Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation

In this paper, our main purpose is to study the soliton solutions of conformable timefractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. New soliton solutions have been obtained by the extended (G′/G)-expansion method, first integral method and complete discrimination system for the polynomial method, respectively. The solutions we obtained mainly include hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. Moreover, we draw its three-dimensional graph.

Definition 1.1. Let ψ : [0, ∞) → R. Then, the conformable fractional derivative of ψ of order α is defined as for all t > 0 and α ∈ (0, 1]. Further, some properties of conformable fractional derivative is given This article is arranged as follows. In Section 2, we employ three different methods to solve the FPRKL equation. In Section 3, we draw three-dimensional graph of Eq (1.1). In Section 4, we give a brief conclusion.

Soliton solutions of Eq (1.1)
Making the complex transformation Substituting Eq (2.1) into Eq (1.1), separating into real and imaginary parts yields Integrating Eq (2.3) once, we have Since the function U satisfies both Eq (2.3) and Eq (2.4), the following constraint condition is obtained a + 3kγ So, k and c in Eq (2.5) can be obtained

Extended (G /G)-expansion method
Balancing u 3 and u in Eq (2.4), we have N = 1. So, the solution form of Eq (2.4) is Here G = G(ξ) satisfies the following nonlinear ordinary differential equation where A, B, C are real parameters, Eq (2.8) satisfies the following equation (2.9) Then we can obtain a nonlinear algebraic equations.
Next, we get the following the results: where C 1 and C 2 are arbitrary constants.

First integral method
Now, Eq (2.4) is equivalent to the following of two dimensional system Then, based on the division theorem, there exists a polynomial g(X) Since a i (X)(i = 0, 1) are polynomials, then we obtain when deg(g(X)) = 1 Substitutinga 0 (X), a 1 (X), g(X) into Eq (2.23) and setting all the coefficients of powers X to be zero, then, we obtain (2.28) Using the conditions Eq (2.28) in Eq (2.21), we obtain Combining Eq (2.29) with Eq (2.20), we obtain the exact solution for FPRKL equation which can be written as Comparing the coefficients of Y i (ξ) = (0, 1, 2, 3) on both sides of Eq (2.22), we obtain Balancing the degrees of g(X) and a 1 (X), we get deg(g(X)) = 1, deg(a 1 (X)) = 2, then where A 1 , A 0 , B 0 are all constants to be determined. Now, Eq (2.37) becomes where d is the constant of integration. Substituting a 0 (X), g(X), a 1 (X) into Eq (2.36) and setting all the coefficients of powers X to be zero, we obtain (2.43) From Eq (2.43) into Eq (2.21), we obtain This shows that the two cases m = 1 and m = 2 give the same solutions.

Complete discrimination system for the polynomial method
Multiplying u on both sides of Eq (2.4), and again integrating it on ξ, we can get , and a 0 is the constant.

Physical explanations
In this section, the numerical simulations of some remarkable solutions for the FPRKL equation are presented. By the (G /G)-expansion method, we obtained the solution φ 1 1 (x, t) and φ 1 2 (x, t) shown in Figure 1. The graphical solutions φ 10 (x, t) and φ 11 (x, t) are shown in Figure 2. Moreover, the graphical solutions φ 16 (x, t) and φ 17 (x, t) are shown in Figure 3.

Conclusions
In this article, we have investigated the exact solutions to the FPRKL equation by three different methods. Many exact solutions have been obtained. In the paper, we get all the traveling wave solutions, which have not been seen in other literature. These solutions might be further useful and effective to study more about the various forms of solitary waves in physics. We have noticed that the proposed complete discrimination system for the polynomial method gives much more new and general exact solutions than the other two suggested methods. In future work, we will consider the bifurcation, phase diagrams and exact solutions of the FPRKL equation.