On quadratic Schrödinger equations in R 1+1 : a normal form approach

For the Schrödinger equation ut + iuxx = 〈▿〉β [u2], β ∈ (0, ½), we establish local well-posedness in Hβ−1+ (note that if β=0, this matches, up to an endpoint, the sharp result of Bejenaru–Tao [Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation, J. Funct. Anal. 233 (2006) 228–259.]). Our approach differs significantly from the previous one, 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−it∂x2 u(0) ∈ H−1/2 (R1), even though u(0) ∈ Hβ−1+. In addition, as a byproduct of our normal form analysis, we obtain a Lipschitz continuity property in H−1/2 of the solution operator (which originally acts on Hβ−1+), which is new even in the case β=0. As an easy corollary, we obtain local well-posedness results for ut + iuxx = 〈▿〉βz〈▿〉βz. Finally, we sketch an approach to obtain similar results for the equations ut + iuxx = 〈▿〉β[uū] and ut + iuxx = 〈▿〉β[ū2].