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

## TL;DR

This paper investigates the modulational instability and solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation, revealing how nonparaxial effects influence wave stability and chirping, with implications for nonlinear pulse compression.

## Contribution

It provides the first detailed analysis of modulational instability and exact chirped solitary wave solutions in the cubic-quintic nonlinear Helmholtz system, highlighting nonparaxial effects.

## Key findings

- Nonparaxial parameter suppresses conventional MI gain spectrum.
- Production of ultra-short nonparaxial pulse trains observed.
- Exact chirped elliptic and solitary wave solutions derived.

## Abstract

We study the formation and propagation of chirped elliptic and solitary waves in cubic-quintic nonlinear Helmholtz (CQNLH) 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 (SVEA) 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 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 behaviour from the standard nonlinear Schr\"odinger (NLS) system. We then study the MI dynamics by direct numerical simulations which demonstrate production of ultra-short 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 anti-dark, 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. Especially, the bright and dark solitary waves exhibit unusual chirping behaviour which will have applications in nonlinear pulse compression process.