3�3 Lax pairs for the fourth, fifth and sixth Painlevé equations

N. Joshi, A.V. Kitaev, P.A. Treharne

Abstract

We obtain 3�3 matrix Lax pairs for systems of ODEs that are solvable in terms of the fourth, fifth and sixth Painleve equations by considering similarity reductions of the scattering Lax pair for the (2+1)-dimensional three-wave resonant interaction system. These results allow us to construct new 3�3 Lax representations for the fourth and fifth Painleve equations, together with the previously known 3�3 Lax representation for the sixth Painleve equation. By comparing these Lax pairs we obtain explicit formulas for the self-similar solutions of the three-wave system in terms of the associated Painleve equations. Finally, we give a practical application of the 3�3 system associated with the fifth Painleve equation by using it to derive an Okamoto-type Backlund transformation for P5.

Keywords Painlevé equations, Lax pairs, Backlund transformations.

AMS Subject Classification Primary 34A05; secondary 34M55.