2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116441We establish that the initial value problem for the quadratic non-linear Schrödinger equation $$ iu_t - Δu = u^2$$ where $u: \R^2 \times \R \to \C$, is locally well-posed in $H^s(\R^2)$ when $s > -1$. The critical exponent for this problem is $s_c=-1$ and previous work in \cite{c1} established local well-posedness for $s > -3/4$.Analysis of PDEs35K55text