Talk:Garnier integrable system

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This page should be renamed as something like "The Garnier integrable system"

This system was discovered in:

Garnier (1919) Sur une classe de systèmes différentiels abéliens déduits de la théorie des équations linéaires, (Rendiconti del Circolo Matematico di Palermo 43, pp.155-191).

by taking the "Painlevé simplification" or "autonomous limit" of the Schlesinger system.

Several of the key ideas of Garnier’s paper were rediscovered as an offshoot of soliton theory, before Garnier’s work was rediscovered and widely disseminated, around 1980.

E.g. there is a section on it in the well-known paper of Flaschka-Newell on isomonodromy (Comm. Math. Phys. 76 (1980), 65-116), and it is mentioned in Dubrovin’s 1981 paper on theta functions, the 1980 Krichever-Novikov review ( Russian Math. Surveys 35:6 (1980), 53-79 ) and in the footnote p.156 of the 1980 paper of Jimbo-Miwa-Mori-Sato.

D.V. Chudnovsky wrote a paper on it (Let. Nuovo Cimento 26 (14) 1979), and M. Gaudin cited that in his 1983 book (La fonction d’onde de Bethe), having discovered the quantum version in 1976. Akira the east wind (talk) 15:57, 23 March 2023 (UTC)[reply]

Notation in "Schlesinger equations" Section

It is not clear what the square brackets in [A_i, A_j] mean. The A_i are complex vector valued, so I assume [.,.] is some sort of Lie Bracket. However this is not stated in the article. This should be clarified, and if there is an explicit form for [.,.] then this should be stated. 131.111.5.201 (talk) 12:42, 13 August 2024 (UTC)[reply]