Exact solutions and bifurcation for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution

Abstract

ABSTRACT The resonant nonlinear Schrödinger equation (RNLSE) with competing weakly nonlocal nonlinearity and fractional temporal evolution, which describes the propagation of optical solitons along the nonlinear optical fibers, is investigated in this paper. Dynamic behavior of such equation with the parabolic-type nonlinearity is discussed. The relationship between the orbits of the dynamic system and traveling wave solutions is demonstrated. Particularly, a family of hyperbolic curves near the saddle points implies the existence of one family of the braking wave solutions correspondingly. Furthermore, the -expansion method and Jacobi elliptic function rational expansion method are conducted to get the exact traveling wave solutions, such as periodic wave solutions, shock wave solutions, and breaking wave solutions.