Non-integrability of some higher-order Painlevé equations in the sense of Liouville

DOI10.3842/SIGMA.2015.045zbMATH Open1377.70035arXiv1412.2867MaRDI QIDQ2353228

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Publication date: 8 July 2015

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Abstract: In this paper we study the equation ${\displaystyle }$ w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (lambda z + alpha)w + gamma, ${\displaystyle }$ which is one of the higher-order Painlev'e equations (i.e., equations in the polynomial class having the Painlev'e property). Like the classical Painlev'e equations, this equation admits a Hamiltonian formulation, B"acklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $gamma/lambda=3k$, $gamma/lambda=3k-1$, $kinmathbbZ$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the $mathrm{P}_{mathrmII}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.

Full work available at URL: https://arxiv.org/abs/1412.2867

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