# On the radius of spatial analyticity for cubic nonlinear Schr\"{o}dinger equation

## TL;DR

This paper proves that the spatial analyticity radius of solutions to the cubic nonlinear Schrödinger equation in 1D, 2D, and 3D cannot decay faster than inversely proportional to time, given initial analytic data.

## Contribution

It establishes a lower bound on the decay rate of the spatial analyticity radius for solutions to the cubic nonlinear Schrödinger equation.

## Key findings

- The radius of spatial analyticity cannot decay faster than 1/|t| as |t| approaches infinity.
- The result applies to solutions in 1D, 2D, and 3D cubic NLS equations.
- Initial data with fixed radius of analyticity leads to this decay bound.

## Abstract

It is shown that the uniform radius of spatial analyticity $\sigma(t)$ of solutions at time $t$ to the 1d, 2d and 3d cubic nonlinear Schr\"{o}dinger equations cannot decay faster than $1/|t|$ as $|t| \to \infty$, given initial data that is analytic with fixed radius $\sigma_0$.