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

Area of Science:

• Nonlinear Optics
• Wave Propagation Physics
• Mathematical Physics

Background:

• The cubic-quintic nonlinear Helmholtz equation models nonparaxial pulse propagation in planar waveguides.
• This system accounts for Kerr-like, quintic nonlinearities, and spatial dispersion beyond the slowly varying envelope approximation.

Purpose of the Study:

• Investigate modulational instability (MI) in the nonlinear Helmholtz equation.
• Analyze the formation and characteristics of chirped elliptic and solitary waves.
• Explore the impact of nonparaxial effects on wave dynamics and stability.

Main Methods:

• Linear stability analysis for modulational instability gain spectra.
• Direct numerical simulations to observe MI dynamics.
• Integration methods with chirped traveling wave ansatz for exact solutions.

Main Results:

• Nonparaxial parameter suppresses conventional MI gain and creates unique spectral features.
• Numerical simulations show ultrashort pulse train generation with oscillations.
• Exact solutions reveal diverse chirped solitary waves (antidark, bright, gray, dark).

Conclusions:

• Nonparaxial effects significantly alter MI and lead to novel solitary wave solutions.
• The system supports unique chirped solitary waves with potential applications.
• Chirping behavior is inversely proportional to optical wave intensity, particularly for bright and dark solitons.