Lie symmetry analysis , exact solutions and conservation laws for the time fractional modified Zakharov – Kuznetsov equation

Dumitru Baleanu, Mustafa Inc, Abdullahi Yusuf , Aliyu Isa Aliyu Department of Mathematics, Cankaya University, Öǧretmenler Cad. 1406530, Ankara, Turkey dumitru@cankaya.edu.tr Institute of Space Sciences, Magurele, Bucharest, Romania Science Faculty, Firat University, 23119, Elaziǧ, Turkey minc@firat.edu.tr Science Faculty, Federal University Dutse, 7156, Jigawa, Nigeria yusufabdullahi@fud.edu.ng; aliyu.isa@fud.edu.ng


Introduction
LSA is one of the most efficient method for investigating the exact solutions of nonlinear partial differential equations (NLPDEs) arising in mathematics, physics and many other fields of science and engineering.It is well known that there is no general method for solving NLPDEs, however, LSA is one of the more powerful method for reaching new exact and explicit solutions for NLPDEs [9, 11, 28, 31-33, 42, 44, 45, 55, 56, 60].
Furthermore, CLs possess an important role in the analysis of NLPDEs from physical viewpoint [55].If the considered system has CLs, then its integrability will be possible [29].More detail about CLs and their construction can be found in [2,30,37,53].
Time fractional NLPDEs comes from classical NLPDEs by replacing its time derivative with fractional derivative.In the present work, we study Lie symmetry analysis, exact traveling wave solutions using fractional D α ξ G/G-expansion method and Ibragimov's nonlocal CLs [30] of the time fractional mZK equation given by where 0 < α 1, and α is the order of the fractional time derivative, while u(x, y, t) implies the electrostatic wave potential in plasmas, that is, a function of the spatial variables x, y and the temporal variable t.
The term ∂ α u/∂t α in (1) is the fractional temporal evolution (FTE) term.FTE is very significant since it yields a history of the problem, and also it gets rid of the slowness of evolution.For instance, when the order of the fractional derivative parameter can be controlled externally, the evolution of the soliton can be manifested artificially [4,16,17,21,[48][49][50][51].This important feature is being globally applied in several areas of physics and engineering.One immediate example is the temporal evolution of solitons in optical fibers, which can be slowed to address Internet bottleneck that is a growing problem in Internet industry [6,14,19,20,22,39].

Preliminaries
Basic definitions and theorems for the RL fractional derivative can be found in [61,66].Consider a FNPDEs of the form Given a one-parameter Lie group of infinitesimal transformations are as follows: https://www.mii.vu.lt/NA where Here The corresponding Lie algebra of symmetries consists of a set of vector fields defined by The vector field ( 5) is a Lie point symmetry of (2), provided Also, the invariance condition [27] reads and the αth extended infinitesimal related to RL fractional time derivative with (6) is given by [27,62] where 3 Lie symmetries and reduction for (1) (1) (1) Suppose that (1) is an invariant under (3), we have that thus, u = u(x, t, y) satisfies (1).Using (3) in ( 8), we obtain the invariant equation Substituting the values of η 0 α , η x and η xxx from ( 4) and ( 7) into ( 9) and isolating coefficients in partial derivatives with respect to x and power of u, the determining equations are obtained.Solving the obtained determining equation, we get where c 1 , c 5 and c 6 are arbitrary constants.Thus, infinitesimal symmetry group for ( 1) is spanned by the three vector fields In particular, for the symmetry X 3 , the similarity variables for the infinitesimal generator X 3 can be obtained from the following equation: which yields the similarity transformation and similarity variable as required.
Theorem 1.The similarity transformation (10) reduces (1) to the nonlinear ODE of fractional order as below: with the EK fractional differential operator [38] where is the EK fractional integral operator [18,63].

Exact traveling wave solutions
In this section, the exact traveling wave solutions for (1) will be presented.For this aim, we consider the fractional derivative appears in (1) in the sense of modified RL derivative [35,36].
4.1 Description of the D α ξ G/G-expansion method The description of the D α ξ G/G-expansion method is presented in detail in [67].Using the same steps as in [67], we present the application of D α ξ G/G-expansion method to (1) in the following subsections.

Application
In this subsection, we apply D α ξ G/G-expansion method to (1).Using the travelling wave transformation ξ = x + y − ct, we reduce (1) to the nonlinear fractional ODE below: Balancing the highest order derivative term D 3α ξ u with nonlinear term u 2 D α ξ u in (18), we get n = 1 [13] and therefore Substitute ( 19) into (18) collecting the coefficients of (D α ξ G(ξ)/G(ξ)) i , i = (0, ±1, ±2, . ..) and equating them to zero.A set of algebraic systems is obtained in a 0 , a −1 , a 1 .Solving the obtained algebraic systems with Mathematica 9, we obtain the following cases: .
https://www.mii.vu.lt/NA 4.3 Graphical and physical explanation of the obtained solutions Herein, we present some three-dimensional, two-dimensional and contour plots of some of the obtained results.The construction of the figures is carried out by taking suitable values of the parameters in order to see the mechanism of the original (1).One can see that the obtained solutions possess solutions like periodic wave, bell-shaped, kink-type and singular solitons.We take solutions ( 20) and ( 21) and illustrate them in Figs. 1, 2, respectively.For more details on CLs and their constructions, see [30,37,53].Using the descriptions and basics definitions for CLs presented in [3], the CLs for (1) corresponding to the order of α are given as follows: Case 1.When α ∈ (0, 1), the components of the conserved vectors are x (W i )v, where i = 1, 2, 3, and the functions W i are given by

Concluding remarks
In this research, Lie symmetry analysis of (1) using RL derivative was investigated.We obtained point symmetries for the governing equation and reduced it to a fractional nonlinear ODE.An exact solutions for the reduced ODE are obtained using D α ξ G/G-expansion method.The obtained solutions include hyperbolic, trigonometric and rational solutions.We computed the CLs for the governing equation by using new conservation theorem.The obtained solutions and CLs might be very vital for interpreting some physical phenomena in different fields of applied mathematics.https://www.mii.vu.lt/NA