Self-similar blow-up profiles for slightly supercritical nonlinear Schr\"odinger equations

TL;DR

This paper constructs self-similar blow-up profiles for the supercritical nonlinear Schrödinger equation, extending understanding of wave collapse near the mass critical case across various dimensions.

Contribution

It introduces a novel bifurcation analysis for self-similar profiles near the mass critical point, utilizing a matched asymptotics approach in a degenerate setting.

Findings

Profiles bifurcate from the ground state solitary wave

Method handles exponentially small terms in bifurcation equations

Applicable to all space dimensions d ≥ 1

Abstract

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schr\"odinger equation $i\partial_t u + \Delta u + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schr\"odinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due to the presence of exponentially small terms in the bifurcation equation related to the log-log blow-up law observed in the mass critical case.