Smooth parametric dependence of asymptotics of the semiclassical focusing NLS

We consider the one dimensional focusing (cubic) Nonlinear Schr"odinger equation (NLS) in the semiclassical limit with exponentially decaying complex-valued initial data, whose phase is multiplied by a real parameter. We prove smooth dependence of the asymptotic solution on the parameter. Numerical results supporting our estimates of important quantities are presented.