The bifurcations of solitary and kink waves described by the Gardner equation

Abstract

In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation ut + auux + bu 2 ux + γuxxx = 0. We obtain some new results as follows: For arbitrary given parameters b and γ, we choose the parameter a as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The first phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

“…Various types of exact solutions of the Gardner equation ( 1) have been extensively studied [5][6][7][8]. For example, by applying the theory of dynamical systems and the bifurcation method, Chen and Liu [9] obtained the solitary wave solutions and kink wave solutions of (1). Recently, increasingly more interest has been paid to the traveling waves of singularly perturbed mathematical physics models [10][11][12][13][14][15].…”

“…Note that when δ = 0, Equation (4) becomes Equation (1). Compared with the methods and results in [9], the process of obtaining the exact solutions of Equation (4) with δ > 0 is more complicated because the degree of the Hamiltonian function is five, and the system (8) admits a higher-order singular point.…”

Bifurcation analysis and dynamics of abundant traveling waves of the modified Novikov equation

“…Furthermore, how about the dynamics of the waves under general $$$ b $$$? Based on these motivations, we further study Equation () with arbitrary $$$ b\in \mathbb{R} $$$ from the perspective of dynamical systems 24–34 and show that Equation () possesses solitary waves, periodic waves, compactons, kink (antikink), and kink‐like (antikink‐like) waves. The detailed differences of the results between Meng and He 23 and this paper are given in Remark 1.…”

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

“…If the parameter , then Equation becomes the famous Gardner equation which is a generic mathematical physics model for weakly nonlinear and weakly dispersive wave propagation when the effects of higher‐order nonlinearity become significant. 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 …”

“…Equation (2) has broad application in various branches of physics, such as fluid physics and plasma physics, and the solutions of Equation 2and their dynamical behavior have been extensively studied from different aspects. [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).…”