Non-radial implosion for the defocusing nonlinear Schr\"odinger equation in $\mathbb{T}^d$ and $\mathbb{R}^d$

TL;DR

This paper constructs smooth, non-radial solutions to the defocusing nonlinear Schrödinger equation that develop finite-time singularities in both periodic and full space settings.

Contribution

It introduces the first known examples of non-radial solutions with finite-time implosion in the defocusing nonlinear Schrödinger equation.

Findings

Existence of smooth, non-radial solutions with finite-time singularities

Finite-time implosion occurs in both $ ext{T}^d$ and $ ext{R}^d$ settings

Advances understanding of singularity formation in nonlinear Schrödinger equations

Abstract

In this paper we construct smooth, non-radial solutions of the defocusing nonlinear Schr\"odinger equation that develop an imploding finite time singularity, both in the periodic setting and the full space.