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

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)-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)-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.

Fractional traveling wave solutions of the (2 + 1)‐dimensional fractional complex Ginzburg–Landau equation via two methods

“…Recently, fractional NLSEs (FNLSEs) and the related models become research hot topics to describe the evolution of fractional optical pulse and beams 27,28 . Diverse methods including F‐expansion method, 29 extended sinh‐Gordon equation method, 27 and (G'/G)‐expansion method 28 have been put forward to study FNLSEs.…”

“…Recently, fractional NLSEs (FNLSEs) and the related models become research hot topics to describe the evolution of fractional optical pulse and beams 27,28 . Diverse methods including F‐expansion method, 29 extended sinh‐Gordon equation method, 27 and (G'/G)‐expansion method 28 have been put forward to study FNLSEs. These methods in the previous studies 19,20,27–29 simplified FPDEs into ordinary differential equations that are easier to solve based on Jumarie's basic formulae 30 .…”

“…Diverse methods including F‐expansion method, 29 extended sinh‐Gordon equation method, 27 and (G'/G)‐expansion method 28 have been put forward to study FNLSEs. These methods in the previous studies 19,20,27–29 simplified FPDEs into ordinary differential equations that are easier to solve based on Jumarie's basic formulae 30 . However, some researchers 31 have pointed that these transformations used in these methods 20,29,30 are not true because Jumarie's basic formulae used in the previous studies 20,27–29 are not correct.…”

“…These methods in the previous studies 19,20,27–29 simplified FPDEs into ordinary differential equations that are easier to solve based on Jumarie's basic formulae 30 . However, some researchers 31 have pointed that these transformations used in these methods 20,29,30 are not true because Jumarie's basic formulae used in the previous studies 20,27–29 are not correct. In order to make Jumarie's basic formulae work, the Mittag–Leffler (ML) function was introduced to construct exact solutions 32,33 …”

Periodic solution of fractal Phi-4 equation

“…The Phi-4 equation can model many nonlinear phenomena arising in optics, thermal science, nanofluid and nonlinear vibration [1][2][3][4]. Much literature focused on its solitary solutions, while its periodic solution was rarely studied.…”