On solutions for several systems of complex nonlinear partial differential equations with two variables

Abstract

This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations

$$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}$$

where $a,b,c,d\in {\mathbb {C}}$, and g is a polynomial in ${\mathbb {C}}^2$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.