Explicit solution of the problem of equivalence for some Painleve equations

TL;DR

This paper develops a test to determine if a second order differential equation is equivalent to Painleve I, II, or III with zero parameters, providing explicit variable transformations using differential invariants.

Contribution

It introduces a method to verify equivalence to Painleve equations and explicitly constructs variable changes for Painleve I and II cases.

Findings

A test for equivalence to Painleve equations is constructed.

Explicit transformations are provided for Painleve I and II.

The method uses differential invariants for variable substitution.

Abstract

For an arbitrary ordinary second order differential equation a test is constructed that checks if this equation is equivalent to Painleve I, II or Painleve III with three zero parameters equations under the substitutions of variables. If it is true then in case the Painleve equations I and II an explicite change of variables is given that is written using the differential invariants of the equation.