Dynamical behavior and exact solutions for time-fractional nonlinear Schrödinger equation with parabolic law nonlinearity

The time-fractional nonlinear Schrödinger equation with parabolic law nonlinearity is studied. Under the travelling wave transformations, Schrödinger equation is reduced to plane system, which is analyzed by theory of planar dynamical systems. The periodic-wave solutions, kink-shaped and bell-shaped solitary-wave solutions that corresponds to periodic orbit, heteroclinic orbit and homoclinic orbit are given. Then some other exact solutions of time-fractional nonlinear Schrödinger equation are constructed applying complete discrimination for polynomial.