Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel'd-Sokolov System

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov system u t + ( v m ) x = 0 , v t + a ( v n ) x x x + b u x v + c u v x = 0 called D ( m , n ) system. We reveal some interesting bifurcation phenomena as follows. (1) For D ( 2 , 1 ) system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) For D ( 1 , 2 ) system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) For D ( 2 , 2 ) system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.