), (5) where 𝑁 is a positive integer and 𝑎₀,𝑎_𝑖,𝑏_𝑖, and 𝑐_𝑖 are constants, while the new variable 𝜙=𝜙(𝜉) satisfies the Riccati equation [21]: 𝜙^′=𝑑𝜙𝑑𝜉=𝑅+𝜙²(𝜉). (6)

Step 3. We define the degree of 𝑢(𝜉) as 𝐷[𝑢(𝜉)]=𝑁. The degree of 𝑢^(𝑝)(𝜉) can be calculated by 𝐷𝑢^(𝑝)𝐷𝑢=𝑁+𝑝,_𝑝𝑢^(𝑞)𝑠=𝑁𝑝+𝑁+𝑞𝑠. (7)

Step 4. Substituting (5) along with (6) into (4) and setting all coefficients of 𝜙^𝑗(𝜉)𝑅+𝜙²(𝜉)^𝑚 (𝑗=0,1,…,𝑚=0,1) to zero, we get a system of nonlinear algebraic equations.

Step 5. Solving the overdetermined system in Step 4 by the symbolic computation system Mathematica, we obtain travelling wave solutions of (1).

Remark 1. One of the main steps in the method is to determine the value of 𝑁 in (5). The value of 𝑁 can be determined by considering the homogeneous balance between the highest-order derivative and nonlinear term appearing in (4). If 𝑁 is not a positive integer, then we first make the transformation 𝑢=𝜓^𝑁.

2.2. The First Integral Method

The nonlinear partial differential equation in two independent variables 𝑥,𝑡𝐹𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑡𝑡,𝑢_𝑡𝑥,𝑢_𝑥𝑥,…=0 (8) can be transformed into a nonlinear ODE 𝐺𝑢,−𝑘𝑣𝑢^′,𝑘𝑢^′,𝑘²𝑣²𝑢^′′,−𝑘²𝑣𝑢^′′,𝑘²𝑢^′′,…=0, (9) under the wave transformation 𝑢(𝑥,𝑡)=𝑢(𝜉), where the wave variable 𝜉=𝑘(𝑥−𝑣𝑡).

Next, we introduce a new independent variable 𝑋(𝜉)(𝜉)(𝜉)=(𝜉)=𝑢,𝑌𝑑𝑋=𝑋̇(𝜉),𝑑𝜉 (10) which leads (9) to a system of nonlinear ODEs 𝑋̇(𝜉)(𝜉),𝑌̇(𝜉)(𝑋(𝜉)(𝜉.=𝑌=𝑃,𝑌)) (11) By the qualitative theory of ordinary differential equations, if we can find the integrals to (11), then the general solutions to (11) can be obtained directly. In order to obtain one first integral of (11), we will apply the Division Theorem.

3. Application

In this section, we will illustrate three methods mentioned above and obtain the exact travelling wave solutions of (1).

Assume that the solution of (1) can be written as 𝜓(𝑥,𝑡)=𝑢(𝜉)exp[𝑖(𝑘𝑥−𝜔𝑡)],𝜉=𝑡−𝜆𝑥+𝜉₀, (12) where 𝑘, 𝜔, and 𝜆 are constants to be determined. Substituting (12) into (1) and separating the real and imaginary parts, we obtain the ODEs for 𝑢(𝜉): 𝛼₃𝑢^′′′+3𝛼₄+2𝛼₅𝑢²𝑢^′−3𝛼₃𝜔²−𝜆−2𝛼₁𝜔𝑢^′𝛼=0,₁−3𝛼₃𝜔𝑢^′′+𝛼₂−𝛼₄𝜔𝑢³+𝛼₃𝜔³−𝛼₁𝜔²−𝑘𝑢=0. (13)(14) Integrating (13) once and setting the constant of integration to zero, we get 𝛼₃𝑢^′′−3𝛼₃𝜔²−𝜆−2𝛼₁𝜔𝑢+3𝛼₄+2𝛼₅3𝑢³=0. (15) Under the conditions 𝛼𝜔=¹3𝛼₄+2𝛼₅−3𝛼₂𝛼₃6𝛼₃𝛼₄+𝛼₅,𝑘=8𝛼₁𝜔²−8𝛼₃𝜔³−2𝛼²₁𝜔+𝛼₁𝜆𝛼₃+3𝜆𝜔, (16) (14) and (15) are transformed into the following equation: 𝑢^′′+𝐴𝑢+𝐵𝑢³=0, (17) where 𝐴 and 𝐵 are defined by 𝐴=(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃, and 𝐵=(3𝛼₄+2𝛼₅)/3𝛼₃.

3.1. Using the Auxiliary Equation Method to Solve (1)

Considering the homogeneous balance between 𝑢^′′ and 𝑢³ in (17) yields 𝑁=1. So we assume that (1) has the following formal solutions: 𝑢(𝜉)=𝑎₀+𝑎₁𝜙(𝜉)+𝑏₁𝑅+𝜙²(𝜉)+𝑐₁𝜙⁻¹(𝜉), (18) where 𝑎₀, 𝑎₁, 𝑏₁, and 𝑐₁ are constants to be determined.

Substituting (18) along with (6) into (17), collecting the coefficients of 𝜙^𝑖(𝜉)(𝑅+𝜙²(𝜉))^𝑗 (𝑖=−3,−2,−1,0,1,2,3,𝑗=0,1), and setting each coefficient to zero, one obtains the system of algebraic equations. Then solving the obtained algebraic equations, we have a set of nontrivial solutions.

Case  1. One has 𝑎₀=0,𝑏₁=0,𝑐₁𝑎=0,₁=±−2𝐵,2𝑅+𝐴=0. (19)

Case  2. One has 𝑎₀=0,𝑐₁=0,𝑎₁=±−1,𝑏2𝐵₁=∓−12𝐵,𝑅+2𝐴=0. (20)

Case  3. One has 𝑎₀=0,𝑏₁=0,𝑎₁=±−2𝐵,𝑐₁𝐴=±2−12𝐵,𝐴−4𝑅=0. (21)

From the expression in (12), (19), (20), (21) and the solutions of (6), we derive the following travelling wave solutions of (1).

Family  1. If 𝑅<0, 𝐴>0, 𝛼₃(3𝛼₄+2𝛼₅)<0, then 𝜓_1,1=±−𝐴𝐵coth𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓exp𝑘𝑥−𝜔𝑡)]_1,2=±−𝐴𝐵tanh𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓exp𝑘𝑥−𝜔𝑡)]_1,3=±−𝐴𝐵tanh√2𝐴𝑡−𝜆𝑥+𝜉₀+𝑖sech√2𝐴𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓×exp𝑘𝑥−𝜔𝑡)]_1,4=±−𝐴𝐵coth√2𝐴𝑡−𝜆𝑥+𝜉₀+csch√2𝐴𝑡−𝜆𝑥+𝜉₀×exp[𝑖(𝑘𝑥−𝜔𝑡)], (22) where 𝜓_1,1 and 𝜓_1,2 are the known solitary wave solution found by Liu (formulae (30) and (31) in [23]).

Family  2. If 𝑅>0, 𝐴>0, 𝛼₃(3𝛼₄+2𝛼₅)<0, then 𝜓_1,5=±−𝐴2𝐵cot𝐴4𝑡−𝜆𝑥+𝜉₀+tan𝐴4𝑡−𝜆𝑥+𝜉₀exp[𝑖(𝑘𝑥−𝜔𝑡)]. (23)

Family  3. If 𝑅>0, 𝐴<0, 𝛼₃(3𝛼₄+2𝛼₅)<0, then 𝜓_1,6=±𝐴𝐵tan−𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓exp𝑘𝑥−𝜔𝑡)]_1,7=±𝐴𝐵cot−𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓exp𝑘𝑥−𝜔𝑡)]_1,8=±𝐴𝐵tan√−2𝐴𝑡−𝜆𝑥+𝜉₀−sec√−2𝐴𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓×exp𝑘𝑥−𝜔𝑡)]_1,9=±𝐴𝐵cot√−2𝐴𝑡−𝜆𝑥+𝜉₀+csc√−2𝐴𝑡−𝜆𝑥+𝜉₀×exp[𝑖(𝑘𝑥−𝜔𝑡)], (24) where 𝜓_1,6 and 𝜓_1,7 are the known periodic wave solution obtained in [23] (formulae (34) and (35) in [23]).

Family  4. If 𝑅<0, 𝐴<0, 𝛼₃(3𝛼₄+2𝛼₅)<0, then 𝜓_1,10=±𝐴2𝐵tanh−𝐴4𝑡−𝜆𝑥+𝜉₀−coth−𝐴4𝑡−𝜆𝑥+𝜉₀exp[𝑖(𝑘𝑥−𝜔𝑡)]. (25)

Family  5. If 𝑅=0, 𝛼₃(3𝛼₄+2𝛼₅)<0, then we obtain the rational solution of (1): 𝜓_1,11=±−6𝛼₃3𝛼₄+2𝛼₅1𝑡−3𝛼₃𝜔²−2𝛼₁𝜔𝑥+𝜉₀×exp[𝑖(𝑘𝑥−𝜔𝑡)]. (26)

3.2. Using the First Integral Method to Solve (1)

The first integral of (17) can be assumed in the following form: 𝐹(𝑋(𝜉),𝑌(𝜉))=^𝑚_𝑖=0𝑎_𝑖(𝑋(𝜉))𝑌^𝑖(𝜉)=0, (27) where 𝑎_𝑖(𝑋) (𝑖=0,1,2,…,𝑚) are polynomials of 𝑋 and 𝑎_𝑚(𝑋)≠0. We start our analysis by assuming 𝑚=1 in (27).

According to the Division Theorem, there exists a polynomial 𝛼(𝑋)+𝛽(𝑋)𝑌 in 𝐶[𝑋,𝑌] such that 𝑎̇₀(𝑋)+𝑎̇₁(𝑋)𝑌𝑌+𝑎₁(𝑋)𝑌̇=𝛼(𝑋)+𝛽(𝑋)𝑌𝑎₀(𝑋)+𝑎₁(𝑋)𝑌. (28)

By equating the coefficients of 𝑌^𝑗 (𝑗=0,1,2) on both sides of (28), we have 𝑎̇₁(𝑋)=𝛽(𝑋)𝑎₁(𝑋),𝑎̇₀(𝑋)=𝛼(𝑋)𝑎₁(𝑋)+𝛽(𝑋)𝑎₀(𝑋),𝛼(𝑋)𝑎₀(𝑋)=𝑎₁(𝑋)−𝐴𝑋−𝐵𝑋³. (29)(30)(31) Through careful analysis, we find that 𝑎₁(𝑋)=1,𝛼(𝑋)=𝛼₁𝑋+𝛼₂,𝑎₀(𝑋)=𝛼₁2𝑋²+𝛼₂𝑋+𝛼₃, (32) where 𝛼₃ is an arbitrary constant of integration. Substituting 𝑎₀(𝑋), 𝑎₁(𝑋), and 𝛼(𝑋) into (31) and setting all the coefficients of 𝑋^𝑗 to zero, one obtains a system of nonlinear algebraic equations. Then, solving the algebraic equations, we get the following.

Case  1. One has 𝛼₁=√−2𝐵,𝛼₂=0,𝛼₃𝐴=−√.−2𝐵 (33)

Case  2. One has 𝛼₁=−√−2𝐵,𝛼₂=0,𝛼₃=𝐴√.−2𝐵 (34) Subsequently, we obtain the following solutions: 𝜓_2,1(𝑥,𝑡)=−𝐴𝐵coth𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(𝜓×exp𝑘𝑥−𝜔𝑡)]_2,2(𝑥,𝑡)=𝐴𝐵tan−−𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓×exp𝑘𝑥−𝜔𝑡)]_2,3=−6𝛼₃3𝛼₄+2𝛼₅1𝑡−3𝛼₃𝜔²−2𝛼₁𝜔𝑥+𝜉₀[𝑖(.𝜓×exp𝑘𝑥−𝜔𝑡)]_2,4(𝑥,𝑡)=−𝐴𝐵tanh−𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓×exp𝑘𝑥−𝜔𝑡)]_2,5(𝑥,𝑡)=𝐴𝐵cot−−𝐴2𝑡−𝜆𝑥+𝜉₀[𝑖(,𝜓×exp𝑘𝑥−𝜔𝑡)]_2,6=−−6𝛼₃3𝛼₄+2𝛼₅1𝑡−3𝛼₃𝜔²−2𝛼₁𝜔𝑥+𝜉₀×exp[𝑖(𝑘𝑥−𝜔𝑡)]. (35)

3.3. Using the Direct Integral Method to Solve (1)

Multiplying (17) by 𝑢^′ and integrating (17) with respect to 𝜉 yield 𝑢^′2𝐴=𝐶−2𝑢²−𝐵4𝑢⁴, (36) where 𝐶 is an integration constant. In [24], Lai et al. have given the solutions of the following nonlinear ODE with four-degree term: 𝐹^′2=𝐶₀+𝐶₂𝐹²+𝐶₄𝐹⁴. (37)

Comparing the coefficient of (36) with that of (37), we obtain the exact travelling wave solutions for (1) as follows.

When 𝐶=1,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=(1+𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=−2𝑚²𝜓_3,1(𝑥,𝑡)=𝑠𝑛(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (38)

When 𝐶=1−𝑚²,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=(1−2𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=2𝑚², 𝜓_3,2(𝑥,𝑡)=𝑐𝑛(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (39)

When 𝐶=𝑚²−1,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=(𝑚²−2),(3𝛼₄+2𝛼₅)/3𝛼₃=2, 𝜓_3,3(𝑥,𝑡)=𝑑𝑛(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (40)

When 𝐶=𝑚²,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=(1+𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=−2, 𝜓_3,4(𝑥,𝑡)=𝑛𝑠(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (41)

When 𝐶=−𝑚²,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2𝑚²−1),(3𝛼₄+2𝛼₅)/3𝛼₃=−2(𝑚²−1), 𝜓_3,5(𝑥,𝑡)=𝑛𝑐(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (42)

When 𝐶=−1,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2−𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=−2(𝑚²−1), 𝜓_3,6(𝑥,𝑡)=𝑛𝑑(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (43)

When 𝐶=1,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2−𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=−2(1−𝑚²), 𝜓_3,7(𝑥,𝑡)=𝑠𝑐(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (44)

When 𝐶=1,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2𝑚²−1),(3𝛼₄+2𝛼₅)/3𝛼₃=−2𝑚²(𝑚²−1), 𝜓_3,8(𝑥,𝑡)=𝑠𝑑(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (45)

When 𝐶=1−𝑚²,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2−𝑚²),(3𝛼₄+2𝛼₅)/3𝛼₃=−2, 𝜓_3,9(𝑥,𝑡)=𝑐𝑠(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (46)

When 𝐶=−𝑚²(1−𝑚²),(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(2𝑚²−1),(3𝛼₄+2𝛼₅)/3𝛼₃=−2, 𝜓_3,10(𝑥,𝑡)=𝑑𝑠(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (47)

When 𝐶=−(𝑚²−1)²/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²+1)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=1/2, 𝜓_3,11(𝑥,𝑡)=𝑚𝑐𝑛(𝜉,𝑚)±𝑑𝑛(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (48)

When 𝐶=1/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(1−2𝑚²)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=−1/2, 𝜓_3,12()=()𝑥,𝑡1±𝑐𝑛𝜉,𝑚()[𝑖(.𝑠𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (49)

When 𝐶=(1−𝑚²)/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²+1)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=(𝑚²−1)/2, 𝜓_3,13()=()𝑥,𝑡1±𝑠𝑛𝜉,𝑚()[𝑖(.𝑐𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (50)

When 𝐶=𝑚²/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²−2)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=−1/2, 𝜓_3,14()=()𝑥,𝑡1±𝑑𝑛𝜉,𝑚()[𝑖(.𝑠𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (51)

When 𝐶=𝑚²/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²−2)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=−𝑚²/2, 𝜓_3,15(𝑥,𝑡)=𝑠𝑛(𝜉,𝑚)±𝑖𝑐𝑛(𝜉,𝑚)exp[𝑖(𝑘𝑥−𝜔𝑡)]. (52)

When 𝐶=1/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²−2)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=−𝑚²/2, 𝜓_3,16()=()𝑥,𝑡𝑠𝑛𝜉,𝑚()[𝑖(.1±𝑑𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (53)

When 𝐶=(𝑚²−1)/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²+1)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=(1−𝑚²)/2, 𝜓_3,17()=()𝑥,𝑡𝑑𝑛𝜉,𝑚()[𝑖(.1±𝑚𝑠𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (54)

When 𝐶=(1−𝑚²)/4,(𝜆+2𝛼₁𝜔−3𝛼₃𝜔²)/𝛼₃=−(𝑚²+1)/2,(3𝛼₄+2𝛼₅)/3𝛼₃=(𝑚²−1)/2, 𝜓_3,18()=()𝑥,𝑡𝑐𝑛𝜉,𝑚()[𝑖(.1±𝑠𝑛𝜉,𝑚exp𝑘𝑥−𝜔𝑡)] (55)

Remark 2. When the modulus 𝑚→1, Jacobi elliptic function 𝑠𝑛(𝜉,𝑚)→tanh(𝜉), 𝑐𝑛(𝜉,𝑚)→sech(𝜉), and 𝑑𝑛(𝜉)→sech(𝜉). Hence, we can obtain the hyperbolic function solutions. For example, 𝜓_3,1 and 𝜓_3,2 are turned into the kink type envelope wave solution 𝜓_3,1,1=exp[𝑖(𝑘𝑥−𝜔𝑡)]tanh𝑡−𝜆𝑥+𝜉₀,𝐴=2,𝐵=−2 (56) and the bell type envelope wave solution 𝜓_3,2,1=exp[𝑖(𝑘𝑥−𝜔𝑡)]sech𝑡−𝜆𝑥+𝜉₀,𝐴=−1,𝐵=2. (57) Formula (57) is the same as the solution 𝜓_1,2.

In particular, 𝜓_3,4, 𝜓_3,12, and 𝜓_3,15 degenerate into 𝜓_3,4,1=exp[𝑖(𝑘𝑥−𝜔𝑡)]coth𝑡−𝜆𝑥+𝜉₀,𝜓𝐴=2,𝐵=−2,_3,12,1=coth𝑡−𝜆𝑥+𝜉₀+csch𝑡−𝜆𝑥+𝜉₀[𝑖(1×exp𝑘𝑥−𝜔𝑡)],𝐴=21,𝐵=−2,𝜓_3,15,1=tanh𝑡−𝜆𝑥+𝜉₀+𝑖sech𝑡−𝜆𝑥+𝜉₀[𝑖(1×exp𝑘𝑥−𝜔𝑡)],𝐴=21,𝐵=−2, (58) respectively, which are in full agreement with the solutions 𝜓_1,1, 𝜓_1,4, and 𝜓_1,3 obtained above.

In addition, 𝜓_3,3 and 𝜓_3,9 degenerate into the known solutions (32) and (33) in [23], respectively.

Remark 3. When 𝑚→0, 𝑠𝑛(𝜉,𝑚)→sin(𝜉), 𝑐𝑛(𝜉,𝑚)→cos(𝜉), and 𝑑𝑛(𝜉)→1. In this case, we mention 𝜓_3,7, 𝜓_3,9, 𝜓_3,12, and 𝜓_3,13 which degenerate into 𝜓_3,7,0=exp[𝑖(𝑘𝑥−𝜔𝑡)]tan𝑡−𝜆𝑥+𝜉₀,𝜓𝐴=−2,𝐵=−2,_3,9,0=exp[𝑖(𝑘𝑥−𝜔𝑡)]cot𝑡−𝜆𝑥+𝜉₀,𝜓𝐴=−2,𝐵=−2,_3,12,0=csc𝑡−𝜆𝑥+𝜉₀+cot𝑡−𝜆𝑥+𝜉₀[𝑖(1×exp𝑘𝑥−𝜔𝑡)],𝐴=−21,𝐵=−2,𝜓_3,13,0=sec𝑡−𝜆𝑥+𝜉₀−tan𝑡−𝜆𝑥+𝜉₀[𝑖(1×exp𝑘𝑥−𝜔𝑡)],𝐴=−21,𝐵=−2, (59) respectively, which are in agreement with the solutions 𝜓_1,6, 𝜓_1,7, 𝜓_1,9, and 𝜓_1,8. It is worth mentioning that 𝜓_3,4 and 𝜓_3,5 degenerate into the known solutions obtained by Liu (formulae (37) and (36) in [23]).

Remark 4. With the aid of Mathematica, we have checked the solutions 𝜓_𝑖,𝑗 (𝑖=1,2,3,𝑗=1,2,…) by putting them back into (1).

Remark 5. Compared with the work in [4, 23], we obtained more soliton solutions, triangle periodic solutions, and Jacobi elliptic solutions including some known solutions in [4, 23].

4. Discussion and Conclusion

In summary, we investigate the travelling wave solutions to nonlinear Schrödinger equation with a higher-order dispersive term by using three methods, namely, the auxiliary equation method, the first integral method, and the direct integral method. As a result, exact solutions including Jacobi elliptic function solution, the kink type envelope wave solution, and the bell type envelope wave solution are obtained. In particular, Jacobi elliptic function solutions degenerate into the hyperbolic function solutions and the triangular function solutions obtained via the auxiliary equation method and the first integral method. Because the nonlinear Schrödinger equation with a higher-order dispersive term is an important model in physics science, these results are expected to help understand wave dynamics in the study of ultrashort femtosecond pulses in optical fibres.

Acknowledgments

The authors would like to express their thanks to the referees for their useful comments which led to some improvements of the current paper. This work is supported in part by the National Natural Science Foundation of China (no. 11071141) and the Natural Science Foundation of Shandong Province of China (ZR2011AL018 and ZR2011AQ008).