Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schr & ouml;dinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non- vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schr & ouml;dinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift-Zhou nonlinear steepest-descent method for Riemann- Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.