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

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.

Qualitative analysis and new exact solutions for the extended space-fractional stochastic (3 + 1)-dimensional Zakharov-Kuznetsov equation

“…expansion method [18], the generalized exponential rational function method [19], the Φ 6 − model expansion and the Hirota bilinear methods [20], the extended generalized ( ) ¢ -G G expansion method [21], and the method of bifurcation theory for planar systems [22][23][24][25]. For more details about qualitative analysis utilizing the bifurcation theory, you can see, e.g.…”