Bifurcations and nonlinear wave solutions for the generalized two-component integrable Dullin–Gottwald–Holm system

Abstract

In this paper, we study the bifurcations and nonlinear wave solutions for a generalized twocomponent integrable Dullin-Gottwald-Holm system. Through the bifurcations of phase portraits, we not only show the existence of several types of nonlinear wave solutions, including solitary waves, peakons, periodic cusp waves, periodic waves, compacton-like waves and kink-like waves, but also obtain their implicit expressions. Additionally, the numerical simulations of the nonlinear wave solutions are given to show the correctness of our results.

Abundant explicit periodic wave solutions and their limit forms to space‐time fractional Drinfel'd–Sokolov–Wilson equation

“…(vii3) when < 0 and g = 0, we have (vii3a) if h → H(0, 0) − 0, then the periodic wave solutions v 15 (x, t) and v 16 (x, t) converge to the two symmetric solitary wave solutions (18); (vii3b) if h → H(0, 0) + 0, then the periodic wave solutions v 17 (x, t) converge to the two symmetric solitary wave solutions (18).…”

“…It is worth mentioning that the bifurcation method is an effective and efficient method to find exact solutions of PDEs and it has made great achievements in finding exact solutions of PDEs and analyzing their dynamical behaviors. [15][16][17][18][19][20][21][22][23][24][25][26] Very recently, the bifurcation method has been generalized to study exact solutions of space-time fPDEs. 27,28 As an illustration, Wen 27 studied the exact solutions of the following space-time fractional Drinfel'd-Sokolov-Wilson equation (DSWE)…”

On existence of kink and antikink wave solutions of singularly perturbed Gardner equation

“…[2][3][4][5][6][7] Especially, Chen and Liu 4 found the heteroclinic orbits and corresponding kink and antikink wave solutions by the method of dynamical systems. [8][9][10][11][12][13][14][15][16][17][18][19][20][21] Note that Equation (1) can be viewed as perturbation of Equation (2). To study the kink and antikink wave solutions of Equation (1), we first give some necessary results about Equation (2).…”

“…Equation has broad application in various branches of physics, such as fluid physics and plasma physics, and the solutions of Equation and their dynamical behavior have been extensively studied from different aspects . Especially, Chen and Liu found the heteroclinic orbits and corresponding kink and antikink wave solutions by the method of dynamical systems …”

Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

“…These solutions will play an important role in soliton theory. In order to get exact solutions directly, many powerful methods have been introduced such as inverse scattering method [1], bilinear transformation [2], Bäklund and Darboux transformation [3][4][5], tanh-sech method [6,7], extended tanh method [8], Exp-function method [9][10][11], the sine-cosine method [12][13][14], the Jacobi elliptic function method [15], -expansion method [16,17], auxiliary equation method [18,19], bifurcation method [20][21][22], homotopy perturbation method [23], and homogeneous balance method [24,25]. Recently, Wang et al [26] introduced a new approach, namely, the ( / )-expansion method, for a reliable treatment of the nonlinear wave equations.…”