Qualitative Description of the Particle Trajectories for the $N$-Solitons Solution of the Korteweg-de Vries Equation

TL;DR

This paper qualitatively analyzes particle trajectories in N-soliton solutions of the KdV equation, highlighting depth dependence and boundary condition issues through numerical simulations.

Contribution

It introduces a stream function-based approach to recover particle trajectories and examines the limitations of first order velocity fields in modeling free surface dynamics.

Findings

Particle trajectories depend accurately on depth for solitary waves.

Failure of the free surface kinematic boundary condition is observed.

Numerical simulations validate the qualitative behavior of trajectories.

Abstract

The qualitative properties of the particle trajectories of the $N$-solitons solution of the KdV equation are recovered from the first order velocity field by the introduction of the stream function. Numerical simulations show an accurate depth dependance of the particles trajectories for solitary waves. Failure of the free surface kinematic boundary condition for the first order type velocity field is highlighted.