# Dynamics of internal envelope solitons in a rotating fluid of a variable depth

## TL;DR

This paper studies the behavior of internal envelope solitons in a rotating, variable-depth fluid, showing how they evolve as they move into shallower coastal zones, including amplitude, velocity, and shape changes.

## Contribution

It provides a theoretical framework for describing internal envelope solitons in variable-depth, rotating fluids using the nonlinear Schrödinger equation and analyzes their transformation in coastal zones.

## Key findings

- Solitons are most likely to have the frequency with maximum modulation instability growth.
- Solitons increase in amplitude and speed while decreasing in duration when approaching shallow regions.
- Solitons can transform into breathers as they evolve in shoaling conditions.

## Abstract

We consider the dynamics of internal envelope solitons in a two-layer rotating fluid with a linearly varying bottom. It is shown that the most probable frequency of a carrier wave which constitutes the solitary wave is the frequency where the growth rate of modulation instability is maximal. An envelope solitary wave of this frequency can be described by the conventional nonlinear Schrodinger equation. A soliton solution to this equation is presented for the time-like version of the nonlinear Schrodinger equation. When such envelope soliton enters a coastal zone where the bottom gradually linearly increases, then it experiences an adiabatical transformation. This leads to an increase of soliton amplitude, velocity, and period of a carrier wave, whereas its duration decreases. It is shown that the soliton becomes taller and narrower. At some distance it looks like a breather, a narrow nonstationary solitary wave. The dependences of soliton parameters on the distance when it moves towards the shoaling are found from the conservation laws and analysed graphically. Estimates for the real ocean are presented.