Local well-posedness for quadratic Schr\"odinger equations in $\mathbf{R^{1+1}}$: a normal form approach

TL;DR

This paper establishes local well-posedness for a class of quadratic Schrödinger equations in one spatial dimension using a novel normal form transformation approach, improving understanding of solution regularity and stability.

Contribution

It introduces a normal form method to analyze quadratic Schrödinger equations, achieving well-posedness results in low regularity spaces and establishing Lipschitz continuity of the solution map.

Findings

Proves local well-posedness in $H^{eta-1+}$ for $0<eta<1/2$.

Shows the solution differs from the linear evolution by an element in $H^{-1/2}$.

Establishes Lipschitz continuity of the solution operator in $H^{-1/2}$.

Abstract

For the Schr\"odinger equation $u_t+i u_{xx}=\nab^\be[u^2]$, $\be\in (0,1/2)$, we establish local well-posedness in $H^{\be-1+}$ (note that if $\be=0$, this matches, up to an endpoint, the sharp result of Bejenaru-Tao, \cite{BT}). Our approach differs significantly from the previous ones in that we use normal form transformation to analyze the worst interacting terms in the nonlinearity and then show that the remaining terms are (much) smoother. In particular, this allows us to conclude that $u-e^{-i t \p_x^2} u(0)\in H^{-\f{1}{2}}(\rone)$, even though $u(0)\in H^{\be-1+}$. In addition and as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in $H^{-\f{1}{2}}$ of the solution operator (which originally acts on $H^{\be-1+}$), which is new even in the case $\be=0$. As an easy corollary, we obtain local well-posedness results for $u_t+ i u_{xx} = [\nab^{\beta} u]^2$. Finally, we sketch an approach to obtain similar statements for the equations $u_t+i u_{xx}=\nab^\be[u\bar{u}]$ and $u_t+i u_{xx}=\nab^\be[\bar{u}^2]$.