Classical solutions of the degenerate fifth Painlevé equation

Abstract

In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to be expressible in terms of solutions of the third Painlevé equation. The classification and description of the classical solutions of the degenerate fifth Painlevé equation is done using the Hamiltonian associated with third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.