Peakon, Periodic Peakons, Compactons and Bifurcations of nonlinear Schrödinger’s Equation with Kudryashov’s Law of Refractive Index

In this paper, we consider the nonlinear Schrödinger’s equation with Kudryashov’s law of refractive index. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the traveling wave system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including peakon, periodic peakon, solitary wave solutions and compactons) under different parameter conditions.