Abstract
Here, two applicable methods, namely, the -expansion technique and modified -expansion technique are being applied on the time-fractional coupled Jaulent–Miodek equation. Materials such as photovoltaic-photorefractive, polymer, and organic contain spatial solitons and optical nonlinearities, which can be identified by seeking from energy-dependent Schrödinger potential. Plentiful exact traveling wave solutions containing unknown values are constructed in the sense of trigonometric, hyperbolic, exponential, and rational functions. Different arbitrary constants acquired in the solutions help us to discuss the dynamical behavior of solutions. Moreover, the graphical representation of solutions is shown vigorously in order to visualize the behavior of the solutions acquired for the mentioned equation. We obtain some periodic, dark soliton, and singular-kink wave solutions which have considerably fortified the existing literature on the time-fractional coupled Jaulent–Miodek equation. Via three-dimensional plot, density plot, and two-dimensional plot by utilizing Maple software, the physical properties of these waves are explained very well.
1. Introduction
The main idea of this article focused on how we can get the new exact and numerical solutions for time-fractional coupled Jaulent–Miodek equation by using the new analytical methods, that is to say, the -expansion method and the modified -expansion method. Several authors studied the various nonlinear models through two different methods which may be applied to recent results such as the Biswas–Milovic equation for Kerr law nonlinearity [1], the Kerr law and dual power law Schrödinger equations [2], the fractional Bogoyavlenskii equations with conformable derivative [3], the Vakhnenko–Parkes equation [4], the Caudrey–Dodd–Gibbon and Pochhammer–Chree equations [5], some nonlinear fractional physical model [6], the time-fractional Kuramoto–Sivashinsky equation [7], new double-chain model of DNA, and a diffusive predator-prey system [8].
Authors usually use the fractional differential equations to explain a few attractive physical phenomena. Many terms we use naturedly or randomly in daily life often have a fractional structure. Verbal and numerical expressions that we use when describing something, explaining an event, and commanding and in many other cases contain behavior physical of fractional equations with wide range of nonlinear physical phenomena in the vast areas of scientific disciplines which are explained by nonlinear evolution equations. Owing to the importance of studying this class of equations, several distinct techniques have been proposed such as the fractional Davey–Stewartson equation with power law nonlinearity [9], the Calogero–Bogoyavlenskii–Schiff equation in (2 + 1) dimensions with time-fractional conformable derivative [10], the fractional mitigating Internet bottleneck with quadratic cubic nonlinearity [11], the density-dependent conformable fractional diffusion-reaction equation [12], the space-time fractional Klein–Gordon equation with symmetry analysis [13], and the space-time fractional advection-diffusion equation with convergence analysis [14].
The further analytical methods for the (2 + 1)-dimensional Jaulent–Miodek (JM) evolution equation have been well scrutinized by a lot of researchers containing the results such as the Hirota’s bilinear method [15], the exp-function method [16], the symmetry reductions method [17], the homotopy perturbation method [18], the optimal hidden symmetries [19], and the related topics of the Jaulent–Miodek equation with various topics, e.g., the (G′/G)-expansion method [20], the bifurcation and exact traveling wave solutions [21], the integrating factors method in an unbounded domain [22], the Adomian’s decomposition method [23], the finite-band solution method [24], N-fold Darboux transformation method [25], the Hermite wavelets method [26], and the modified Riemann–Liouville derivative and exterior derivatives [27]. In the following, we take the nonlinear time-fractional coupled Jaulent–Miodek (FCJM) equation, firstly introduced by Jaulent and Miodek [27–29], that is,
Gupta and Ray [30] used the wavelet method based on the Hermite wavelet expansion and found the numerical solution to a coupled system of nonlinear time-fractional Jaulent–Miodek equations. The Lie symmetry analysis has been performed on a coupled system of nonlinear time-fractional Jaulent–Miodek equations associated with energy-dependent Schrödinger potential to find the exact solution using Erdelyi–Kober fractional derivatives [31]. Moreover, the authors of [32–34] obtained new exact solutions for some of PDEs using the Hirota bilinear method.
The foremost objective in this article is discovering some periodic, soliton, singular, and singular-kink solutions by using two methods, namely, -expansion method and modified -expansion method [35], for the time-fractional coupled JM equations. In [36], Boulkhemair et al. used geometrical variations of a state-constrained functional on star-shaped domains. Nachaoui et al. investigated parallel numerical computation of an analytical method for solving an inverse problem in [37]. Use the modified Riemann–Liouville derivative of order [38] as follows:with the below relations [39–41]where denotes the gamma function.
In numerous scientific and engineering problems, to find exact analytical solutions of nonlinear partial differential equations (NLPDEs) for understanding many physical phenomena has great significance. So, establishing explicit traveling wave solutions of many physical systems has attracted considerable attention in the last few decades. Fortunately, in the literature, most of these methods are based on the hypothesis that the exact solution can be expressed as finite expansion of a function and use the following methodology in general to determine it. The method is efficiently used for several kinds of differential equations with respect to subsidiary conditions, types, etc. It includes initial value problems, boundary value problem, partial and ordinary differential equations, linear and nonlinear equations, and also nonlinear stochastic system as well as deterministic differential equations. In contrast to previously defined methods, this method extends more solutions containing periodic, hyperbolic, rational, kink, singular, and singular-kink solutions. In the continuation, we utilize concepts of local fractional calculus and its properties in Section 2. By utilizing of the symbolic calculation and employing the used TEM and MEEM, we want to solve the CFJM system in Sections 3 and 4. As a consequence, we plot some periodic, soliton, singular, and singular-kink solutions in Section 5. At the end, some concluding remarks about the obtained results are given.
2. Description of Local Fractional Calculus and Its Properties
Definition 1. (see [42–44]). Assume that is defined throughout some interval containing and all points near , then is said to be local fractional continuous at , symbolized by ; then to each positive number and also to a positive constant corresponding some positive , we havewhenever , and . Therefore, the function is called local fractional continuous on the ,in which is the fractal dimension.
Definition 2. (see [42–44]). A function is called to be nondifferentiable function of exponent which satisfies Hölder function of exponent , then for , one gets
Definition 3. (see [42–44]). A function is said to be local fractional continuous of order , or in short, -local fractional continuous, when we havefor any and .
Definition 4. (see [42–44]). A function is said to be in the space if and only if it can be expressed asfor any and .
Definition 5. (see [42–44]). Get . Hence, the local fractional derivative of having order at is defined aswhere .
Remark 1. (see [42, 4, 45, 46]). More important properties of the local fractional derivative are as follows:
Remark 2. (see [42–47]). Moreover, the chain rule of the local fractional derivative is in the following:when and exist.
3. Summarization of the -Expansion Method
In this section, we will describe the main steps of the -expansion method for finding traveling wave solutions of the time-fractional coupled Jaulent–Miodek equation in order to furnish its exact solutions: Step 1. Given a nonlinear physical model governed by partial differential equation, where fractional derivative is explained in local sense and is a polynomial in and its different partial derivatives can be converted into an ordinary differential equation (ODE), by utilizing the suitable fractional complex change where and are the unknown values which would be computed subsequently. Also, the chain rule [44] is utilized as where and are the fractal indexes [43, 48], without loss of generality, consider , in which is a constant. Step 2. Suppose that the solution of ODE (13) can be expressed by a polynomial in as follows: where and are constants to be found such that and satisfies the following ordinary differential equation: We will take the below particular solutions of equation (17): Family 1: if and , afterwards Family 2: if and , afterwards Family 3: if , , and , afterwards Family 4: if , , and , afterwards Family 5: if , , and , afterwards Family 6: if and , afterwards Family 7: if and , afterwards Family 8: if , afterwards Family 9: If , afterwards Family 10: if and , afterwards Family 11: if , afterwards Family 12: if , afterwards Family 13: if , afterwards Family 14: if , afterwards Family 15: if and , afterwards Family 16: if and , afterwards Family 17: if and , afterwards Family 18: if and , afterwards Family 19: if , afterwards where , , , and are free constants to be determined later. To determine the positive integers and , it, usually, can be accomplished by the nonlinear terms of highest-order in equation (13) with the highest-order linear terms. Substituting equations (16) and (17) into equation (13) yields an equation of powers of . Step 3. With N determined, we then collect all coefficients of powers of in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters , and .
3.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by TEM
Take the fractional CJM equation as
By using the transformation (14), equation (18) will be reduced to the following ODE:
Balancing with and with in equation (19) gives and . We substitute the values of and in (16). To find the periodic waves of the fractional CJM equations, we would like to start from an ansatz as follows:where , and are free parameters. Inserting (20) into equation (19) and then collecting the coefficients in the front of the various functions including and give the following:
By solving the nonlinear algebraic system of equations can conclude the following results: Set I: Putting (22) into (20), we obtain a periodic solution of equation (18) by utilizing Family 1 as follows: where and . Appending (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows: where and . Substituting (22) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows: where and . Inserting (22) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows: where . Set II: Substituting (31) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows: where and . Appending (31) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows: where and . Substituting (31) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows: where and . Inserting (31) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 12 as follows: where and . Set III: Inserting (36) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows: where and . Appending (36) into (20), we obtain a hyperbolic solution of equation (18) by utilizing Family 2 as follows: where and . Inserting (36) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows: where and . Appending (36) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 12 as follows: where and . Set IV:
Substituting (41) into (20), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows:where and .
Inserting (41) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 2 as follows:where and .
Plugging (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 8 as follows:where and .
Inserting (41) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 12 as follows:where and .
4. Summarization of the Modified -Expansion Method
In this section, we will describe the main steps of the modified -expansion method for finding traveling wave solutions of NLPDEs to the time-fractional coupled Jaulent–Miodek equation in order to furnish its exact solutions: Step 1. Given a nonlinear physical model governed by partial differential equation of fractional order is expressed as the above equation can be converted into an ordinary differential equation (ODE), by utilizing the suitable fractional complex transform where and are free parameters which would be computed subsequently. Also, the chain rule [44] is used as where and are the fractal indexes [43, 48]; without loss of generality, take , where is a constant. Step 2. Suppose that the solution of ODE (47) can be expressed by a polynomial in as follows: where and are constants to be found such that , and satisfy the below ordinary differential equation: the solutions of equation (51) is given by Family 1: if and , afterwards Family 2: if and , afterwards Family 3: if and , afterwards Family 4: if and , afterwards Family 5: if and , afterwards where , , and are also the free constants. Step 3. To determine the positive integer , it, usually, can be accomplished by the nonlinear terms of highest-order in equation (47) with the highest-order linear terms. Substituting equations (50) and (51) into equation (13) yields an equation of powers of . Step 4. With determined, we then collect all coefficients of powers of in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters , , and .
4.1. Periodic, Soliton, Singular, and Kink-Singular Solutions of the FCJM Equation by MEM
Balancing with and with in equation (19) gives the following cases:
Putting the values of (52) and to search for different wave solutions of the fractional CJM equations, we would like to start from ansatz as the following form:where , and are unknown values. Inserting (53) into equation (56) and then collecting the coefficients in the front of the various functions containing are given as follows:
Solving the nonlinear algebraic system of equations can be concluded as the following solutions: Set I: Plugging (55) into (53), we perceive a periodic solution of equation (18) by utilizing Family 1 as follows: Set II: Appending (57) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 3 as follows: Set III: Appending (59) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows: Set IV: Appending (61) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 3 as follows: Set V: Appending (63) into (20), we receive a hyperbolic solution of equation (18) by utilizing Family 3 as follows: Set VI: Appending (65) into (20), we gain a hyperbolic solution of equation (18) by utilizing Family 1 as follows: where . Plugging (65) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows: where . Set VII:
Appending (68) into (20), we get a hyperbolic solution of equation (18) by utilizing Family 1 as follows:where . Appending (68) into (20), we perceive a hyperbolic solution of equation (18) by utilizing Family 2 as follows:where .
5. Interpretation Solutions with Physical Concepts
This section shows some soliton solutions of fractional form of the coupled Jaulent–Miodek equation.
Remark 3. The three-dimensional dynamic plots of the wave and corresponding density designs, contour designs, and two-dimensional designs were successfully plotted in Figures 1–7 by utilizing the Maple.
In order to analyze the dynamics characteristics briefly, we would like to discuss the evolution characteristic. By taking appropriate parameters of these parameters in the following, we setWe can see that the trigonometric function is as periodic solution, hyperbolic function is as soliton wave solution, exponential function is as kink solution, rational exponential function is as singular-kink solution, and rational function is as singular solution.
6. Conclusion
We have investigated in detail the time-fractional coupled Jaulent–Miodek equation. By utilizing the -expansion technique and modified -expansion technique, we find bright, dark soliton, analytically periodic, singular, and kink-soliton solutions with the constraints under which these solutions are obtained. The -expansion technique and modified -expansion technique are helpful to obtain solutions in the form of hyperbolic and trigonometric forms which are exact and helpful in understanding the fractional forms of it. Finally, a transformation is used to draw soliton solutions of system (1) by the use of Maple software. So, this gives the efficient applications of aforementioned techniques for fractional PDEs.
By utilizing Maple, the evolution phenomenon of these waves is seen in Figures 1–7, respectively. Moreover, we can discover the physical structure and properties of the new periodic wave, soliton, singular-kink, and bell soliton solutions for the time FCJM equations. These new solutions can be utilized to perceive their physical morphology and density curves by computer computations, which will be extensively used in optics, DWDM systems, metasurfaces, and metamaterials, and so forth.
Data Availability
The datasets supporting the conclusions of this article are included in the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.