Propagating Wave Patterns in a Derivative Nonlinear Schr\"odinger System with Quintic Nonlinearity

Exact expressions are obtained for a diversity of propagating patterns for a derivative nonlinear Schr\"odinger equation with a quintic nonlinearity. These patterns include bright pulses, fronts and dark solitons. The evolution of the wave envelope is determined via a pair of integrals of motion, and reduction is achieved to Jacobi elliptic cn and dn function representations. Numerical simulations are performed to establish the existence of parameter ranges for stability. The derivative quintic nonlinear Schr\"odinger model equations investigated here are important in the analysis of strong optical signals propagating in spatial or temporal waveguides.