## Advances in Differential Equations

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

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

Didier Pilod and Roger Peres de Moura

#### 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

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 35B45: A priori estimates 76B55: Internal waves

#### 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