Small data blow-up of L^{2}-solution for the nonlinear Schr\"odinger equation without gauge invariance

TL;DR

This paper demonstrates that solutions to the nonlinear Schrödinger equation can experience blow-up in their L^{2}-norm without size restrictions, under certain sign conditions on initial data.

Contribution

It proves L^{2}-norm blow-up for the nonlinear Schrödinger equation without gauge invariance, under specific sign conditions on initial data.

Findings

L^{2}-norm blow-up occurs for solutions with suitable initial data.

No size restrictions are needed on initial data for blow-up.

Sign conditions on initial data are crucial for the blow-up result.

Abstract

We study the initial value problem for the nonlinear Schr\"odinger equation. We will prove that the blow-up of the L^{2}-norm of solutions with suitable initial data. We impose a condition related to the sign of the data but put no restriction on their size.