Surfaces de Bonnet et \'equations de Painlev\'e

TL;DR

This paper demonstrates that the moving frame equations of Bonnet surfaces can be extended to derive a second-order, isomonodromic matrix Lax pair for the sixth Painlevé equation, linking differential geometry and integrable systems.

Contribution

It introduces a novel connection between Bonnet surface equations and the Lax pair formulation of Painlevé VI, expanding understanding of geometric and integrable system relationships.

Findings

Derived a Lax pair from Bonnet surface equations

Linked differential geometry to Painlevé VI integrability

Extended geometric methods to integrable systems

Abstract

Nous montrons que les \'equations du rep\`ere mobile des surfaces de Bonnet conduisent \`a une paire de Lax matricielle isomonodromique d'ordre deux pour la sixi\`eme \'equation de Painlev\'e. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixth Painlev\'e equation.