2026/02/10 by Peter A. Clarkson, Clare Dunning, Ben Mitchell · 1 voice
Mathematics · Physics and Astronomy · #Discrete system #Discrete time and continuous time #Elliptic rational functions #Fractional Differential Equations Solutions #Laguerre polynomials #Laguerre's method #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Rational function
paper · pdf · doi:10.1007/s11040-026-09571-1
openalex created_date 2026/07/16 · openalex publication_date 2026/07/16 · openalex updated_date 2026/08/05
In this paper discrete equations are derived from Bäcklund transformations of the fifth Painlevé equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is non-uniqueness of some rational solutions of the fifth Painlevé equation. Pairs of non-unique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.