Shielding of breathers for the focusing nonlinear Schrödinger equation

We study a deterministic gas of breathers for the Focusing Nonlinear Schrödinger 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 Bertola et al. (2023) for a soliton gas also to a breather gas.