New solitary wave solutions for the conformable Klein-Gordon equation with quantic nonlinearity

We present new exact solutions in the form of solitary waves for the conformable Klein-Gordon equation with quintic nonlinearity. We use functional variable method which converts a conformable PDE to a second-order ordinary differential equation through a traveling wave transformation. We obtain periodic wave and solitary wave solutions including particularly kinkprofile and bell-profile type solutions. The present method is a direct and concise technique which has the potential to be applicable to many other conformable PDEs arising in physics and engineering.


Introduction
Nonlinear conformable evolution equations (NLCEEs) became significantly useful tools in the modeling of many problems in sciences and technology. Exact wave solutions of these models are very important and active research area. NLCEEs are getting the attention of researchers and becoming phenomenal subject in the contemporary science. Many systems in mathematical physics and fluid dynamics are modeled via fractional differential equations. Exact wave solutions of these models are quite active and important research area in science. For the numerical and exact solutions of NLCEEs, there are some efficient techniques in the literature such as method of (G /G)−expansion, extended sinh-Gordon equation expansion, Kudryashov, exp-function, exponential rational function, modified Khater, functional variable, improved Bernoulli sub-equation function, sub-equation, tanh, Jacobi elliptic function expansion, auxiliary equation, extended direct algebraic, etc., see . The functional variable (FV) method was introduced in [28] and was further developed in the studies [29][30][31][32][33]. FV method treats nonlinear PDEs with linear techniques and constructs interesting type of soliton solutions (kink, black, white, pattern, etc). The conformable fractional derivatives don't have a physical meaning as the Caputo or Riemann-Liouville derivatives. This situation is a general open problem for fractional calculus. Despite this many physical applications of conformable fractional derivative appear in the literature. Dazhi Zhao and Maokang Luo generalized the conformable fractional derivative and give the physical interpretation of generalized conformable derivative. In addition, with the help of this fractional derivative and some important formulas, one can convert conformable fractional partial differential equations into integer-order differential equations by travelling wave transformation [39].
The aim of the present paper is present new exact solutions to conformable Klein-Gordon (KG) equation with quintic nonlinearity by employing FV method. Nonlinear conformable Klein-Gordon equation has the form (for α = 1, see [34]) in which u represents wave profile, and k, γ, λ, σ 0 are real valued constants. KG equation arises in theoretical physics, particularly in the area of relativistic quantum mechanics and it is used in modeling of dislocations in crystals. For n = 3, Eq (1.1) is known as conformable Klein-Gordon equation with quintic nonlinearity [24] In particular, if σ = 0, then Eq (1.2) reduces to some other PDEs including the ones in [35,36].
Next, we overview method of functional variable.

Method of functional variable
Consider the NLCEE: in which F is a polynomial function in terms of unknown function u, and D α t u is defined as [37] where 0 < t, α ∈ (0, 1]. Now, let us define the wave variable [38] u(x, t) = U(ξ), in which ω is a parameter which will be determined later. Hence, we can write that By writing Eq (2.3) in Eq (2.1), we get ordinary differential equations: Now, define a transformation: from which, we obtain in which " " stands for d dU . Using Eq (2.6) in Eq (2.3), ordinary differential Eq (2.3) can be reduced to: Now, let us consider the equation in which a, b, c are parameters. Next, we present a set of exact wave solutions of (2.8), see e.g., [39]: Case 1. If a > 0, then (2.8) admits hyperbolic function solution: (2.9) Case 2. If a, c > 0, then (2.8) admits the following hyperbolic function solution (2.14) Case 3. If a > 0 and b 2 − 4ac > 0, then (2.8) admits the following hyperbolic function solution Case 4. If a > 0 and b 2 − 4ac < 0, then (2.8) admits the following hyperbolic function solution (2.16) Case 5. If a > 0 and b 2 − 4ac = 0, then (2.8) admits the following hyperbolic function solution Case 6. If a < 0 and c > 0, then (2.8) admits the following triangular function solution Case 7. If a > 0and b = 0, then (2.8) admits the following hyperbolic function solution (2.31) Case 10. If a < 0 and c = 0, then (2.8) admits the following triangular function solution (2.33)
By using the relations (16-40), we obtain exact solutions of conformable KG equation with quintic nonlinearity (1.2).

Graphical representations
In this part, some graphical representations of exact wave solutions of conformable KG equation are presented in three different forms. 3D plots of exact solutions |u 3 | , |u 3 | , |u 3 | are displayed in Figures  1(a)

Conclusions and outlook
We presented new exact solutions of conformable Klein-Gordon equation with quantic nonlinearity by using method of functional variable. Solutions were expressed in terms of solitary waves such as kink-profile and bell-profile. Moreover, we obtain exact periodic solutions of the KG equation. Computational results show that FV method is a highly efficient technique in the solutions of conformable PDEs. In a future research work, we will investigate the applicability of these results to some fractional-stochastic differential equations.