Non-integrability of the fourth Painlevé equation in the Liouville–Arnold sense

Abstract

In this paper we are concerned with the integrability of the fourth Painlevé equation (PIV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlevé equation0.1with parameters a = m, b = −2(1 + 2n + m) where , is not integrable in the Liouville–Arnold sense by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and the Ziglin–Ramis–Morales-Ruiz–Simó method yield the non-integrability results.