Phase-modulated solitary waves controlled by a boundary condition at the bottom

Area of Science:

• Fluid dynamics
• Nonlinear wave phenomena
• Mathematical physics

Background:

• Shallow-water wave propagation is often modeled using the Korteweg-de Vries (KdV) equation.
• The influence of bottom topography on nonlinear waves is a complex phenomenon.
• Understanding these interactions is crucial for coastal and ocean engineering.

Purpose of the Study:

• To derive a forced Korteweg-de Vries (KdV) equation for weakly nonlinear, shallow-water surface waves.
• To investigate how nontrivial bottom boundary conditions affect wave evolution.
• To analyze the impact of bottom conditions on solitary wave characteristics.

Main Methods:

• Derivation of a forced KdV equation based on shallow-water theory.
• Analytical investigation of solitary wave solutions.
• Analysis of phase and amplitude modulation due to bottom boundary conditions.

Main Results:

• Different bottom boundary conditions self-consistently yield distinct forced KdV equations.
• Solitary wave solutions were obtained analytically.
• The phase of solitary waves is modulated by the bottom boundary condition, while the amplitude remains constant.

Conclusions:

• The derived forced KdV equation accurately describes wave propagation over complex seabeds.
• Bottom topography plays a significant role in shaping nonlinear shallow-water waves.
• The findings offer insights into the behavior of solitary waves in realistic oceanic environments.