Shielding of breathers for the focusing nonlinear Schr\"odinger equation

TL;DR

This paper investigates the behavior of a large ensemble of breathers in the focusing nonlinear Schrödinger equation, extending the shielding effect known for soliton gases to breather gases through a limiting spectral distribution approach.

Contribution

It introduces a novel limiting process for a gas of breathers, demonstrating the shielding effect in breather gases similar to that previously observed in soliton gases.

Findings

The breather gas behaves as finite breather solutions under certain conditions.

The shielding effect extends from soliton gases to breather gases.

A new spectral limit approach for breather ensembles is developed.

Abstract

We study a deterministic gas of breathers for the Focusing Nonlinear Schr\"odinger equation. The gas of breathers is obtained from a $N$-breather solution in the limit $N\to \infty$.\\ The limit is performed at the level of scattering data by letting the $N$-breather spectrum to fill uniformly a suitable compact domain of the complex plane in the limit $N\to\infty$. The corresponding norming constants are interpolated by a smooth function and scaled as $1/N$. For particular choices of the domain and the interpolating function, the gas of breathers behaves as finite breathers solution. This extends the shielding effect discovered in "M. Bertola, T. Grava, and G. Orsatti - Physical Review Letters, 130.12 (2023): 1" for a soliton gas also to a breather gas.