New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

New soliton solutions of fractional Jaulent-Miodek (JM) system are presented via symmetry analysis and fractional logistic function methods. Fractional Lie symmetry analysis is unified with symmetry analysis method. Conservation laws of the system are used to obtain new conserved vectors. Numerical simulations of the JM equations and efficiency of the methods are presented. These solutions might be imperative and significant for the explanation of some practical physical phenomena. The results show that present methods are powerful, competitive, reliable, and easy to implement for the nonlinear fractional differential equations.


Introduction
Integral and derivative operators of any arbitrary order are the basis of fractional calculus, which has been of great interest for researchers due to its dynamic behavior and exact description of nonlinear complex phenomena in numerous fields in science and engineering [1][2][3][4][5][6]. Analytical methods have played an essential role for Fractional partial differential equations (FPDEs) [1][2][3][4]. Lie symmetry analysis also gives a powerful and effectual implement for generating invariant solutions. The theory of symmetry analysis is based on the invariance of variables [7][8][9][10][11][12][13][14]. Hence, the study of symmetry analysis has been made a huge interest for researchers during past decades.

Local Fractional-Order Derivative
Assume h( x ) ∈ C α (m, n), where C α (m, n) denotes α times differentiable with each derivative continuous in (m, n). Then, the derivative with fractional order α at x = x 0 is defined as [42,43] where And has following property [42,43]:

Brief Description of the Proposed Method
The section emphasizes describing a comparatively new analytic method for getting solutions for the FPDEs. The procedure for the proposed method has been described in the following manner: Step 1: The FPDE is given as: where u(x, t) is a function.
Step 2: Solution of Equation (7) is presented as where γ and k are parameters. Then, (6) [44,45] can reduce the fractional derivative into the following form Then, the Equation (7) can be reduced by using Equation (7), by the following form: Step 3: Here, the exact solution of Equation (7) is mentioned in terms of the polynomial in ϕ(ξ) as follows: where ϕ(ξ) is considered as the sigmoid function or logistic function [46,47], is defined as follows: ϕ(ξ) = e ξ 1+e ξ and satisfies the following Riccati equation: and the value of n can be evaluated by using the homogenous balancing principle [48,49]. Moreover, the derivatives of different order for the function U(ξ) can be determined by using Equation (11).
Step 4: Now, the coefficients a i are determined by putting Equation (11) into Equation (9) and solving the acquired algebraic equations obtained by equating coefficients of ϕ i to 0.
Step 5: Unknowns obtained in step 4 are written into Equation (10) to get the solutions for Equation (7).

Soliton Solutions for JM System
The logistic function method is employed for solving Equation (1). By using Equation (8) in Equation (1), we have: and Similar to Equation (10), let us consider the solutions of the governing system are presented by following mathematical equations as By means of homogenous balance principle [48,49], we get n = 2 and m = 1. Thus, the solutions are: U(ξ) = a 0 + a 1 ϕ + a 2 ϕ 2 and V(ξ) where ϕ follows satisfies Equation (11). Putting Equation (15) with Equation (11) into Equations (12) and (13), equating the obtained coefficient of ϕ i to 0, we get: For set 1, the following hyperbolic solutions can be obtained as where ξ = kx − k 3 t α 4Γ(α+1) . Set 2: For set 2, the following hyperbolic solutions can be obtained as where ξ = kx − k 3 t α 4Γ(α+1) .

Set 3:
For set 3, the following hyperbolic solutions can be obtained as For set 4, the following hyperbolic solutions can be obtained as

Numerical Simulations
This part emphasizes on numerical simulation for the Equations (1) and (2) by the fractional logistic equation method. Furthermore, the Equations (16) and (18)

Numerical Simulations
This part emphasizes on numerical simulation for the Equations (1) and (2) by the fractional logistic equation method. Furthermore, the Equations (16) and (18) have been used here for generating solutions graphs. The in Equation (16)

Theory of Symmetry Analysis Method
In this part, the general method for generating the symmetries of FPDEs is discussed by means of fractional Lie symmetry analysis. Consider

Theory of Symmetry Analysis Method
In this part, the general method for generating the symmetries of FPDEs is discussed by means of fractional Lie symmetry analysis. Consider Let us now consider that the Equations (20) and (21) are invariant in one-parameter Lie group transformation: where ε << 1 is considered as a group parameter, τ, η, ϑ, ξ are infinitesimals. Total expression for η x , η xx , η xxx , ϑ x , ϑ xx and ϑ xxx are: where . ., j, k = 1, 2, 3, . . . and u j = ∂u ∂x j ,v j = ∂v ∂x j , u jk = ∂ 2 u ∂x j ∂x k , v jk = ∂ 2 v ∂x j ∂x k and so on.
V satisfies: here, Pr denotes the prolongation for the given vector and and Now, by considering the usual structure of RL fractional operator, the transformations of system (22) has been formed. We have τ(x, t, u, v) t=0 = 0 (26) By RL derivative, the α-th infinitesimal [50][51][52] with Equation (26) can be presented as follows: where the D α t denotes the total fractional differential operator. We have: We also have We have: Now by using Equations (28) and (30) with f (t) = 1, we have Thus, Equation (29) yields

Lie Symmetry
By third prolongation in Equations (1) and (2), we can obtain infinitesimals: Lie algebra corresponding to infinitesimal symmetry of governing system is spanned by Now, corresponding to Equations (1) and (2), we have following infinitesimal generators given as [7,8]

Case 2:
The following characteristic equation can be obtained by using the infinitesimal generator in Equation (35), given as dx After solving Equation (36), the following similarity variable can be obtained, given as Theorem 1. The transformation (38) and (39) reduces Equations (1) and (2) to the following form of Ordinary differential equations (ODEs) given as: with the Erdélyi-Kober operator P τ,α β : and P τ,α β G := (43) where, the Erdélyi-Kober fractional integral operator can be expressed as: and and

Conclusions
Fractional logistic function technique is proposed for soliton solutions of fractional JM system. Numerical simulation for solutions has been shown for analyzing the physical nature of obtained solutions. Moreover, Lie group analysis technique is proposed for investigation of symmetry properties and conservation laws for fractional Jaulent-Miodek system. Conservation laws for the system are acquired by new theorem and formal Lagrangian. These analyses are relatively new and reliable for finding exact solutions and constructing conservation laws with generating similarity solutions for the FPDEs. Furthermore, this method enriches the solution of the equations, which is of great significance for study of the FPDEs.