SOME COMMON EXPRESSIONS AND NEW BIFURCATION PHENOMENA FOR NONLINEAR WAVES IN A GENERALIZED mKdV EQUATION

Abstract

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation ut + a(1 + bu2)u2ux + uxxx = 0. (i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves. (ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves. We also show that the common expressions include many results given by pioneers.

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

“…(1) has been few discussed and understood. Hence it is the main investigation of our paper by using phase analysis which was used in some lectures, for instance [12], [19], [11], [2] and [3]. In this paper, for arbitrary given parameters b and γ, choosing a as bifurcation parameter, we show that in Eq.…”

Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

“…The Z-K equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [2,3]. There are lots of research for various generalized Z-K equations [4][5][6][7][8][9][10][11][12][13]. For the Z-K equation…”

“…When 𝛿 = 0, Liu and Yan [9] obtained some common expressions and two kinds of bifurcation phenomena for nonlinear waves of (4). Meanwhile, they pointed out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively.…”