On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Abstract

We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3 × 3 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 2 × 2 matrix Fuchs-Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3 × 3 matrix pairs, we found an integral auto-transformation 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 2 × 2 matrix Fuchs-Garnier pairs for the third Painlevé equation.

Symmetries in the third Painlevé equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

“…There are several works concerning the third Pailevé equation based on the theory of integrable systems [8], [9], [13], [18], [21]. However, there have not been any satisfactory theories presented so far which could explain the relationship between Okamoto's theory, especially the symmetry of the affine Weyl group based on a Hamiltonian equation and the τ -function, and the soliton equations realized as representations of affine Lie algebras.…”

Bäcklund transformations for certain rational solutions of Painlevé VI

“…To obtain the Jimbo-Miwa-Okamoto σ -form (1.2) of the Painlevé VI equation from the 3-component KP, the following choice of new variables was used in [1] and a similar choice was made in [5]:…”

On the linearization of the first and second Painlevé equations

“…The linear systems may be written as scalar or matrix equations and are typically referred to as Lax pairs for the Painlevé equations. In our previous paper [9] we suggested calling these systems the "Fuchs-Garnier" pairs for the Painlevé equations to pay tribute to the two scientists who first introduced these systems in the beginning of the XX-th century. In the period since Fuchs and Garnier first wrote their (scalar) pairs the list has expanded so that now, for most of the Painlevé equations, there are several different Fuchs-Garnier pairs associated with the same Painlevé equation.…”

“…The notion of secondary linearized Fuchs-Garnier pairs for the Painlevé equations was introduced in our previous work [9] and refers to Fuchs-Garnier pairs which are linear in the spectral parameter λ, see system (1.5) below. However it is important to mention that the question concerning a relation between the F N -and JM 2 -pairs was one of the main motivations for both our works.…”

“…In [9], using the similarity reductions of the Lax pair for the three-wave resonant interaction (3WRI) system, we obtained secondary linearized Fuchs-Garnier pairs in 3 × 3 matrices for all the Painlevé equations except P 1 and P 2 . As there are no similarity reductions of the 3WRI system to P 1 and P 2 this approach could not be used to obtain secondary linearized pairs for the latter equations.…”