Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space

Abstract

We establish local well posedness for arbitrarily large initial data in the usual Sobolev spaces $H^{s}({\mathbb{R}}),$ $s>\frac{1}{2},$ for the Cauchy problem associated to the integro-differential equation $$ \partial_{t}u+i\alpha\partial^{2}_{x}u=\beta u\left(1+i\mathcal{T}_h\right) \partial_{x}(\left|u\right|^{2})+i\gamma|u|^{2}u, $$ where $u=u(x,t)\in{\mathbb{C}},$ $x, t\,\in{\mathbb{R}}$, and $\mathcal{T}_h$ denotes the singular operator defined by $$ \mathcal{T}_{h}f(x)=\frac{1}{2h}\,\mbox{p.v.} \int^{\infty}_{-\infty}\coth\left(\frac{\pi(x-y)}{2h}\right) f(y)dy, $$ when $0 < h\le \infty$. Note that $\mathcal{T}_{\infty}=\mathcal{H}$ is the Hilbert transform. Our method of proof relies on a gauge transformation localized in positive frequencies which allows us to weaken the high-low frequencies interaction in the nonlinearity.