The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Divyang G. Bhimani

doi:10.3934/dcds.2019259

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2019, Volume 39, Issue 10: 5923-5944. Doi: 10.3934/dcds.2019259

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The nonlinear Schrödinger equations with harmonic potential in modulation spaces

Divyang G. Bhimani

TIFR Centre for Applicable Mathematics, Bangalore 560 065, India

Received: October 31, 2018

Revised: December 31, 2018

Published: July 2019

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Abstract

We study nonlinear Schrödinger $ i\partial_tu-Hu = F(u) $ (NLSH) equation associated to harmonic oscillator $ H = -\Delta +|x|^2 $ in modulation spaces $ M^{p,q}. $ When $ F(u) = (|x|^{-\gamma}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $ M^{p,p}(\mathbb R^d) $ $ (1\leq p < 2d/(d+\gamma), 0<\gamma< \min \{ 2, d/2\}). $ When $ F(u) = (K\ast |u|^{2k})u $ with $ K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $ M^{\infty,1} $ (Sjöstrand's class) or $ M^{1, \infty}, $ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. As a consequence, we can get local and global well-posedness for (NLSH) in a function spaces$ - $which are larger than usual $ L^p_s- $Sobolev spaces.

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Mathematics Subject Classification: Primary: 35Q55, 35L05, 42B35; Secondary: 35A01.

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