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 Schrödinger 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 ([4], [5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface R = R(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 R(x, t), are called breaking curves or nonlinear caustics. The genus of R(x, t) is related to the number of oscillatory phases in the asymptotic solution of the integrable system at the point x, t. An evolution theorem ([9]) guarantees the continuous evolution of the asymptotic solution in space-time away from the breaking curves. In the case of the analytic scattering data f(z;x, t) (in the NLS case, f is a normalized logarithm of the reflection coefficient with time evolution included), the primary role in the breaking mechanism is played by a phase function Ih(z;x, t), which is closely related to the g function. Namely, a break can be caused ([9]) either through the change of topology of zero level curves of Ih(z;x, t) (regular break), or through the interaction of zero level curves of Ih(z;x, t) with singularities of f (singular break). Every time a breaking curve in the x, t plane is reached, one has to prove the validity of the nonlinear steepest descent asymptotics in a region across the curve. 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). Our proof is based on the determinantal formula for h(z;x, t) and its space and time derivatives, obtained in [7], [8]. Although the results are stated and proven for the focusing NLS equation, it is clear ([8]) that they can be reformulated for AKNS systems, as well as for the nonlinear steepest descend method in a more general setting.