Classical Solutions of the Degenerate Fifth Painlev\'e Equation

TL;DR

This paper classifies classical solutions of the degenerate fifth Painlevé equation, including algebraic and Bessel function solutions, and explores their applications to the sine-Gordon equation and polynomial recurrence coefficients.

Contribution

It provides a comprehensive classification of classical solutions for the degenerate fifth Painlevé equation, including hierarchies and special function solutions, with applications to integrable systems.

Findings

Classified algebraic and Bessel function solutions.

Derived exact solutions for the complex sine-Gordon equation.

Connected solutions to recurrence relations of generalized Charlier polynomials.

Abstract

In this paper classical solutions of the degenerate fifth Painlev\'e equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlev\'e equation are known to expressible in terms of the third Painlev\'e 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.