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

TL;DR

This paper compares the modified nonlinear Schrödinger (MNLS) equation with standard NLS variants in the semiclassical limit, revealing diverse dynamic behaviors including subsonic, supersonic, and mixed regimes, with implications for wave and flow phenomena.

Contribution

It demonstrates how the MNLS equation can emulate focusing, defocusing, or mixed NLS dynamics depending on initial data, introducing a sonic line analogy in wave behavior.

Findings

MNLS can behave like focusing or defocusing NLS depending on initial conditions.

Identification of a sonic line separating subsonic and supersonic flow regimes.

The dynamics are analyzed using inverse-scattering and steepest descent methods.

Abstract

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.