On meromorphic solutions of certain partial differential equations

Lü, Feng

Canadian mathematical bulletin, Tome 68 (2025) no. 4, pp. 1177-1191

Résumé

In this article, we describe meromorphic solutions of certain partial differential equations, which are originated from the algebraic equation $P(f,g)=0$, where P is a polynomial on $\mathbb {C}^2$. As an application, with the theorem of Coman–Poletsky, we give a proof of the classic theorem: Every meromorphic solution $u(s)$ on $\mathbb {C}$ of $P(u,u')=0$ belongs to W, which is the class of meromorphic functions on $\mathbb {C}$ that consists of elliptic functions, rational functions and functions of the form $R(e^{a s})$, where R is rational and $a\in \mathbb {C}$. In addition, we consider the factorization of meromorphic solutions on $\mathbb {C}^n$ of some well-known PDEs, such as Inviscid Burgers’ equation, Riccati equation, Malmquist–Yosida equation, PDEs of Fermat type.

Détail

DOI : 10.4153/S0008439525000347

Mots-clés : Meromorphic solution, factorization, partial differential equations, pseudo-prime