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September/October 2010 Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space
Didier Pilod, Roger Peres de Moura
Adv. Differential Equations 15(9/10): 925-952 (September/October 2010).

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.

Citation

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Didier Pilod. Roger Peres de Moura. "Local well-posedness for the nonlocal nonlinear Schrödinger equation below the energy space." Adv. Differential Equations 15 (9/10) 925 - 952, September/October 2010.

Information

Published: September/October 2010
First available in Project Euclid: 18 December 2012

zbMATH: 1203.35266
MathSciNet: MR2677424

Subjects:
Primary: 35B45, 35Q55, 76B55

Rights: Copyright © 2010 Khayyam Publishing, Inc.

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Vol.15 • No. 9/10 • September/October 2010
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