^{1,a)}, A. V. Kitaev

^{2,b)}and P. A. Treharne

^{3,c)}

### Abstract

We extend similarity reductions of the coupled -dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painlevé equations, together with the previously known Fuchs-Garnier pair for the sixth Painlevé equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard matrix Fuchs-Garnier pairs obtained by Jimbo and Miwa [Physica D2, 407–448 (1981)]. As an application of the matrix pairs, we found an integral autotransformation for the standard Fuchs-Garnier pair for the fifth Painlevé equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different matrix Fuchs-Garnier pairs for the third Painlevé equation.

This work was supported by ARC Grant No. DP0559019. The research was carried out during A.V.K.’s visits to the School of Mathematics and Statistics at the University of Sydney, Australia. The authors are also grateful them referee for valuable comments which helped them improve the paper.

I. INTRODUCTION

II. LAX PAIR FOR THE 3WRI SYSTEM

III. SIMILARITY REDUCTION TO THE SIXTH PAINLEVÉ EQUATION

A. The Fuchs-Garnier pair

B. Reduction of the Fuchs-Garnier pair to the pair in Jimbo-Miwa form

C. Similarity solution of the 3WRI system in terms of the sixth Painlevé equation

IV. SIMILARITY REDUCTION TO THE FIFTH PAINLEVÉ EQUATION

A. Fuchs-Garnier pair for the reduced system

B. Fuchs-Garnier pairs for the fifth Painlevé equation

C. Parametrization of solutions in terms of

D. Alternate reduction to the Fuchs-Garnier pair for

E. An Okamoto-type Bäcklund transformation for

V. SIMILARITY REDUCTION TO THE FOURTH PAINLEVÉ EQUATION

A. A Fuchs-Garnier pair for the reduced system

B. Fuchs-Garnier pairs for the fourth Painlevé equation

C. Parametrization of solutions in terms of

D. Relation between the Noumi-Yamada and Jimbo-Miwa Fuchs-Garnier pairs for the fourth Painlevé equation

VI. SIMILARITY REDUCTION TO THE THIRD PAINLEVÉ EQUATION

A. A Fuchs-Garnier pair for the reduced system

B. Fuchs-Garnier pairs for the third Painlevé equation

C. Parametrization of solutions in terms of

D. Alternate Fuchs-Garnier pairs for the third Painlevé equation

### Key Topics

- Laplace equations
- 39.0
- Matrix equations
- 28.0
- Eigenvalues
- 12.0
- Integral transforms
- 11.0
- Integral equations
- 7.0

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