Almost global existence for cubic nonlinear Schr\"odinger equations in one space dimension

TL;DR

This paper proves that solutions to certain cubic nonlinear Schrödinger equations in one dimension exist globally with sharp decay estimates up to an exponentially large time scale, and demonstrates norm growth beyond this period for specific nonlinearities.

Contribution

It establishes almost global existence and decay estimates for non-gauge-invariant cubic NLS in one dimension, extending understanding of long-time behavior.

Findings

Solutions decay sharply up to exponential time scale

Norm growth occurs beyond this time for some nonlinearities

Initial data of size epsilon lead to controlled solutions for long times

Abstract

We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time $\exp(C\varepsilon^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.