Cubic-quintic nonlinear Helmholtz equation: Modulational instability, chirped elliptic and solitary waves

We study the formation and propagation of chirped elliptic and solitary waves in the cubic-quintic nonlinear Helmholtz equation. This system describes nonparaxial pulse propagation in a planar waveguide with Kerr-like and quintic nonlinearities along with spatial dispersion originating from the nonparaxial effect that becomes dominant when the conventional slowly varying envelope approximation fails. We first carry out the modulational instability (MI) analysis of a plane wave in this system by employing the linear stability analysis and investigate the influence of different physical parameters on the MI gain spectra. In particular, we show that the nonparaxial parameter suppresses the conventional MI gain spectrum and also leads to a nontrivial monotonic increase in the gain spectrum near the tails of the conventional MI band, a qualitatively distinct behavior from the standard nonlinear Schrödinger system. We then study the MI dynamics by direct numerical simulations, which demonstrate the production of ultrashort nonparaxial pulse trains with internal oscillations and slight distortions at the wings. Following the MI dynamics, we obtain exact elliptic and solitary wave solutions using the integration method by considering physically interesting chirped traveling wave ansatz. In particular, we show that the system features intriguing chirped antidark, bright, gray, and dark solitary waves depending upon the nature of nonlinearities. We also show that the chirping is inversely proportional to the intensity of the optical wave. In particular, the bright and dark solitary waves exhibit unusual chirping behavior, which will have applications in the nonlinear pulse compression process.