The critical mass of the Townes soliton

Find a closed form for the critical mass N_c ≈ 11.7008965246 of the Townes soliton, the collapse threshold of the 2D cubic NLS.

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1	{
2	"domain": "physics-astronomy",
3	"topics": [
4	"mathematical-physics",
5	"optics-photonics",
6	"analysis"
7	],
8	"type": "construction",
9	"source": {
10	"key": "horizonmath",
11	"name": "HorizonMath",
12	"url": "https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L411-L421",
13	"id": "townes_soliton",
14	"curator": "Erik Y. Wang, Sumeet R. Motwani et al.",
15	"status": "open",
16	"as_of": "2026-10-20",
17	"catalog": "0521c19bbd38"
18	},
19	"review": true,
20	"approval": "first-time"
21	}