Wave-packet formation at the zero-dispersion point in the Gardner-Ostrovsky equation

Area of Science:

• Fluid dynamics
• Nonlinear wave phenomena

Background:

• Internal solitary waves decay into inertia-gravity waves over time due to weak rotation.
• The formation mechanism of the resulting nonlinear wave packets remains unexplained.

Purpose of the Study:

• To investigate the formation mechanism of nonlinear wave packets from decaying solitary waves under rotation.
• To explain the role of the zero-dispersion point in this phenomenon.

Main Methods:

• Analysis of the initial value problem using the Gardner-Ostrovsky equation (rotation-modified extended Korteweg-de Vries).
• Investigation of the linear Gardner-Ostrovsky equation's properties at the zero-dispersion point.
• Numerical comparisons between the Gardner-Ostrovsky equation and a derived nonlinear Schrödinger equation.

Main Results:

• A nonlinear splitting of the wave-number spectrum at the zero-dispersion point is identified as the cause of wave packet formation.
• Energy is shifted into the modulationally unstable regime of the Gardner-Ostrovsky equation.
• Numerical results confirm the spectral splitting and its role in solitary wave decay.

Conclusions:

• Nonlinear spectral splitting at the zero-dispersion point is the key mechanism for nonlinear wave packet formation from decaying solitary waves.
• This finding provides a satisfactory explanation for a previously poorly understood phenomenon in fluid dynamics.