Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

We numerically realize breather gas for the focusing nonlinear Schr\"odinger equation. This is done by building a random ensemble of N $\sim$ 50 breathers via Darboux transform recursive scheme in high precision arithmetics. Three types gases are synthesized according to three prototypical spectral configurations corresponding Akhmediev, Kuznetsov-Ma and Peregrine as elementary quasi-particles respective gases. The interaction properties constructed investigated propagating through them "trial" generic (Tajiri-Watanabe) comparing mean propagation velocity with predictions recently developed kinetic theory (El Tovbis, PRE 2020).