# Rogue Waves and Large Deviations in Deep Sea

## TL;DR

This paper develops a probabilistic and computational framework combining large deviations theory and Monte Carlo methods to predict rogue waves in deep sea, identifying early warning signs and specific wave patterns.

## Contribution

It introduces an efficient optimization-based approach to estimate rogue wave probabilities and precursors using the modified nonlinear Schrödinger equation with random initial conditions.

## Key findings

- Rogue waves are linked to unlikely wave configurations triggering large disturbances.
- Specific wave shape patterns serve as early precursors for rogue waves.
- The method effectively estimates tail probabilities of extreme sea surface elevations.

## Abstract

The appearance of rogue waves in deep sea is investigated using the modified nonlinear Schr\"odinger (MNLS) equation in one spatial-dimension with random initial conditions that are assumed to be normally distributed, with a spectrum approximating realistic conditions of a uni-directional sea state. It is shown that one can use the incomplete information contained in this spectrum as prior and supplement this information with the MNLS dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height but with a very specific shape that is identified explicitly, thereby allowing for early detection. The method proposed here combines Monte Carlo sampling with tools from large deviations theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can be solved efficiently. This approach is transferable to other problems in which the system's governing equations contain random initial conditions and/or parameters.