Particle trajectories in nonlinear Schrodinger models

TL;DR

This paper demonstrates that the velocity potential and particle trajectories in nonlinear Schrödinger models, including higher-order and viscous variants, can be reconstructed efficiently, expanding the understanding of flow fields in wave motion.

Contribution

It introduces a method to reconstruct the velocity potential and particle trajectories in nonlinear Schrödinger models, including higher-order and viscous effects, which was previously underexplored.

Findings

Velocity potential can be reconstructed similarly to surface profile.

Particle trajectories can be described in higher-order models.

The approach applies to models with viscous effects.

Abstract

The nonlinear Schrodinger equation is well known as a universal equation in the study of wave motion. In the context of wave motion at the free surface of an incompressible fluid, the equation accurately predicts the evolution of modulated wave trains with low to moderate wave steepness. While there is an abundance of studies investigating the reconstruction of the surface profile $\eta$, and the fidelity of such profiles provided by the nonlinear Schrodinger equation as predictions of real surface water waves, very few works have focused on the associated flow field in the fluid. In the current work, it is shown that the velocity potential $\phi$ can be reconstructed in a similar way as the free-surface profile. This observation opens up a range of potential applications since the nonlinear Schrodinger equation features fairly simple closed-form solutions and can be solved numerically with comparatively little effort. In particular, it is shown that particle trajectories in the fluid can be described with relative ease not only in the context of the nonlinear Schrodinger equation, but also in higher-order models such as the Dysthe equation, and in models incorporating certain types of viscous effects.