The Yang-Hua theorems in several complex variables

TL;DR

This paper extends classical theorems on meromorphic functions to several complex variables, providing new uniqueness results and explicit solutions for nonlinear PDEs with potential physical applications.

Contribution

It generalizes Yang and Hua's theorems to multiple complex variables and introduces new uniqueness theorems with explicit solutions for nonlinear PDEs.

Findings

Extended Yang-Hua theorems to several complex variables

Established new uniqueness theorems in higher dimensions

Derived explicit solutions with physical interpretations

Abstract

In this paper, we investigate meromorphic solutions in $\mathbb{C}^m$ of the nonlinear differential equation \[\displaystyle f^n\partial_u(f)g^n\partial_u(g)=1,\] where $\partial_u(f)=\sum_{j=1}^mu_j\partial_j(f)$ and $\sum_{j=1}^m u_j\neq 0$. Our results extend those of Yang and Hua [{\sc C. C. Yang} and {\sc X. H. Hua}, Uniqueness and value sharing of meromorphic functions, \textit{Ann. Acad. Sci. Fenn. Math.}, \textbf{22} (1997), 395-406.] to the framework of several complex variables. Moreover, we establish new uniqueness theorems that further generalize their conclusions to higher dimensions. As an application, explicit solutions of certain nonlinear partial differential equations in several variables are derived, and their physical interpretations are summarized in tabular form.