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“Quantizations” of the second Painlevé equation and the problem of the equivalence of its L-A pairs

“…Therefore, according to our results here, it is natural to consider the Flaschka-Newell pair not as a pair for the second Painlevé equation but as a pair for the Painlevé equation of the 34th type. This agrees with the results in [16], where the Painlevé equations were considered from the standpoint of their position in the augmented Garnier-Okamoto diagram (which was incompletely described in [17]). …”

“…reduces these equations to "quantization" (17), which has solution (16). Therefore, (22) and (23) are also solvable in the sense of Remark 2.…”

The Painlevé‐Kowalevski and Poly‐Painlevé Tests for Integrability

“…(See also Airault [20]; Clarkson [78]; Fokas and Ablowitz [102]; Gromak [140]; Murata [214]; Okamoto [219][220][221][222][223]. ) For example, consider the second Painleve equation (2.4.35):…”

“…This relati(lnship has also been exploited to obtain Backlund transformations (re[ating solutions of a given Pain[eve equation to solutions of the same equation, but with different values of the parameters), rational solutions, and one-parameter families of solutions [20,[31][32][33]102]. Classically,Fuchs [3241,Garnier [122], and Schlesinger [320] considered the Painleve equations as the isomonodromic conditions for suitable linear systems with rational coefficients possessing regular and irregular singular points (see also Okamoto [219] 72,73,107,158,159,168,169,321,171,172,173,206,207,213,219,284].…”

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

“…Учет этого условия приводит к явному выражению для H через параметры θ 1 , θ 2 , α, β, λ, µ [3]. Как было показано Окамото [4] (см. также [3], теорема A.5.2.13), при изменении λ(τ ), где λ(τ ) является решением уравнения PVI, уравнение Heun1 испытывает изомонодромную деформацию.…”