Exact meromorphic solutions of Schwarzian differential equations

All meromorphic solutions of the autonomous Schwarzian differential equations

“…As 𝑐 ≠ 0, we must have 𝜎 2 = 1, which is a contradiction. Finally, we can see from the results in [10] that (1.10) has no nonconstant rational solutions, which completes the proof.…”

“…The combination of Theorem 1 and the results presented in [10] leads to the following result. Theorem All transcendental meromorphic solutions of the autonomous Schwarzian differential equation (1.4) can be constructed explicitly and are provided in the Appendix.…”

“…Based on Theorem 1 and the results presented in [10], we list all transcendental meromorphic solutions of the six canonical autonomous Schwarzian differential equations in…”

“…Suppose it has a nonconstant rational solution $$u$$. Then $$u$$ has the following properties [10]: $$u^{\prime }(z)=0$$ if and only if $$u(z)=\tau _j$$, $$j=1, 2, 3, 4;$$ all zeros of $$u^{\prime }$$ are simple; all poles of $$u$$ are simple. Thus, the rational function $$$$\begin{equation} \frac{{u^{\prime }}^2}{(u-\tau _1)(u-\tau _2)(u-\tau _3)(u-\tau _4)} \end{equation}$$$$has neither zeros nor poles, which implies that it must be a nonzero constant, denoted by $$k$$. Therefore, (4.1) can be expressed as $$...$$…”

“…Recently, two of the authors [10] have presented a comprehensive construction of all transcendental meromorphic solutions of Equations (1.5)- (1.8) and (1.10). Specifically, all of them are elliptic functions for Equations (1.5)- (1.8).…”