Exact traveling wave solutions of the space–time fractional complex Ginzburg–Landau equation and the space-time fractional Phi-4 equation using reliable methods

Abstract

Abstract Ultrashort pulse propagation in optical transmission lines and phenomena in particle physics can be investigated via the cubic–quintic Ginzburg–Landau equation and the Phi-4 equation, respectively. The main objective of this paper is to construct exact traveling wave solutions of the (2+1) $(2 + 1)$-dimensional cubic–quintic Ginzburg–Landau equation and the Phi-4 equation of space-time fractional orders in the sense of the conformable fractional derivative. The method employed to solve the Ginzburg–Landau equation and the Phi-4 equation are the modified Kudryashov method and the (G′/G,1/G) $(G'/G,1/G)$-expansion method, respectively. Several types of exact analytical solutions are obtained including reciprocal of exponential function solutions, hyperbolic function solutions, trigonometric function solutions and rational function solutions. Graphical representations and physical explanations of some of the obtained solutions are demonstrated using a range of fractional orders. All of the solutions have been verified by substitution into their corresponding equations with the aid of a symbolic software package. These methods are simple and efficient for solving the proposed equations.

Keywords

• Conformable fractional derivative
• Nonlinear space-time fractional complex Ginzburg–Landau equation
• Nonlinear space-time fractional Phi-4 equation
• Modified Kudryashov method
• ( G ′ / G , 1 / G ) $(G'/G,1/G)$ -expansion method