Linear Instability of the Peregrine Breather: Numerical and Analytical Investigations

TL;DR

This paper demonstrates that the Peregrine breather, a solution to the nonlinear Schrödinger equation, is linearly unstable through analytical derivations and numerical simulations, highlighting its instability in physical systems.

Contribution

It provides the first combined analytical and numerical analysis confirming the linear instability of the Peregrine breather.

Findings

Analytical proof of instability via eigenfunction products.

Numerical simulations show exponential growth of perturbations.

Confirms the Peregrine breather's instability in physical models.

Abstract

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schr\"odinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov-Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.