Math Major Undergraduate Talk

Math Major Undergraduate Talk

Title: "Thresholds for Solutions Existence in the Focusing Nonlinear Schroedinger equation"
Speaker: Changkai Sun (our math major undergraduate)
 

Abstract: We study the nonlinear Schroedinger equation with focusing nonlinearity in various space dimensions. We consider finite energy and finite variance initial data in the so called mass-supercritical regime. One of the goals in such studies is to understand whether solutions evolved by the nonlinear Schroedinger evolution exist globally in time, or could form a singularity, and thus, `break down' (in a certain sense) in finite time. There have been much research done in this direction recently, and there are various theoretical thresholds available now. However, all of them lack the full picture, i.e., there are theoretical gaps in classification of the above initial data. In this numerical work, we consider Gaussian and super Gaussian initial data, we let it evolve by the nonlinear Schroedinger flow with power nonlinearities p=3,5,7 and we are able to obtain numerical thresholds which identify the borderline between the globally existing solutions and solutions which blow up in finite time. We then compare our results with the known theoretical ones and show how the "gaps" in long-time existence of solutions should be addressed.

This research is a part of Changkai Sun's senior thesis.