Ground state solutions of the complex Gross Pitaevskii equation associated to exciton-polariton Bose-Einstein condensates

Hichem Hajaiej; Slim Ibrahim; Nader Masmoudi

doi:10.1016/j.matpur.2021.01.010

Ground state solutions of the complex Gross Pitaevskii equation associated to exciton-polariton Bose-Einstein condensates

Hichem Hajaiej, Slim Ibrahim, Nader Masmoudi

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schrödinger equation and prove some corresponding laws.

Original language	English (US)
Pages (from-to)	1-23
Number of pages	23
Journal	Journal des Mathematiques Pures et Appliquees
Volume	148
DOIs	https://doi.org/10.1016/j.matpur.2021.01.010
State	Published - Mar 1 2021
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2022-06-15
Acknowledgements: S. I. was supported by NSERC grant (371637-2019). N. M was partially supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. The authors would like to thank Peter Markowich for proposing them this problem, and are grateful to staff of King Abdullah University of Science and Technology for their great hospitality.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

ASJC Scopus subject areas

Applied Mathematics
General Mathematics

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10.1016/j.matpur.2021.01.010

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title = "Ground state solutions of the complex Gross Pitaevskii equation associated to exciton-polariton Bose-Einstein condensates",

abstract = "We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schr{\"o}dinger equation and prove some corresponding laws.",

author = "Hichem Hajaiej and Slim Ibrahim and Nader Masmoudi",

note = "KAUST Repository Item: Exported on 2022-06-15 Acknowledgements: S. I. was supported by NSERC grant (371637-2019). N. M was partially supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. The authors would like to thank Peter Markowich for proposing them this problem, and are grateful to staff of King Abdullah University of Science and Technology for their great hospitality. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Hajaiej, Hichem

AU - Ibrahim, Slim

AU - Masmoudi, Nader

N1 - KAUST Repository Item: Exported on 2022-06-15 Acknowledgements: S. I. was supported by NSERC grant (371637-2019). N. M was partially supported by NSF grant DMS-1716466 and by Tamkeen under the NYU Abu Dhabi Research Institute grant of the center SITE. The authors would like to thank Peter Markowich for proposing them this problem, and are grateful to staff of King Abdullah University of Science and Technology for their great hospitality. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schrödinger equation and prove some corresponding laws.

AB - We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schrödinger equation and prove some corresponding laws.

UR - http://hdl.handle.net/10754/660836

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DO - 10.1016/j.matpur.2021.01.010

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JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -

Ground state solutions of the complex Gross Pitaevskii equation associated to exciton-polariton Bose-Einstein condensates

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