Abstract
We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.
Highlights
Introduction and PreliminaryZakharov-Kuznetsov (Z-K) equation [1], ut + auux (∇2u) x = (1)was first derived for describing weakly nonlinear ion-acoustic wave in a strongly magnetized lossless plasma in two dimensions
We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equation ut +ux + γuxxx + δuxyy = 0
The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves
Summary
Was first derived for describing weakly nonlinear ion-acoustic wave in a strongly magnetized lossless plasma in two dimensions. When δ = 0, Liu and Yan [9] obtained some common expressions and two kinds of bifurcation phenomena for nonlinear waves of (4). They pointed out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. In order to investigate the bifurcation phenomena of (4), letting c > 0 be wave speed and substituting u = φ(ξ) with ξ = x + y − ct into (4), it follows that. We obtain three types of explicit expressions of nonlinear wave solutions.
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