Abstract
The complex Ginzburg-Landau model appears in the mathematical description of wave propagation in nonlinear optics. In this paper, the fractional complex Ginzburg-Landau model is investigated using the generalized exponential rational function method. The Kerr law and parabolic law are considered to discuss the nonlinearity of the proposed model. The fractional effects are also included using a novel local fractional derivative of order . Many novel solutions containing trigonometric functions, hyperbolic functions, and exponential functions are acquired using the generalized exponential rational function method. The 3D-surface graphs, 2D-contour graphs, density graphs, and 2D-line graphs of some retrieved solutions are plotted using Maple software. A variety of exact traveling wave solutions are reported including dark, bright, and kink soliton solutions. The nature of the optical solitons is demonstrated through the graphical representations of the acquired solutions for variation in the fractional order of derivative. It is hoped that the acquired solutions will aid in understanding the dynamics of the various physical phenomena and dynamical processes governed by the considered model.
1. Introduction
Nonlinear partial differential equations (NPDEs) govern the majority of physical phenomena and dynamical processes. The study of nonlinear wave propagation problems provides motivations for developing NPDEs. The most significant task in nonlinear science is to obtain solutions for NPDEs, particularly solitary and soliton wave solutions. Nonlinear natural and physical phenomena are better understood with the aid of solutions. In chemistry, mathematical physics, mathematical biology, and many other basic sciences, NPDEs are frequently used to model different phenomena and dynamics. The NPDEs such as the Lakshmanan-Porsezian-Daniel model [1, 2], Navier-Stokes equations [3], shallow water-like equation [4], resonant nonlinear Schrödinger equation [5], and Korteweg-de Vries equation [6] are used to transform numerous natural processes in mathematical form. Numerical simulations, mathematical expressions, and formulations are increasingly used in various problems of science in recent years [7–10].
In this paper, the complex Ginzburg-Landau (CGL) model can be expressed as where represents the profile of wave propagation and it depends upon the space variable and time variable . Moreover, and represent the group velocity dispersion and nonlinearity coefficient, respectively. The real parameters and are included to represent perturbation and detuning effects, respectively. Functional is essential for complex function in two-dimensional linear space . The -times continuously differentiable function is of the form
In scientific studies, fractional order derivatives [11–13] are very important and play an essential role in the investigation of NPDEs. Recently, the fractional CGL model is investigated by many researchers [14–16]. In this paper, the CGL model is investigated using the newly proposed definition of new local fractional derivative (NLFD) of order defined in [17] on the basis of the study of local fractional derivatives introduced in [18].
The purpose of this paper is to examine the fractional CGL model with newly proposed NLFD of order . A generalized exponential rational function (GERF) method is utilized to acquire the analytical solitary wave solutions of the fractional CGL equation. The proposed work involving the GERF method along with the local fractional derivative of order is novel which has not been done previously in literature.
The CGL model is examined using GERF along with the newly proposed local fractional differential operator for the first time in this work. A variety of novel interesting graphical observations are presented to exhibit the dynamical behavior of the CGL model. Bright, dark, and kink optical solitons are obtained which are graphically examined for the effect of variation in the value of the fractional parameter on the traveling wave solutions.
In the following subsection, a description of the NLFD is illustrated.
1.1. The Description of NLFD Order
In this subsection, the basic properties of NLFD of order as defined in [18] are presented as follows.
Definition 1. Consider , and define For with , if there exist , then In this case, is known as -differentiable at with respect to . One significant outcome of the new fractional local derivative is of the form where is differentiable with . The following property which has been proved in [17] is given as The chain rule is defined as Using Equations (6) and (7), it is obtained that where is constant.
The fractional derivative defined in Equation (3) is a novel generalization of the classical integer order derivative which exhibits interesting mathematical properties. The application of this definition is easier due to the chain rule. Moreover, the properties of NLFD make it suitable to evaluate higher order derivatives. These results are essential to extract new solitary wave solutions and soliton solutions of nonlinear evolution equations.
The rest of the research is structured as follows: the GERF method is explained in Section 2. The implementation of the proposed method is given in Section 3. 3D-surface plots, 2D-contour plots, density plots, and comparison of 2D-line plots are given in Section 4. Section 5 includes the conclusion of the research paper.
2. The Description of GERF Method
The GERF method is a relatively new approach to NPDEs. Many researchers utilized this method to extract the solutions of NPDEs [19–23]. The NPDE is considered as where represents the profile of wave propagation and it is dependent on space variable and temporal variable . Moreover, represents a polynomial in and partial derivatives of . Using fractional derivatives as illustrated in Subsection 1.1, Equation (9) is transformed into fractional form as where . The fractional transformations are considered as
Using fractional transformations from Equation (8), Equation (10) is reduced to the following ODE: where , is the polynomial of and its ordinary derivatives.
The general solution for Equation (12) is considered as where , , and are unknown parameters to be evaluated. The function is defined as where and are real or complex numbers . For different values of and , Equation (13) together with Equation (14) provides the different exact soliton solutions of Equation (12). Among these solutions, the nontrivial solutions are of significance for the determination of the nontrivial solutions of Equation (10) by backward substitution.
In formal solution Equation (13), can be determined by the homogeneous balance principle (HBP). There are some instances in which retains the negative or fractional values. The following substitutions are used in such cases:
By substituting the general solution Equation (13) together with Equation (14) into Equation (10) and coefficients of , taken to be zero, in result, the system of linear equations is acquired. The unknown parameters can be determined by resolving the retrieved system of linear equations simultaneously with the aid of any Maple software. By substituting nontrivial values of solutions to Equation (13), the solution of Equation (10) can be obtained.
3. Mathematical Analysis
Using the NLFD of order , NPDE (1) is transformed to the following form: where the time and space fractional derivatives are defined according to the definition of the novel local fractional derivative given by Equation (3) as
Equation (16) is the fractional form of CGL Equation (1) with the newly proposed definition of local fractional derivatives. This is obtained by replacing integer order derivatives with fractional order derivatives. The fractional order derivatives are more generalized than the classical integer order derivatives. This definition of fractional derivative is proposed and utilized to report several interesting and useful results in recent research works [17, 24].
The fractional transformations from Equation (8) are used to transform Equation (16) into ODE having the following real part:
The imaginary part is given as
The following substitution is considered:
Hence, Equation (18) is simplified to the following form:
Equation (19) yields
The solution of Equation (21) is obtained subject to condition (22) by considering the Kerr law nonlinearity in the following subsection.
3.1. Kerr Law Nonlinearity
Due to the propagation of light waves, optical fibers exhibit many definite forms of nonlinearities. One of these nonlinearities is Kerr law nonlinearity which is mathematically described as
The Kerr law of nonlinearity arises due to the nonlinear responses from the nonharmonic motion of electrons bound in molecules faced by a light wave in an optical fiber, under the influence of an external electric field. The Kerr nonlinearity is observed in most optical fibers. The nonlinearity of Equation (21) with the Kerr law is considered as
Using Equation (24), Equation (21) is transformed to the following form:
In Equation (13), formal solution is evaluated by HBL. Then the generalized solution of Equation (25) is transformed as
The formal solution of Equation (1) is written as where , , and are unknown constants to be determined. The solution of Equation (16) is investigated under the following cases.
Case 1. If , , Equation (14) is simplified to the following form:
Equation (26) with Equation (28) is substituted in Equation (25). By taking the coefficients of ’s zero, algebraic linear equations are acquired. By resolving these algebraic equations simultaneously, the following sets of solitary wave solutions are acquired as follows.
Set 1.
Set 2.
Set 3.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the solution of Equation (16) is determined as follows:
By putting the unknown constant values into Equation (27) from corresponding Set 2, the solution of Equation (16) is determined as follows:
By putting the unknown constant values into Equation (27) from corresponding Set 3, the solution of Equation (16) is determined as follows:
Case 2. If , , Equation (14) is reduced to the following form:
Equation (26) with Equation (35) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
Set 3.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the solution of Equation (16) is determined as
By putting the unknown constant values into Equation (27) from corresponding Set 2, the solution of Equation (16) is determined as
By putting the unknown constant values into Equation (27) from corresponding Set 3, the following result for the solution of Equation (16) is obtained:
Case 3. If , , Equation (14) is reduced to the form
Equation (26) with Equation (42) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the following solution of Equation (16) is obtained:
Case 4. If , , Equation (14) is reduced to the form
Equation (26) with Equation (45) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the result for the solution of Equation (16) is obtained as
By putting the unknown constant values into Equation (27) from corresponding Set 2, the following result is determined for the solution of Equation (16):
Case 5. According to this method, Equation (14) allows to select any complex or real values of , which can provide the nontrivial wave solutions of the considered equation. Therefore, with the values and , Equation (14) is taken which reduces Equation (14) to the following form:
Equation (26) with Equation (50) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the following result for the solution of Equation (16) is determined.
By putting the unknown constant values into Equation (27) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
Case 6. According to this method, Equation (14) allows to select any complex or real values of , which can provide the nontrivial wave solutions of the considered equation. Therefore, with the values and , Equation (14) is taken which reduces Equation (14) to the following form:
Equation (26) with Equation (55) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
Case 7. If , , Equation (14) is reduced to the following form:
Equation (26) with Equation (58) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (27) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
Case 8. If , , Equation (14) is reduced to the following form:
Equation (26) with Equation (63) is substituted in Equation (25). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
By putting the unknown constant values into Equation (27) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
3.2. Parabolic Law Nonlinearity
The parabolic law nonlinearity is considered as
Using Equation (66), Equation (21) is reduced to the following form:
The value of is extracted using the homogeneous balance principle. Since the value of is not a positive integer, the following substitution is used:
Hence, Equation (67) is simplified to the following form:
The positive integer value of is obtained using the homogeneous balance principle. The generalized solution Equation (69) is written as
The formal solution of Equation (16) is written as
In Equations (70) and (71), , , and are unknown constants to be determined. The solution of Equation (13) is investigated in the following cases.
Case 1. If , , Equation (14) is simplified to the following form:
Equation (70) with Equation (72) is substituted in Equation (69). The system of algebraic linear equations is acquired by taking the coefficients of ’s zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
Set 3.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 3, the following result for the solution of Equation (16) is determined:
Case 2. If , , Equation (14) reduces to the form
Equation (70) with Equation (79) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The system of algebraic linear equations is acquired by taking the coefficients of zero. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
Case 3. If , , Equation (14) reduces to the following form:
Equation (70) with Equation (82) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The following set of solutions are obtained by resolving the retrieved system of equations simultaneously with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
Case 4. If , , Equation (14) is reduced to the following form:
Equation (70) with Equation (87) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
Case 5. According to this method, Equation (14) allows to select any complex or real values of , which can provide the nontrivial wave solutions of considered equations. Therefore, and are taken which reduce Equation (14) to the following form:
Equation (70) with Equation (92) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
Case 6. According to this method, Equation (14) allows to select any complex or real values of , which can provide the nontrivial wave solutions of considered equations. Therefore, and are taken which reduce Equation (14) to the following form:
Equation (70) with Equation (97) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The following set of solutions are obtained by resolving the retrieved homogeneous equations at once with the aid of Maple software.
Set 1.
Set 2.
Set 3.
Set 4.
Set 5.
Set 6.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 3, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 4, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 5, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 6, the following result for the solution of Equation (16) is determined:
Case 7. If , , Equation (14) is reduced to the following form:
Equation (70) with Equation (110) is substituted in Equation (69). By putting the accumulated terms of different powers of ’s to zero, homogeneous linear equations are acquired. The following set of solutions are obtained by resolving the retrieved system of equations simultaneously with the aid of Maple software.
Set 1.
Set 2.
By putting the unknown constant values into Equation (70) from corresponding Set 1, the following result for the solution of Equation (16) is determined:
By putting the unknown constant values into Equation (70) from corresponding Set 2, the following result for the solution of Equation (16) is determined:
4. Graphical Explanation
This section presents graphical representations of some obtained solutions. The 3D-surface graphs, 2D-contour graphs, density plots, and comparison of 2D-line graphs of retrieved solutions are plotted using Maple software. In plotted graphs, (a), (b), (c), and (d) represent the D-surface graphs, D-contour graphs, density plots, and comparison of D-line graphs, respectively. The graphs are plotted for absolute values of obtained solutions. Using Maple software, both values cannot be plotted simultaneously; therefore, plotting graphs only requires or instead of . The suitable numeric values are assigned to undetermined constants , and fractional derivative order , to generate the well-shaped graphs of obtained solutions. From the corresponding sets of obtained solutions, the values of , , , , , and can be taken for plotting the graphs of acquired solutions.
The acquired solutions are graphically presented in Figures 1–12. The graphical representations include dark, bright, kink, and solitary wave solutions. The graphs exhibit the effect of variation in the fractional parameter on the obtained solutions. The evolution of a dark soliton is shown corresponding to the solution through Figures 1–3 for increasing values of fractional parameter . Figures 1–3 show the graphical simulations for , and , respectively. Graphical simulations for the solution are presented in Figures 4–6. The evolution of a kink soliton is illustrated from Figures 7–9, which has been obtained corresponding to the solution . It is evident that the amplitude of the wave varies from one asymptotic state to another in this case. The bright soliton solution is shown in Figures 9–12 for . The bright soliton is characterized by a local amplitude increase in the wave profile.
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5. Conclusion
In this paper, the complex Ginzburg-Landau model is investigated using novel local fractional derivatives of order . CGL model solutions are acquired using the GERF method. Two laws of nonlinearities, Kerr law and parabolic law, are used to examine the nonlinearity of the proposed model. Many solitary wave solutions are obtained using the proposed method for the CGL model. The trigonometric functions , hyperbolic functions , and exponential functions are included in obtained soliton solutions of the CGL model. Many soliton solutions, such as dark, bright, kink, and many other solitary wave solutions, are extracted using the proposed method. The 3D-surface graphs, 2D-contour graphs, density plots, and comparison 2D-line graphs are included to represent the dynamical behavior of retrieved solutions. The results presented in this paper are novel as compared to the already reported results for the CGL model. The authors of [25] presented the solutions of a CGL model using an iterative technique which was successfully utilized to retrieve the approximate solutions corresponding to specific initial conditions. The dark-singular soliton solutions of the CGL model were constructed in [26], but no other traveling wave solutions were investigated. The CGL model with M-truncated, beta, and conformable derivatives was theoretically examined using different mathematical techniques in previous works [11, 14, 27]. However, this manuscript utilizes a newly proposed local fractional derivative along with the GERF method to report a variety of explicit exact closed form traveling wave solutions of the CGL model in terms of free parameters. Hence, the comparison of the obtained results for the already reported studies of the CGL model confirm the novelty and usefulness of the results presented in this paper [11, 14, 25–27]. The reported results are useful to understand the possible dynamical behaviors exhibited by the CGL model. The soliton solutions will be useful additions to the literature for understanding related physical systems. In the future, the fractional CGL model can also be investigated by other nonlinearity laws and exact methods.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.