The quintic NLS as the mean field limit of a Boson gas with three-body interactions
Abstract
We investigate the dynamics of a boson gas with three-body interactions in dimensions $d=1,2$. We prove that in the limit of infinite particle number, the BBGKY hierarchy of $k$-particle marginals converges to a limiting (Gross-Pitaevskii (GP)) hierarchy for which we prove existence and uniqueness of solutions. Factorized solutions of the GP hierarchy are shown to be determined by solutions of a quintic nonlinear Schrödinger equation. Our proof is based on, and extends, methods of Erdös-Schlein-Yau, Klainerman-Machedon, and Kirkpatrick-Schlein-Staffilani.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2008
- DOI:
- 10.48550/arXiv.0812.2740
- arXiv:
- arXiv:0812.2740
- Bibcode:
- 2008arXiv0812.2740C
- Keywords:
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- Mathematical Physics;
- Mathematics - Analysis of PDEs;
- 35Q55;
- 81V70
- E-Print:
- AMS Latex, 35 pages