Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation

Abstract

In this paper, the dynamical behavior of a time-space fractional Phi-4 equation is investigated by utilizing the bifurcation method of a planar dynamical system. Under the given parameter conditions, phase portraits and bifurcations are obtained with the help of the mathematical software Maple. Moreover, some new exact traveling wave solutions are obtained, such as Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, kink solitary wave solutions, and periodic wave solutions.

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].…”

Phase Portraits and Bounded and Singular Traveling Wave Solution of Stochastic Nonlinear Biswas–Arshed Equation

“…By using the "three-step method" of Professor Li's method together with the phase orbit of system (8) [13][14][15][16][17], the traveling wave solution of (1) can be constructed.…”

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.…”