Necessary and sufficient condition for global existence of $L^2$ solutions for 1D periodic NLS with non-gauge invariant quadratic nonlinearity

TL;DR

This paper establishes precise necessary and sufficient conditions for the global existence of $L^2$ solutions to 1D periodic NLS with non-gauge invariant quadratic nonlinearity, focusing on Fourier mode interactions.

Contribution

It provides the exact criteria for global existence of solutions by analyzing Fourier mode interactions, advancing understanding beyond previous Fourier mode focus.

Findings

Non-trivial solutions do not exist globally under certain conditions.

The criteria precisely distinguish trivial from non-trivial global solutions.

Fourier mode interactions determine the global existence of solutions.

Abstract

We study 1D NLS with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global solutions which are constant with respect to space. The non-existence of global solutions also has been studied only by focusing on the behavior of the Fourier $0$ mode of solutions. However, the earlier works are not sufficient to obtain the precise criteria for the global existence for the Cauchy problem. In this paper, the exact criteria for the global existence of $L^2$ solutions is shown by studying the interaction between the Fourier $0$ mode and oscillation of solutions. Namely, $L^2$ solutions are shown a priori not to exist globally if they are different from the trivial ones.