Low regularity for a quadratic Schr\"odinger equation on the circle

TL;DR

This paper analyzes the dynamics of a quadratic Schrödinger equation on the circle, providing explicit computations of solution iterates and establishing well-posedness in frequency-based Lp spaces.

Contribution

It offers a detailed analysis of the solution dynamics and proves well-posedness in new function spaces for the quadratic Schrödinger equation on the circle.

Findings

Explicit computation of the first Picard iterate

Solution well-posedness in Lp-based frequency spaces

Enhanced understanding of low regularity solutions

Abstract

In this paper we consider a Schrodinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a precision on the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega. We also show that the equation is well-posed in a space based on Lp norms in frequencies.