Admissible solutions of delay Schwarzian differential equations
Shijian Wu
TL;DR
This work analyzes subnormal transcendental meromorphic solutions of the delay Schwarzian equation $f(z+1)f(z-1)+a(z)S(f,z)=R(z,f)=\frac{P(z,f)}{Q(z,f)}$, where $a(z)$ is rational and $P,Q$ are coprime polynomials in $f$ with rational coefficients. Leveraging Nevanlinna theory and a detailed pole-zero analysis for difference-differential polynomials, the authors prove a sharp degree bound $\deg_f R\le 7$ and a refinement $\deg_f P\le \deg_f Q+2$, with a multiplicity bound $k\le 2$ for any rational root $b_1$ of $Q(z,f)$. They provide a full classification of admissible forms according to the multiplicity structure of the roots of $Q$, and they treat a special double-root case $Q(z,f)$ having a root of multiplicity two, showing strong constraints on the zero-multiplicity of $f-b_1$ in that scenario. The results extend the landscape of Painlevé-type classifications to Schwarzian delay equations and illuminate the delicate interplay between degree growth, pole dynamics, and root structure in admissible delay-differential settings.
Abstract
In this paper, we study delay differential equations involving the Schwarzian derivative $S(f,z)$, expressed in the form \begin{equation*} f(z+1)f(z-1) + a(z)S(f,z) =R(z,f(z))= \frac{P(z,f(z))}{Q(z,f(z))} \end{equation*} where $a(z)$ is rational, $P(z,f)$ and $Q(z,f)$ are coprime polynomials in $f$ with rational coefficients. Our main result shows that if a subnormal transcendental meromorphic solution exists, then the rational function $R(z,f)=P(z,f)/Q(z,f)$ satisfies $°_fR\leq 7$ and $°_fP\leq °_fQ +2$, where $°_fR =\max\{°_fP, °_fQ\}.$ Furthermore, for any rational root $b_1$ of $Q(z,f)$ in $f$ with multiplicity $k$, we show that $k \leq 2$. Finally, a classification of such equations is provided according to the multiplicity structure of the roots of $Q(z,f)$. Some examples are given to support these results.