Post-blowup dynamics for the nonlinear Schr\"odinger equation

TL;DR

This paper presents a detailed numerical analysis of the post-blowup behavior of solutions to the 1D focusing critical nonlinear Schrödinger equation, highlighting adiabatic and non-adiabatic regimes and their underlying mechanisms.

Contribution

It introduces a systematic numerical approach to study post-blowup dynamics, revealing the transition from adiabatic to non-adiabatic regimes and the effects of mass influx and defocusing.

Findings

Adiabatic approximation accurately describes initial post-blowup behavior.

Non-adiabatic regime involves rapid defocusing and mass influx.

In the limit of zero dissipation, critical mass is instantly radiated to infinity.

Abstract

In this work we present a systematic numerical study of the post-blowup dynamics of singular solutions of the 1D focusing critical NLS equation in the framework of a nonlinear damped perturbation. The first part of this study shows that initially the post-blowup is described by the adiabatic approximation, in which the collapsing core approaches an universal profile and the solution width is governed by a system of ODEs (reduced system). After that, a non-adiabatic regime is observed soon after the maximum of the solution, in which our direct numerical simulations show a clear deviation from the dynamics based on the reduced system. Our study suggests that such non-adiabatic regime is caused by the increasing influx of mass into the collapsing core of the solution, which is not considered in the derivation of the reduced system. Also, adiabatic theoretical predictions related to the wave-maximum and wave-dissipation are compared with our numerical simulations. The second part of this work describes the non-adiabatic dynamics. Here, numerical simulations reveal a dominant quasi linear regime, caused by the rapid defocusing process. The collapsing core approaches the universal profile, after removing some oscillations resulting from the interference with the tail. Finally, our numerical study suggests that in the limit of vanishing dissipation, and in a free-space domain, the critical mass is radiated to infinity instantly at the collapse time.