On Semiclassical (Zero Dispersion Limit) Solutions to the Focusing NLS for a Certain Family of Rapidly Decaying Initial Data

We derive the expression for the leading order term of the solution (and establish its accuracy) by solving the inverse scattering problem. This is a singularly perturbed problem that can be solved by utilizing the Riemann-Hilbert problem technique. Long time asymptotics of this leading term are also discussed.