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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	2	+ "domain": "physics-astronomy",
	3	+ "topics": [
	4	+ "mathematical-physics",
	5	+ "optics-photonics",
	6	+ "analysis"
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	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	+}