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arXiv · 1710.04944

Generalized Bonnet surfaces and Lax pairs of ${\rm P_{\rm VI}}$

Robert Conte
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Abstract

We build analytic surfaces in $\mathbb{R}cubec$ represented by the most general sixth Painlevé equation $P_{VI}$ in two steps. Firstly, the moving frame of the surfaces built by Bonnet in 1867 is extrapolated to a new, second order, isomonodromic matrix Lax pair of $P_{VI}$, whose elements depend rationally on the dependent variable and quadratically on the monodromy exponents $θ_j$. Secondly, by converting back this Lax pair to a moving frame, we obtain an extrapolation of Bonnet surfaces to surfaces with two more degrees of freedom. Finally, we give a rigorous derivation of the quantum correspondence for $P_{VI}$.

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Robert Conte. 2017-10-13. Generalized Bonnet surfaces and Lax pairs of ${\rm P_{\rm VI}}$. https://doi.org/10.1063/1.4995689

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