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. We consider a one-dimensional, trapped, focusing Bose gas where \ (N\) bosons interact with each other via both a two-body interaction potential of the form \ (aN^ -1 U (N^ (x-y) ) \) and an attractive three-body interaction potential of the form \ (-bN^2 -2 W (N^ (x-y, x-z) ) \), where \ (a R\), \ (b, 0\), \ (0 1\), \ (U, W 0\), and \ (ₑU (x) \;d x=1 = ₑℂ W (x, y) \;d x \;d y\). The system is stable either for any \ (a R\) as long as \ (b b: = 3²/2\) —the critical strength of the one-dimensional focusing quintic nonlinear Schrödinger (NLS) equation— or for \ (a 0\) when \ (b=b\). In the former case, fixing \ (b (0, b) \), we prove that in the mean-field limit the many-body system exhibits the Bose–Einstein condensation on the cubic-quintic NLS ground states. When assuming \ (b=bN b\) and \ (a=aN 0\) as \ (N \), with the former convergence being slow enough and "not faster" than the latter, we prove that the ground state of the system is fully condensed on the (unique) solution to the quintic NLS equation. In the latter case, \ (b=b\) fixed, we obtain the convergence of many-body energy for small \ (\) when \ (a 0\) is fixed. Finally, we analyze the behavior of the many-body ground states when the convergence \ (bN b\) is "faster" than the slow enough convergence \ (0 aN 0\). KeywordsBose gasthree-body interactionBose–Einstein condensationnonlinear SchrödingerMSC codes81V7035J1035Q5582B1082D05
Nguyen et al. (Fri,) studied this question.