Asymptotic Analysis of Transcendental Solutions of Nonlinear Systems

Abstract

Asymptotic analysis is central to mathematical physics and applied mathematics, offering insight into the behaviour of complex systems in limiting regimes. This thesis investigates the asymptotic properties of transcendental solutions of nonlinear systems, focusing on Painlevé equations in both their continuous and q-difference forms. Painlevé equations are a class of nonlinear ODEs notable for the Painlevé property: their solutions have no movable branch points or essential singularities. These equations arise in diverse contexts— from statistical mechanics and quantum field theory to integrable systems and special functions—and their transcendental solutions exhibit rich asymptotic structures. Beyond their classical formulations, q-difference Painlevé equations have become prominent in discrete integrable systems, with links to orthogonal polynomials, lattice models, and combinatorics. These discrete analogues mirror their continuous counterparts' structural and integrable properties, extending the scope of Painlevé analysis.