Semiclassical (zero dispersion limit) solutions to the focusing NLS for a special class of the initial data with both soliton and solitonless cases
We calculate the leading order term of the solution of the focusing Nonlinear (cubic) Schroedinger Equation (NLS) in the semi-classical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t ≥ 0. We utilize the Riemann-Hilbert Problem formulation of the inverse scattering problem to obtain the leading order term of the solution. Error estimates are provided. We also find the long time behavior of semi-classical solutions for solitonless cases within some range of time-small parameter dependence