Research paperDynamic behaviors for a perturbed nonlinear Schrödinger equation with the power-law nonlinearity in a non-Kerr medium

- A perturbed nonlinear Schrödinger equation with the power law nonlinearity is studied from the planar-dynamic system viewpoint.
- Through the qualitative and bifurcation analysis, phase portraits of the dynamic system are given.
- Relations among the Hamiltonian, orbits of the dynamic system and types of the analytic solutions are discussed.
- The equation possesses the periodic-wave solutions, kink- and bell-shaped solitary-wave solutions.
- A periodic wave can reduce to a solitary wave under a Hamiltonian-dependent limiting condition.

Effects of quantic nonlinearity on the propagation of the ultrashort optical pulses in a non-Kerr medium, like an optical fiber, can be described by a perturbed nonlinear Schrödinger equation with the power law nonlinearity, which is studied in this paper from a planar-dynamic-system view point. We obtain the equivalent two-dimensional planar dynamic system of such an equation, for which, according to the bifurcation theory and qualitative theory, phase portraits are given. Through the analysis of those phase portraits, we present the relations among the Hamiltonian, orbits of the dynamic system and types of the analytic solutions. Analytic expressions of the periodic-wave solutions, kink- and bell-shaped solitary-wave solutions are derived, and we find that the periodic-wave solutions can be reduced to the kink- and bell-shaped solitary-wave solutions.