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

Abstract

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov systemut+(vm)x=0,vt+a(vn)xxx+buxv+cuvx=0calledD(m,n)system. We reveal some interesting bifurcation phenomena as follows. (1) ForD(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) ForD(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) ForD(2,2)system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

Study of some system of nonlinear partial differential equations by LDM and MLDM

“…By using equation ( 16) and (17) Which is exact solution of equation ( 14). and 𝑣(𝑥, 𝑡) = ∑ 𝑣 𝑛 (𝑥, 𝑡)…”

“…F. Zhang, J.Qi and W. Yuan employed complex method to derive general meromorphic solutions of DS system [16]. A bifurcation phenomenon of nonlinear waves for DS system was studied by H. Cai, C. Pan and Z. Liu [17]. Jing Wang gave different properties of DS system like Hamiltonian, symplectic, cosymplectic, recursion operator, scaling symmetry and roots of symmetries [18].…”