Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Abstract

In this paper, we investigate the dynamics of a boson gas with three-body interactions in T 2 . We prove that when the particle number N tends to infinity, the BBGKY hierarchy of k-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized N -body initial datum, we show that this N → ∞ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.2000 Mathematics Subject Classification. Primary: 35L15, 35L45; Secondary: 35Q40.

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“…These three-body interactions can influence various properties of the system, such as its equation of state, phase transitions. In the pass few years, the BECs with three-body interactions received attention of many researchers and many mathematical research works dealt with the three-body problem [6,7,47,46,8,36,26,33,32,34,39]. In particular, it was consider in [46,26] a system of combined scaled few-body interactions with the same scaled parameter.…”