Bifurcation and exact solutions for the ($2+1$)-dimensional conformable time-fractional Zoomeron equation

Abstract

In this paper, the bifurcation and new exact solutions for the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are investigated by utilizing two reliable methods, which are generalized $(G'/G)$ ( G ′ / G ) -expansion method and the integral bifurcation method. The exact solutions of the ($2+1$ 2 + 1 )-dimensional conformable time-fractional Zoomeron equation are obtained by utilizing the generalized $(G'/G)$ ( G ′ / G ) -expansion method, these solutions are classified as hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Giving different parameter conditions, many integral bifurcations, phase portraits, and traveling wave solutions for the equation are obtained via the integral bifurcation method. Graphical representations of different kinds of the exact solutions reveal that the two methods are of significance for constructing the exact solutions of fractional partial differential equation.

New Exact Solutions of the Fractional Complex Ginzburg–Landau Equation

“…In order to better analyze the complex optical phenomena and further study their essence, the best ways are to find the exact traveling solutions [8][9][10][11][12][13][14][15] to the Ginzburg-Landau equation describing the nonlinear optical phenomena. In recent years, a variety of powerful mathematical approaches have been developed to derive the exact solutions to Ginzburg-Landau equation, such as the (G ′ /G 2 )-expansion method [16], the Modified simple equation method [17], the F-expansion [18], the sine-Gordon expansion method [19], the extended direct algebraic method [20], and the dynamical system method [21][22][23].…”

Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

“…Remark 2. Obviously (see [16][17][18][19][20]), all traveling wave solutions of ( 1) can be obtained from the phase orbits of the Hamiltonian system (9) according to Lemma 1.…”

Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation

“…A variety of analytical methods were used by researchers in recent studies to develop new exact solutions to the (2+1)-dimensional time fractional Zoomeron equation. Several techniques have been employed, including the extended exp(−ϕ)-expansion method [39], generalized G ′ /G-expansion method [40], and the improved Bernoulli sub-equation function method [41]. In this study, we utilize the intricate traveling wave transformation to simplify the (2 + 1)-dimensional fractional Zoomeron equation into an ordinary differential equation.…”