On meromorphic solutions of Malmquist type difference equations

TL;DR

This paper classifies meromorphic solutions of Malmquist type difference equations using Nevanlinna theory, completes previous classifications, and explores their connections to differential equations and integrable maps.

Contribution

It completes the classification of solutions for a key case and links difference equations to differential equations and integrable systems.

Findings

Identified a previously omitted equation in the classification.

Derived continuum limits connecting difference and differential equations.

Showed relations between difference equations and integrable maps.

Abstract

Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form $f(z+1)^n=R(z,f)$ $(\dag)$ that have transcendental meromorphic solutions, where $R(z,f)$ is rational in both arguments. In this paper, we first complete the classification for the case $\deg_{f}(R(z,f))=n$ of~$(\dag)$ by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on~$(\dag)$. Then, we study the relations between $(\dag)$ and its differential counterpart $(f')^n=R(z,f)$. We show that most autonomous equations, singled out from~$(\dag)$ with $n=2$, have a natural continuum limit to either the differential Riccati equation $f'=a+f^2$ or the differential equation $(f')^2=a(f^2-\tau_1^2)(f^2-\tau_2^2)$, where $a\not=0$ and $\tau_i$ are constants such that $\tau_1^2\not=\tau_2^2$. The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.