On the modified nonlinear Schrödinger equation in the semiclassical limit: supersonic, subsonic, and transsonic behavior

The purpose of this paper is to present a comparison between the modified nonlinear Schrödinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schrödinger (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.