On the modified nonlinear Schr\"odinger equation in the semiclassical limit: supersonic, subsonic, and transsonic behavior

The purpose of this paper is to present a comparison between the modified nonlinear Schr\"odinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr\"odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.Comment: 30 pages, 2 figures. Submitted to Acta Mathematica Scientia (special issue in honor of Peter Lax's 85th birthday

The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation Theory and Spectral Analysis

We study an integrable modification of the focusing nonlinear Schroedinger equation from the point of view of semiclassical asymptotics. In particular, (i) we establish several important consequences of the mixed-type limiting quasilinear system including the existence of maps that embed the limiting forms of both the focusing and defocusing nonlinear Schroedinger equations into the framework of a single limiting system for the modified equation, (ii) we obtain bounds for the location of discrete spectrum for the associated spectral problem that are particularly suited to the semiclassical limit and that generalize known results for the spectrum of the nonselfadjoint Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family of initial data for which we solve the associated spectral problem in terms of special functions for all values of the semiclassical scaling parameter. We view our results as part of a broader project to analyze the semiclassical limit of the modified nonlinear Schroedinger equation via the noncommutative steepest descent procedure of Deift and Zhou, and we also present a self-contained development of a Riemann-Hilbert problem of inverse scattering that differs from those given in the literature and that is well-adapted to semiclassical asymptotics.Comment: 56 Pages, 21 Figure