# On the nonlinear Schr\"odinger equation for waves on a nonuniform current

## TL;DR

This paper derives a variable-coefficient nonlinear Schrödinger equation to model surface waves on large-scale steady nonuniform currents, accurately capturing rogue wave formation without assuming small current velocities.

## Contribution

It introduces a new derivation of the nonlinear Schrödinger equation for nonuniform currents without smallness assumptions, enhancing wave stability analysis.

## Key findings

- Equation accurately predicts wave modulation stability loss.
- Numerical simulations confirm theoretical estimates.
- Model describes rogue wave formation on nonuniform currents.

## Abstract

A nonlinear Schr\"odinger equation with variable coefficients for surface waves on a large-scale steady nonuniform current has been derived without the assumption of a relative smallness of the velocity of the current. This equation can describe with good accuracy the loss of modulation stability of a wave coming to a counter current, leading to the formation of so called rogue waves. Some theoretical estimates are compared to the numerical simulation with the exact equations for a two-dimensional potential motion of an ideal fluid with a free boundary over a nonuniform bottom at a nonzero average horizontal velocity.