On the one dimensional cubic NLS in a critical space

TL;DR

This paper investigates the initial value problem for the one-dimensional cubic nonlinear Schrödinger equation in a critical space, focusing on initial data composed of equispaced Dirac deltas with small amplitudes, motivated by a geometric flow problem.

Contribution

It introduces a novel analysis of the cubic NLS with Dirac delta initial data in a critical space, linking it to a geometric problem involving curve flows in three dimensions.

Findings

Establishes well-posedness under smallness conditions for Dirac delta initial data.

Connects the NLS problem to a geometric flow of curves in three dimensions.

Provides insights into the behavior of solutions with singular initial data.

Abstract

In this note we study the initial value problem in a critical space for the one dimensional Schr\"odinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.