We consider the long-time properties of the an obstruction in the Riemann-Hilbert approach to one dimensional focusing Nonlinear Schrodinger equation in the semiclassical limit for a one parameter family of initial conditions. For certain values of t
he parameter a large number of solitons in the system interfere with the $g$-function mechanism in the steepest descent to oscillatory Riemann-Hilbert problems. The obstruction prevents the Riemann-Hilbert analysis in a region in $(x,t)$ plane. We obtain the long time asymptotics of the boundary of the region (obstruction curve). As $ttoinfty$ the obstruction curve has a vertical asymptotes $x=pm ln 2$. The asymptotic analysis is supported with numerical results.
We consider the one dimensional focusing (cubic) Nonlinear Schrodinger 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.
A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $zlog z$ type singularity on the jump contour is considered. The jum
p function and the jump contour are assumed to depend on a vector of external parameters $vecbeta$. We prove that if the RHP has a solution at some value $vecbeta_0$ then the solution of the RHP is uniquely defined in a some neighborhood of $vecbeta_0$ and is smooth in $vecbeta$. This result is applied to the case of semiclassical focusing NLS.
