Abstract

In this study, the exact traveling wave solutions of the time fractional complex Ginzburg-Landau equation with the Kerr law and dual-power law nonlinearity are studied. The nonlinear fractional partial differential equations are converted to a nonlinear ordinary differential equation via a traveling wave transformation in the sense of conformable fractional derivatives. A range of solutions, which include hyperbolic function solutions, trigonometric function solutions, and rational function solutions, is derived by utilizing the new extended -expansion method. By selecting appropriate parameters of the solutions, numerical simulations are presented to explain further the propagation of optical pulses in optic fibers.

1. Introduction

The Ginzburg-Landau (GL) equation [1–7] is one of the most important partial differential equations in the field of mathematics and physics, which was introduced into the study of superconductivity phenomenology theory in the 20th century by Ginzburg and Landau. The GL equation usually describes the optical soliton [8–15] propagation through optical fibers over longer distances. Therefore, it is very important to analyze the dynamic behavior of the GL equation. In particular, searching for the approximate analytical solutions or exact traveling solutions of the GL equation has been more and more widely followed with interest because they can help to explain the complexity of nonlinear optics. So, in recent years, many powerful methods were used for finding the exact traveling solutions of the GL equation, which include the first integral method [16], dynamical system method [17], -expansion method [18], Hirota’s method [19], and harmonic balance method [20].

In recent years, the fractional GL equation [21–24] has been more and more widely followed with interest because it can be used to accurately model some nonlinear optical phenomena. In particular, constructing the exact traveling solutions of the fractional GL equation is very important work because it can be better explained the dynamics of soliton propagation through optical fibers over longer distances in nonlinear optics. So far, many important methods have been proposed to find the exact solutions of the fractional GL equation. In [25], Arshed extracted the hyperbolic, trigonometric, and rational function solutions of the fractional GL equation by the -expansion method. In [26], Raza obtained the periodic and hyperbolic soliton solutions of the conformable time fractional GL equation by using the solitary wave ansatz method and -expansion method. In [27], Abdou et al. employed Jacobi’s elliptic function expansion approach to retrieve doubly periodic function solutions for the time-space fractional GL equation. In [28], Sirisubtawee et al. investigated the cubic-quintic GL equation by the modified Kudryashov method and -expansion method. In [29], Ghanbari and Gómez-Aguilar obtained the periodic and hyperbolic soliton solutions of the conformable GL equation by using the generalized exponential rational function method. Although exact traveling wave solutions of the fractional GL equation have been successfully obtained by utilizing the above methods, it is far from enough on searching for the exact solution of a more general fractional complex GL equation. The main purpose of this paper is to construct exact traveling wave solutions of the time fractional complex GL equation with the Kerr law and dual-power law nonlinearity by using the new extended -expansion method, and a range of solutions which include hyperbolic function solutions, trigonometric function solutions, rational function solutions, and negative power solutions is derived.

In this paper, we will consider the following time fractional complex GL equation [26, 29]: where represents distance across the fiber in dimensionless form. and denote the coefficient of the group velocity dispersion and the coefficient of nonlinearity. The terms with constants and arise from the perturbation effects. In particular, comes from the detuning effect. , , , , and are real-valued constants. is the complex conjugate numbers of . Equation (1) is used to model various physical phenomena, such as nonlinear waves, Bose-Einstein condensation, second-order phase transitions, superconductivity, superfluidity, liquid crystals, and strings.

In Equation (1), is a real-valued algebraic function. Let be times continuously differentiable. Then, where is a complex plane of two-dimensional linear space in .

The rest of this paper is organized as follows. In Section 2, we will give the definition of the conformable fractional derivative and its properties. In Section 3, we will introduce the extended -expansion method. In Section 4, we will discuss the exact traveling solutions of the fractional complex GL equation with the Kerr law and dual-power law nonlinearity by using the extended -expansion method and plot the profiles of several representative exact traveling solutions. In Section 5, we will present the concluding remarks.

2. Preliminaries

Definition 1 [28]. Let . Then, the conformable fractional derivative of of order is defined as

If the limit in Equation (3) exists, then we say that is -conformable differentiable at a point .

Theorem 2 [28]. Let and and be -conformable differentiable at a point . Then,

Theorem 3 [28]. Let be a function such that is differentiable and -conformable differentiable. Further, let be a differentiable function defined in the range of . Then, where the prime notation denotes the classical derivative.

3. Overview of the Extended -Expansion Method

Let us consider a general fractional partial differential equation in the form where is an unknown function and is a polynomial of and its partial fractional order derivatives, in which the highest order derivatives and nonlinear terms are involved.

Step 1. By using the traveling wave transformation , where , is a nonzero arbitrary constant, , we can rewrite Equation (6) as the following nonlinear ordinary differential equation (ODE) of the integer order with respect to :

If it is possible, we should integrate Equation (7) term by term one or more times; the constants of integration are considered to be zero.

Step 2. Assume that the traveling wave solution of Equation (7) can be expressed as follows: where either or may be zero, but both and cannot be zero at a time, and and are arbitrary constants, where satisfies the following nonlinear ODE: where , , and are real parameters.

Step 3. The value of the positive integer in Equation (8) can be computed by using the homogeneous balance between the highest order nonlinear term and the highest order derivative in appearing in Equation (8).

Step 4. Substituting Equations (8) and (9) into Equation (7) with the value of obtained in Step 3, we obtain polynomials in and . Then, we collect all coefficients of the resulted polynomial to zero, and we obtain a set of algebraic equations for and .

Step 5. By solving the algebraic equations obtained in Step 4, we can obtain the values of the constants and . Replacing the values into Equation (8), we construct a more general type and new exact traveling wave solutions of Equation (3).

The solutions of Equation (9) can be categorized into the following three cases.

Case 1 (hyperbolic function solutions). When and , then

Case 2 (trigonometric function solutions). When and , then

Case 3 (rational function solutions). When and , then

Remark 4. In fact, by taking , , and in Equation (9), the second nonlinear ODE reduces into the following second-order linear ODE:

4. Exact Solutions of the Time Fractional Complex Ginzburg-Landau Equation

In order to obtain the traveling wave solution of Equation (1), we assume that the exact solution has the following form: where and .

Substituting (14) into Equation (1), then collecting the real and imaginary parts, we get the following equations: where .

By taking , Equation (15) becomes

4.1. Kerr Law Nonlinearity

The Kerr law nonlinearity is the case when ; then, Equation (17) reduces to

By using the homogeneous balance principle, we select . Then, we suppose that the solution of Equation (18) satisfies where , , and are constants to be determined.

Substituting (19) together with (9) into (18) yields a polynomial in () and (). Collecting the coefficient of the resulted polynomial and setting them to zero, we obtain a nonlinear algebraic equation in , , and . Solving the algebraic equation using the computer algebraic software of Maple, we get the following results:

Case 1. , , , , and .

Case 2. , , , , and , where , , , , , , and are free parameters.

Substituting these solutions into (19), using (14), we obtain the traveling wave solution as follows: where is defined in Section 3, and .

Family 1. The hyperbolic function solutions can be obtained as and . Then, where , and and are arbitrary constants.

In particular, if we assume that , , , and in (21), then we have and if , , , and in (21), then we obtain

The fixed values are , , , , , , and , and the fractional parameter is used to plot the positive real part of the exact solution as in Figures 1 and 2.

Family 2. The trigonometric function solutions can be obtained as and . Then, where , and and are arbitrary constants.

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(b)

(c)

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Figure 1 

The positive real part of in Equation (23) for differential values of the fractional parameter .

Figure 2 

The positive real part of at in Equation (23) for differential values of the fractional parameter

Specifically, if we suppose that , , , and in (25), then we obtain and if , , , and in (25), then we have

The fixed values are , , , , , , and , and the fractional parameter is used to plot the positive real part of the exact solution as in Figures 3 and 4.

Family 3. The rational function solutions can be obtained as and . Then, where and are arbitrary constants.