Ring-type multisoliton dynamics in shallow water

Area of Science:

• Fluid Dynamics
• Nonlinear Wave Phenomena
• Mathematical Physics

Background:

• Nonlinear ring waves in shallow water are crucial for understanding complex fluid dynamics.
• The cylindrical Korteweg-de Vries equation (cKdVE) models weakly nonlinear, weakly dispersive ring waves in ideal fluids.
• Previous research has explored localized solutions, but a clear distinction between analytical and numerical findings in cylindrical geometry is needed.

Purpose of the Study:

• To theoretically investigate the propagation of nonlinear ring waves as multisolitons in shallow water.
• To solve the cylindrical Korteweg-de Vries equation (cKdVE) both analytically and numerically.
• To compare the spatiotemporal evolution and characteristics of analytically derived tilted solitons versus numerically obtained standard solitons.

Main Methods:

• Analytical solutions of the cKdVE to identify localized, tilted soliton solutions with oblique asymptotes.
• Numerical simulations of the cKdVE to obtain localized, standard ring soliton structures.
• Detailed analysis of wave dynamics, including break-up, multiplet formation, pulse overlapping, and amplitude-width complementarity for both solution types.

Main Results:

• Analytically found tilted solitons exhibit an oblique asymptote, while numerically found standard solitons have vanishing wings.
• Both analytical and numerical solutions demonstrate soliton-like behavior, preserving their character throughout propagation.
• Tilted solitons maintain a constant amplitude-width product, whereas standard solitons show a decreasing product due to faster amplitude decay than width increase.

Conclusions:

• The study distinguishes between two types of localized ring solitons in shallow water based on boundary conditions.
• The 'amplitude-width' complementarity is observed in both cases, but its mathematical preservation differs between analytical and numerical solutions.
• The findings offer new insights into nonlinear wave propagation in cylindrical geometries, contrasting with planar models and prior investigations.