Monodromy Preserving Deformation, Painlevé Equations and Garnier Systems

Abstract

The most important non-linear ordinary differential equations are the following six Painlevé equations:

$$ \begin{array}{*{20}c} {P_I :\frac{{d^2 \lambda }} {{dt^2 }} = 6\lambda ^2 + t,} \\ {P_{II} :\frac{{d^2 \lambda }} {{dt^2 }} = 2\lambda ^3 + t\lambda + \alpha ,} \\ {P_{III} :\frac{{d^2 \lambda }} {{dt^2 }} = \frac{1} {\lambda }\left( {\frac{{d\lambda }} {{dt}}} \right)^2 - \frac{1} {\lambda }\frac{{d\lambda }} {{dt}} + \frac{1} {t}(\alpha \lambda ^2 + \beta ) + \gamma \lambda ^3 + \frac{\sigma } {\lambda },} \\ {P_{IV} :\frac{{d^2 \lambda }} {{dt^2 }} = \frac{1} {{2\lambda }}\left( {\frac{{d\lambda }} {{dt}}} \right)^2 - \frac{3} {2}\lambda ^3 + 4t\lambda ^4 + 2(t^2 - \alpha )\lambda + \frac{\beta } {\lambda },} \\ {P_V :\frac{{d^2 \lambda }} {{dt^2 }} = \left( {\frac{1} {{2\lambda }} + \frac{1} {{\lambda - 1}}} \right)\left( {\frac{{d\lambda }} {{dt}}} \right)^2 - \frac{1} {t}\frac{{d\lambda }} {{dt}} + \frac{{(\lambda - 1)^2 }} {t}(\sigma \lambda + \frac{\beta } {\lambda }) + \gamma \frac{\lambda } {t} + \sigma \frac{{\lambda (\lambda + 1)}} {{\lambda - 1}},} \\ {P_{VI} :\frac{{d^2 \lambda }} {{dt^2 }} = \frac{1} {2}\left( {\frac{1} {\lambda } + \frac{1} {{\lambda - 1}} + \frac{1} {{\lambda - t}}} \right)\left( {\frac{{d\lambda }} {{dt}}} \right)^2 - \left( {\frac{1} {t} + \frac{1} {{t - 1}} + \frac{1} {{\lambda - t}}} \right)\frac{{d\lambda }} {{dt}} + \frac{{\lambda (\lambda - 1)(\lambda - 1)}} {{t^2 (t - 1)^2 }}\left[ {(\sigma - \beta \frac{t} {{\lambda ^2 }} + \gamma \frac{{t - 1}} {{(\lambda - 1)^2 }} + \left( {\frac{1} {2} - \delta } \right)\frac{{t(t - 1)}} {{(\lambda - t)^2 }}} \right],} \\ \end{array} $$

where α, β, γ, δ are complex constants. (Warning: The parameters of P _VI are slitely different from those customarily used; -β and ½ - δ have been denoted by β and δ. The reason of our choice will turn out to be clear in the text.) We study, in this chapter, these differential equations first classically (§1) and secondly in the framework of the monodromy preserving deformation. After introducing the concept of monodromy preserving deformation (§2, §3), we derive the Gamier system written in the form of Hamiltonian system, which governs such deformation of a second order Fuchsian equation with n+3 singularities (§4).