Low regularity for a quadratic Schrödinger equation on $\mathbb{T}$

Abstract

In this paper we consider a Schrödinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a better description of the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega [15]. We also show that the equation is well posed in a space $\mathcal H^{s,p}(\mathbb T)$ which contains the Sobolev space $H^{s}(\mathbb T)$ when $p\geq 2$.