# Effect of local Peregrine soliton emergence on statistics of random waves in the 1-D focusing Nonlinear Schr\"odinger equation

## TL;DR

This paper demonstrates how the local emergence of Peregrine solitons influences the statistical behavior of random waves in the 1-D focusing NLS equation, linking deterministic phenomena with statistical properties.

## Contribution

It introduces a numerical approach connecting deterministic Peregrine soliton events with the statistical evolution of random waves, offering new insights into rogue wave formation.

## Key findings

- Local Peregrine solitons affect wave statistics
- Experimental evidence supports the numerical results
- Provides a new tool for understanding rogue wave dynamics

## Abstract

The Peregrine soliton is often considered as a prototype of the rogue waves. After recent advances in the semi-classical limit of the 1-D focusing Nonlinear Schr\"odinger (NLS) equation this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local appearance of Peregrine solitons on the evolution of statistical properties of random waves. Evidences of this effect are found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present approach can serve as a powerful tool for the description of the transient dynamics of random waves and provide new insights into the problem of the rogue waves formation.