New solitons and conditional stability to the high dispersive nonlinear Schrödinger equation with parabolic law nonlinearity

The complete discrimination system was proposed to search traveling wave solutions for a (1 + 1)-dimensional nonlinear Schrodinger equation with parabolic law nonlinearity. As a consequence, we obtained a series of solutions which contain dark soliton, bright soliton, triangular solitary wave solution, kink soliton, periodic solutions and implicit analytical solutions. At the same time, numerical simulations were demonstrated to visualize the mechanics of these solutions. Lastly, linear stability of dark soliton $$u_{25}$$ in Lyapunov’s sense was discussed.