Wave solutions of the time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity

In this paper, the bifurcation theory of dynamical system is applied to investigate time-space fractional complex Ginzburg-Landau equation with Kerr law nonlinearity. We mainly consider case α ≠ 2β which not discussed in previous work. By overcoming some difficulties aroused by singular traveling wave system, such as analysis nonanalytic vector field, tracking orbits near full degenerate equilibrium and calculation complicated elliptic integrals, we give a total 20 explicit exact solutions classify them into 11 categories. Some new are obtained including compactons bounded corresponding manifolds.