Shielding of breathers for the focusing nonlinear Schrödinger equation

The authors investigate the focusing nonlinear Schrödinger equation through the lens of breather dynamics. They construct a deterministic breather gas by considering the N-breather solution in the limit N→∞, where the discrete scattering spectrum fills a compact domain of the complex plane and the norming constants are smoothly interpolated and scaled as 1/N. This framework generalizes recent results on soliton gases to the breather setting. A central contribution is the demonstration of a shielding effect: under suitable choices of spectral domains (including quadrature domains) and interpolating functions, the breather gas reproduces a finite breather configuration, effectively ``hiding'' the infinite background. The analysis is carried out using Riemann-Hilbert techniques, and connections are drawn with special classes of breather solutions such as Kuznetsov-Ma and Akhmediev breathers.