Two kinds of important bifurcation phenomena of nonlinear waves in a generalized Novikov–Veselov equation

Abstract

This paper considers a generalized Novikov–Veselov equation $$u_{t}+ a u_{xxx}+b u_{yyy}+c (u^{2}\partial _{y}^{-1}u_{x})_{x}+d(u^{2}\partial _{x}^{-1}u_{y})_{y}=0$$ . By utilizing the bifurcation method and qualitative theory of dynamical systems, we show that there exist two kinds of important bifurcation phenomena when the bifurcation parameters vary. One of the bifurcation phenomena is that the fractional solitary waves can be bifurcated from three types of nonlinear waves which are the trigonometric periodic waves, the elliptic periodic waves and the hyperbolic solitary waves. Another bifurcation phenomenon is that the kink waves can be bifurcated from two types of nonlinear waves which are the solitary waves and the singular waves. Moreover, we carry out many numerical simulations of bifurcation processes of these nonlinear waves to verify our main results.