Advances in Differential Equations

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.

Article information

Source
Adv. Differential Equations Volume 15, Number 9/10 (2010), 925-952.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854616

Mathematical Reviews number (MathSciNet)
MR2677424

Zentralblatt MATH identifier
1203.35266

Citation

de Moura, Roger Peres; Pilod, Didier. Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space. Adv. Differential Equations 15 (2010), no. 9/10, 925--952. https://projecteuclid.org/euclid.ade/1355854616