On matrix Painlevé-4 equations

Abstract

Using the Painlevé–Kovalevskaya test, we find several polynomial matrix systems, which can be regarded as non-commutative generalisations of the Painlevé-4 equation. For these systems isomonodromic Lax pairs are presented. Limiting transitions that reduce them to known matrix Painlevé-2 equations are found.

On classification of non-abelian Painlevé type systems

“…is a Lax representation for system (8) 3 and therefore the operators L and M k = H k M define a Lax pair for system (9). The hierarchy described above looks completely trivial: all integrals are powers of only one integral H, the symmetries and corresponding M -operators are proportional if we fix a value of H. However, all these objects become non-trivial after the non-abelianization [16,7].…”

“…One of them, Hamiltonian, was found in [12], the second has the Okamoto integral [6], and the third one is presented in Appendix A.3. All of them also arose in the paper [7], where the Painlevé-Kovalevskaya test was used for classification (see also [1]).…”

“…We assume that if m = 1 then the system (1) coincides with one of the six Painlevé systems P 1 − P 6 . Some examples of integrable non-abelian systems of the form (1) are contained in [12,4,3,7,15,1,2]. Some of them were found using the existence of special isomonodromy representations or the Painlevé-Kovalevskaya test while several of the systems have been derived from integrable PDEs and lattices by reductions.…”

“…Classification papers [4,3,7] devoted to systems of P 1 , P 2 , and P 4 were based on the matrix Painlevé-Kovalevskaya test. Unfortunately, for the P 3 , P 5 , and P 6 systems this approach turns out to be too laborious.…”

“…It turns out that known examples of non-abelian Painlevé systems satisfy some of these conditions (for P 1 and P 2 see [8]). The hierarchy of Assumption 1 can be obtained from the trivial hierarchy of the scalar auxiliary system by the non-abelianization procedure [16,7] described in Section 2.…”