On the global asymptotic analysis of a q-discrete Painlevé equation

Abstract

In this thesis we make effective the global asymptotic analysis of a nonlinear q-difference Painlevé equation, whose initial value space is a rational surface of type A1(1) according to Sakai's classification. This equation can be thought of as an eight parameter generalisation of the celebrated sixth Painlevé equation, where the reduction to the differential equation goes via the continuum limit of its symmetric form. The first part of the thesis is concerned with the local asymptotic analysis of solutions near the critical points of the q-difference equation, the origin and infinity. A conjecturally complete list of possible asymptotic behaviours is found near both critical points. It is shown that, upon taking the continuum limit, the list essentially coincides with that of critical behaviours of solutions of the sixth Painlevé equation, obtained by Guzzetti. In the second part of the thesis, the integrability of the equation under consideration is exploited, to solve the nonlinear connection problem, which entails explicitly relating the critical behaviours of solutions near the two different critical points. This is done by employing a q-analog of the isomonodromic deformation method to a q-difference Lax pair devised by Yamada. The direct monodromy problem is solved, both for critical behaviours near the origin and infinity, by showing that near the critical points, the connection problem of Yamada's system factorises in two copies of a simpler connection problem, which can be solved explicitly. Comparison of the results, leads to explicit parametric connection formulae for critical behaviours of the q-difference Painlevé equation.