Modulational instability in the full-dispersion Camassa-Holm equation

Area of Science:

• Fluid dynamics
• Nonlinear wave phenomena
• Mathematical physics

Background:

• Traveling waves in nonlinear dispersive equations are crucial for understanding fluid dynamics.
• Previous models like the Camassa-Holm and Whitham equations have limitations in capturing complex wave behaviors.
• Surface tension significantly influences wave dynamics, particularly for smaller wavelengths.

Purpose of the Study:

• To determine the stability and instability of small, periodic traveling waves under long-wavelength perturbations.
• To extend the analysis to a nonlinear dispersive equation incorporating comprehensive water wave dispersion and medium-amplitude wave nonlinearities.
• To investigate the impact of surface tension on wave stability.

Main Methods:

• Analysis of a nonlinear dispersive equation extending Camassa-Holm and Whitham equations.
• Investigating stability and instability to long-wavelength perturbations.
• Comparing results with and without surface tension effects.

Main Results:

• In the absence of surface tension, results qualitatively match the Benjamin-Feir instability of Stokes waves.
• With surface tension, results align with formal asymptotic expansions and improve upon Whitham equation predictions.
• The study predicts the critical wave number in the strong surface tension limit.

Conclusions:

• The developed model provides a more accurate description of traveling wave stability, especially under surface tension.
• Findings contribute to a deeper understanding of nonlinear wave phenomena in fluid dynamics.
• The study highlights the importance of incorporating both dispersion and nonlinearity for accurate wave modeling.