Asymptotic analysis for rarefaction problem of the focusing nonlinear Schrödinger equation with discrete spectrum

The long-time asymptotic behaviors of the rarefaction problem for the focusing nonlinear Schrödinger equation with discrete spectrum are analyzed via the Riemann–Hilbert formulation. It is shown that for the rarefaction problem with pure step initial condition there are three asymptotic sectors in time–space: the plane wave sector, the 1-phase elliptic wave sector and the vacuum sector, while for the rarefaction problem with general initial data there are five asymptotic sectors in time–space: the plane wave sector, the sector of plane wave with soliton transmission, the sector of plane wave with phase shift, the sector of 1-phase elliptic wave with phase shift and the vacuum sector with phase shift. The leading-order term of each sector along with the corresponding error estimate is given by adopting the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. The asymptotic solutions match very well with the results from Whitham modulation theory and the direct numerical simulations.