Soliton interaction with external forcing within the Korteweg-de Vries equation

Area of Science:

• Nonlinear physics
• Fluid dynamics
• Mathematical physics

Background:

• The Korteweg-de Vries (KdV) equation models shallow water waves and other phenomena.
• Previous research on the forced KdV equation focused on limited width ratios between solitons and perturbations.
• Understanding resonant interactions is crucial for various physical systems.

Purpose of the Study:

• To revise and extend solutions for the forced KdV equation.
• To investigate solitary wave interactions with external pulse-type perturbations for arbitrary width relationships.
• To analyze cases beyond the narrow forcing or narrow soliton approximations.

Main Methods:

• Application of an asymptotic method to derive approximate equations.
• Derivation of second-order approximate equations based on a small parameter for external force amplitude.
• Analysis of exact solutions for the derived approximate equations.
• Comparison with direct numerical modeling.

Main Results:

• Developed a method to handle arbitrary width relationships between solitons and perturbations.
• Obtained approximate equations and analyzed their exact solutions.
• Graphical illustrations of theoretical outcomes.
• Demonstrated good agreement between asymptotic method results and numerical modeling.

Conclusions:

• The asymptotic method provides accurate approximations for forced KdV equation solutions.
• The study expands the understanding of resonant solitary wave interactions.
• Findings are validated by direct numerical simulations, confirming theoretical predictions.