A new class of critical solutions for 1D cubic NLS

TL;DR

This paper establishes the existence of a new class of solutions for the 1D cubic nonlinear Schrödinger equation with initial data involving Dirac masses at critical regularity, using scattering and oscillatory integral techniques.

Contribution

It introduces a novel class of solutions for 1D cubic NLS with critical initial data related to Dirac masses, expanding understanding at critical regularity levels.

Findings

Existence of solutions with initial data as sum of Dirac masses

Solutions belong to $ ext{dot} H^s$ for all $s < -1/2$

Use of scattering approach and oscillatory integral estimates

Abstract

The aim of this article is to prove the existence of a new class of solutions of 1D cubic NLS with an initial data related to a sum of Dirac masses, of critical regularity $F(L^\infty)$, and belonging to $\dot H^s$ for any $s <-1/2$. This problem is motivated by the lack of result for critical regularity initial condition. Our result is based on a scattering approach, after performing a pseudo-conformal transformation, and on fine estimations of oscillatory integrals.