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Nonlinear steepest descent asymptotics for semiclassical limit of integrable systems: Continuation in the parameter space

The initial value problem of an integrable system, such as the Nonlinear Schr\" odinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent, supplemented by the $g$-function mechanism, is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface $\Rscr=\Rscr(x,t)$ in the spectral variable, where the space-time variables $(x,t)$ play the role of external parameters. The curves in the $x,t$ plane, separating regions of different genuses of $\Rscr(x,t)$, are called breaking curves or nonlinear caustics. The genus of $\Rscr(x,t)$ is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point $x,t$. In this paper we prove that in the case of a regular break, the nonlinear steepest descent asymptotics can be "automatically" continued through the breaking curve (however, the expressions for the asymptotic solution will be different on the different sides of the curve). Although the results are stated and proven for the focusing NLS equation, it is clear that they can be reformulated for AKNS systems, as well as for the nonlinear steepest descend method in a more general setting.