Application of the 𝐺^′/𝐺 Expansion Method in Ultrashort Pulses in Nonlinear Optical Fibers

Abstract

With the increasing input power in optical fibers, the dispersion problem is becoming a severe restriction on wavelength division multiplexing (WDM). With the aid of solitons, in which the shape and speed can remain constant during propagation, it is expected that the transmission of nonlinear ultrashort pulses in optical fibers can effectively control the dispersion. The propagation of a nonlinear ultrashort laser pulse in an optical fiber, which fits the high-order nonlinear Schrödinger equation (NLSE), has been solved using the 𝐺^′/𝐺 expansion method. Group velocity dispersion, self-phase modulation, the fourth-order dispersion, and the fifth-order nonlinearity of the high-order NLSE were taken into consideration. A series of solutions has been obtained such as the solitary wave solutions of kink, inverse kink, the tangent trigonometric function, and the cotangent trigonometric function. The results have shown that the 𝐺^′/𝐺 expansion method is an effective way to obtain the exact solutions for the high-order NLSE, and it provides a theoretical basis for the transmission of ultrashort pulses in nonlinear optical fibers.

1. Introduction

It is understood that a soliton is excited by a nonlinear field, and its energy is relatively concentrated in a small area. The elastic scattering phenomenon occurs during the interaction between two solitons. The energy is nondispersed, so that the shape and speed can remain unchanged during its propagation. The history of the studies of solitons can be traced back to 1834, when James Scott Russell, a British scientist, accidentally observed that a bulge of water in Edinburgh-Glasgow Canal was propagating undistorted over several kilometers [1]. In 1973, Hasegawa and Tappert [2] first proposed the idea of applying “optical solitons” to photonic communication. After rigorous theoretical and mathematical deduction, he predicted that both bright and dark soliton pulses were present in an optical fiber. He also proved that any nondestructive optical pulse could travel as stable as a soliton during its transmission in an optical fiber. As an intrinsically nonlinear phenomenon, a soliton is the product of the optical fiber dispersion and the nonlinear interaction. It obeys the NLSE and is controlled by the optical fiber dispersion and the nonlinear effects of self-phase modulation. In 1980, bright solitons were observed for the first time in an optical fiber by Mollenaure [3] and his colleagues during their experiments at the Bell Laboratory of the United States. In 1987, Emplit observed dark solitons in a fiber by using an amplitude technique and a phase filtering technique. In mathematics, the progress of the soliton theory has been achieved mainly by finding a large number of soliton solutions for nonlinear partial differential equations, so as to gradually establish a systematic theory of solitons. The special properties of a soliton, the related mathematical description of its rich structure, and the deep physical origins have profoundly expanded the applications of soliton theory and methods. The study of the exact solutions for the NLSEs has become an important research and development direction. The 𝐺^′/𝐺 expansion method [4], one of the methods for obtaining exact solutions for NLSE, has attracted much attention recently [4–6]. In particular, it plays an important role in deriving the propagation wave solution of NLSE. Therefore, a good understanding of the 𝐺^′/𝐺 expansion method is of great significance to the transmission application of ultrashort pulses in nonlinear optical fibers. This paper presents the method in detail with examples for obtaining multiple soliton solutions.

2. 𝐺^′/𝐺 Expansion Method

The nonlinear equation with two variables 𝑥 and 𝑡 could be written as 𝑃𝑢(𝑥,𝑡),𝜕𝑢,𝜕𝑥𝜕𝑢,𝜕𝜕𝑡²𝑢𝜕𝑥²,𝜕²𝑢,𝜕𝜕𝑥𝜕𝑡²𝑢𝜕𝑡²,…=0. (1)

The first step of the 𝐺^′/𝐺 expansion method is to assume a variable 𝜉=𝑥−𝑣𝑡; thus, 𝑢=𝑢(𝑥,𝑡)=𝑢(𝜉); it can then be expressed as a 𝐺^′/𝐺 power polynomial: 𝑢(𝜉)=𝛼_𝑚𝐺^′𝐺^𝑚+𝛼_𝑚−1𝐺^′𝐺^𝑚−1+⋯+𝛼₀, (2) in which 𝐺=𝐺(𝜉) satisfies the second-order linear ordinary differential equation: 𝐺^′′+𝜆𝐺^′+𝜇𝐺=0, (3) where 𝛼_𝑚,𝛼_𝑚−1,…,𝛼₀,𝜆,𝜇 in (2) and (3) are undetermined parameters, in which, 𝛼_𝑚≠0, and the general solution of (3) is expressed as 𝐺(𝜉)=𝐴e^{(−𝜆+𝜆2−4𝜇)𝜉/2}+𝐵e^{(−𝜆−𝜆2−4𝜇)𝜉/2}. (4)

The derivative of 𝐺(𝜉) can be derived from (4): 𝐺^′=−𝜆2e^{(−𝜆/2)𝜉}×(𝐴+𝐵)cosh𝜆²−4𝜇2𝜉+(𝐴−𝐵)sinh𝜆²−4𝜇2𝜉+e^{(−𝜆/2)𝜉}(𝐴+𝐵)sinh𝜆²−4𝜇2𝜉+(𝐴−𝐵)cosh𝜆²−4𝜇2𝜉×𝜆²−4𝜇2. (5)

The ratio of 𝐺(𝜉) and 𝐺^′(𝜉) is expressed as [6] 𝐺^′𝐺𝜆=−2+𝜆²−4𝜇2×(𝐴+𝐵)sinh𝜆²−4𝜇/2𝜉+(𝐴−𝐵)cosh𝜆²−4𝜇/2𝜉(𝐴+𝐵)cosh𝜆²−4𝜇/2𝜉+(𝐴−𝐵)sinh𝜆²𝜉.−4𝜇/2 (6)

3. Applying the𝐺^′/𝐺Expansion Method to Solve NLSE for Ultrashort Pulses in Optical Fibers

Many problems in the study of natural science can be related to nonlinear evolution equations, and the optical soliton is one of them. It is always problematic to find soliton solutions for nonlinear evolution equations by physicists and mathematicians. With the development of modern computers, many methods have been developed, such as the homogeneous balance method and the hyperbolic function method [7–9]. This research uses the 𝐺^′/𝐺 expansion method to solve the high-order NLSE.

The generalized NLSE is given by ^∞_𝑙=0i^𝑙𝛼_𝑙𝜕^𝑙𝜕𝑥^𝑙𝜓+^∞_𝑚=0i^𝑚𝛽_𝑚𝜕^𝑚𝑚!𝜕𝑡^𝑚𝜓+^∞_𝑛=0𝛾_2𝑛𝜓^2𝑛𝜓=0. (7) Considering the case when 𝑙=1, 𝑚=2,3,4, and 𝑛=1,2, the corresponding coefficients are nonzero and the rest of the coefficients are zero. Assuming the coefficient 𝛼₁=1 and allowing the variable 𝑥 to be replaced by 𝑧 and the function 𝜓 to be replaced by 𝑞, the high-order NLSE (with items included, namely, the group velocity dispersion, the self-phase modulation effect, the fourth-order dispersion, and the fifth-order nonlinear term) can then be expressed as i𝜕𝑞−𝛽𝜕𝑧2²𝜕²𝑞𝜕𝑡²+𝛾₂𝑞²𝑞=i𝛽₃6𝜕³𝑞𝜕𝑡³+𝛽₄𝜕24⁴𝑞𝜕𝑡⁴−𝛾₄𝑞⁴𝑞, (8) in which 𝛽₂, 𝛽₃, and 𝛽₄ are the second-, third-, and fourth-order dispersion coefficients and 𝛾₂ and 𝛾₄ are the third- and fifth-order nonlinear coefficients. The second-order dispersion coefficient 𝛽₂ is also known as the group velocity dispersion coefficient. With increasing optical power in a fiber, it is necessary to take the third-order nonlinear term into account, which represents the phase equilibrium between the group velocity dispersion and the nonlinear effect. This will allow the maintaining of unchanged properties for soliton pulses in an optical fiber, which is necessary for long distance transmission, and is also beneficial to WDM.

Other methods, such as the extended tanh-function method, the subsidiary ordinary differential method, and the extended hyperbolic auxiliary equation method, have been used in order to find the exact solutions for (8) [10–12]. It has been found that there is no solution for (8) when 𝛽₃−𝛽₄𝜈≠0. In order to obtain the exact solution, 𝜈=𝛽₃/𝛽₄ is assumed and (8) is then converted to a real equation after its real part and imaginary part are separated. The 𝐺^′/𝐺 expansion method is then used for solving the equation.

3.1. Reduction Travelling Wave

Assume the form of the exact solution is 𝑞(𝑧,𝑡)=𝑢(𝜉)e^{i(𝐿𝑧−𝑣𝑡)},𝜉=𝑘𝑧−𝜔𝑡. (9)

The parameters of 𝑘,𝐿,𝜔,𝑣 are undetermined parameters. A series of expressions can be obtained based on (9). The key equation (10) can be obtained after the real part and the imaginary part of (8) are separated: 𝛽₄𝜔⁴𝑢^′′′′+6𝛽₃𝜔²𝛽₃𝛽₄+12𝛽₂𝜔²𝑢^′′+−12𝛽₂𝛽²₃𝛽²₄𝛽+24𝐿−3⁴³𝛽³₄𝑢−24𝛾₂𝑢³−24𝛾₄𝑢⁵=0. (10)

3.2. The Exact Solution Obtained Using the 𝐺^′/𝐺 Expansion Method

The 𝐺^′/𝐺 expansion method was proposed for the exact solutions by Wang et al. [4]. The method is effective to solve the exact solutions for the nonlinear partial differential equation.

The form of the exact solution for (10) is 𝑢(𝜉)=^𝑛_𝑖=0𝛼_𝑖𝐺^′𝐺^𝑖, (11) where 𝐺=𝐺(𝜉) and it satisfies (3).

The result of 𝑛=1 can be obtained by balancing the highest derivative 𝑢^′′′′ and the highest derivative with high nonlinearity 𝑢⁵ in (10). The general expression for (10) can be expressed as 𝑢(𝜉)=𝛼₁𝐺^′𝐺+𝛼₀. (12)

Based on (12) and (10), and assuming the coefficients of (𝐺^′/𝐺)^𝑖, 𝑖=0,1,2,3,4,5 are zero, a set of six algebraic equations can be obtained: 𝛼₁𝛽₄𝜔⁴−𝛾₄𝛼⁴₁=0,𝛼₁𝛽₄𝜔⁴𝜆−2𝛾₄𝛼³₁𝛼₀=0,𝛼₁𝛽₄𝜔⁴25𝜆²+20𝜇+6𝛽₃𝜔²𝛽₃𝛽₄+12𝛽₂𝜔²−12𝛾₂𝛼²₁−120𝛾₄𝛼²₁𝛼²₀=0,𝛼₁𝛽₄𝜔⁴5𝜆³+20𝜆𝜇+6𝛽₃𝜔²𝛽₃𝛽₄+12𝛽₂𝜔²𝜆−24𝛾₂𝛼₁𝛼₀−80𝛾₄𝛼₁𝛼³₀=0,𝛼₁𝛽₄𝜔⁴𝜆⁴+22𝜆²𝜇+16𝜇²+6𝛽₃𝜔²𝛽₃𝛽₄+12𝛽₂𝜔²𝜆²−+2𝜇12𝛽₂𝛽²₃𝛽²₄𝛽−24𝐿+3⁴³𝛽³₄−72𝛾₂𝛼²₀−120𝛾₄𝛼⁴₀=0,𝛽₄𝜔⁴𝜇𝜆³+8𝜆𝜇²𝛼₁+6𝛽₃𝜔²𝛽₃𝛽₄+12𝛽₂𝜔²𝜆𝜇𝛼₁+−12𝛽₂𝛽²₃𝛽²₄𝛽+24𝐿−3⁴³𝛽³₄𝛼₀−24𝛾₂𝛼³₀−24𝛾₄𝛼⁵₀=0. (13)(14)(15)(16)(17)(18)

The parameter 𝛼₁ can be obtained from (13): 𝛼₁=⁴𝛽₄𝛾₄𝜔. (19)

It was mistakenly written as 𝛼₁=⁵(𝛽₄/𝛾₄)𝜔 in [6].

3.2.1. The First Type Exact Solution

From (14) when 𝜆=0, the result is 𝛼₀=0: 3𝜇=10𝛽₄𝜔³2𝛾₂𝛽₄𝛾₄−𝛽²₃𝛽₄+2𝛽₂𝜔,𝐿=2𝛾₂𝛽₄𝛾₄−𝛽²₃𝛽₄+2𝛽₂𝜔×−3𝛾₂25𝛽₄𝜔²𝛽₄𝛾₄+350𝛽₄𝜔𝛽²₃𝛽₄+3𝛽₂25𝛽₄𝜔−3𝛽20𝜔𝛽³₄²−3𝛽₂10𝛽₄𝜔+12𝛽₂𝛽²₃𝛽²₄+18𝛽⁴₃𝛽³₄. (20)

The exact solution of (21) based on (6) is 𝑞₁=⁴𝛽₄𝛾₄𝜔√⋅()−𝜇𝐴+𝐵sinh−4𝜇/2𝜉+(𝐴−𝐵)cosh−4𝜇/2𝜉(𝐴+𝐵)cosh−4𝜇/2𝜉+(𝐴−𝐵)sinh𝜉×−4𝜇/2e^{i(𝐿𝑧−𝑣𝑡)}. (21)

Consider the following.(a)When 𝜇<0 and 𝐴=𝐵=1, the kink wave solution is 𝑞₁₁=⁴𝛽₄𝛾₄𝜔√−𝜇tanh−4𝜇2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (22)(b)When 𝜇<0 and 𝐴=−𝐵=1, the inverse kink wave solution is 𝑞₁₂=⁴𝛽₄𝛾₄𝜔√−𝜇coth−4𝜇2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (23)(c)When 𝜇≥0 and 𝐴=𝐵=1, the solitary wave solution of the tangent trigonometric function is 𝑞₁₃=−𝜔⁴𝜇𝛽₄𝛾₄tan√𝜇𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (24)(d)When 𝜇≥0 and 𝐴=−𝐵=1, the solitary wave solution of the cotangent trigonometric function is 𝑞₁₄=⁴𝜇𝛽₄𝛾₄𝜔cot√𝜇𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (25)

3.2.2. The Second Type Exact Solution

From (14) when 𝜆≠0, the result is 𝛼₁=⁴(𝛽₄/𝛾₄)𝜔, 𝛼₀=(𝛽₄𝜔⁴𝜆/2𝛾₄𝛼³₁)=(𝜆/2)⁴(𝛽₄/𝛾₄)𝜔=(𝜆/2)𝛼₁, 3𝜇=−𝛽10𝛽³₄𝜔²−35𝛽₂𝛽₄𝜔²+3𝛾₂5𝜔²1𝛽₄𝛾₄+𝜆²4;𝐿=𝛽₂𝛽²₃2𝛽²₄+𝛽⁴₃8𝛽³₄+𝛾₂3𝜆²4𝜔²𝛽₄𝛾₄𝜆+5⁴𝛽16⁴−𝛽₄𝜔⁴𝜆⁴−1244𝛽₃𝜔²𝛽₃𝛽₄𝜆²−12𝛽₂𝜔²𝜆²−11𝛽12⁴𝜔⁴𝜆²+12𝛽₃𝜔²𝛽₃𝛽₄+𝛽₂𝜔²𝜇−23𝛽₄𝜔⁴𝜇². (26)(27)

The exact solution of (26) based on (6) is 𝑞₂=⁴𝛽₄𝛾₄𝜔×−𝜆2+𝜆²−4𝜇2⋅(𝐴+𝐵)sinh𝜆²−4𝜇2𝜉+(𝐴−𝐵)cosh𝜆²−4𝜇2𝜉×(𝐴+𝐵)cosh𝜆²−4𝜇2𝜉+(𝐴−𝐵)sinh𝜆²−4𝜇2𝜉⁻¹+𝜆2⁴𝛽₄𝛾₄𝜔e^{i(𝐿𝑧−𝑣𝑡)}. (28)

The item of −(𝜆/2) in [6] was lost and the item of 𝛼₀ was not compensated.

Consider the following.(a)When 𝜆²−4𝜇≥0 and 𝐴=𝐵=1, the kink wave solution is 𝑞₂₁=⁴𝛽₄𝛾₄𝜔𝜆²−4𝜇2tanh𝜆²−4𝜇2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (29)(b)When 𝜆²−4𝜇≥0 and 𝐴=−𝐵=1, the inverse kink wave solution is 𝑞₂₂=⁴𝛽₄𝛾₄𝜔𝜆²−4𝜇2coth𝜆²−4𝜇2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (30)(c)When 𝜆²−4𝜇<0 and 𝐴=𝐵=1, the solitary wave solution of the tangent trigonometric function is 𝑞₂₃=−𝜔⁴𝛽₄𝛾₄4𝜇−𝜆²2tan4𝜇−𝜆²2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (31)(d)When 𝜆²−4𝜇<0 and 𝐴=−𝐵=1, the solitary wave solution of the cotangent trigonometric function is 𝑞₂₄=⁴𝛽₄𝛾₄𝜔4𝜇−𝜆²2cot4𝜇−𝜆²2𝜉e^{i(𝐿𝑧−𝑣𝑡)}. (32)

The application of the feature equations can be found in [13–17].

4. Conclusions

A 𝐺^′/𝐺 expansion method was applied to obtain the exact solutions for NLSE which is the governing equation for ultrashort pulse transmission in a nonlinear fiber. The items of the group velocity dispersion, self-phase modulation effect, the fourth-order dispersion, and the fifth-order nonlinearity were considered. Two kinds of the kink wave solution, inverse kink wave solutions, the solitary wave solution of the tangent trigonometric function, and the solitary wave solution of the cotangent trigonometric function were found by using the 𝐺^′/𝐺 expansion method.

These results have important significance in optical fiber communication and its applications.

Acknowledgments

This work is supported by Open Issues of Jiangsu Key Lab for Solar Cell materials and technology, Changzhou University (201207). The authors acknowledge the support from the open issues of State Key laboratory of Satellite Ocean Environment Dynamics (no. SOED1201).