The description of entire solutions for some class of complex nonlinear partial differential equation (systems) in C2C²

This article focuses on describing the entire solutions of complex nonlinear partial differential equation (systems) with two variables. Utilizing Nevanlinna theory along with the Hadamard factorization theory of meromorphic functions, we derive several results that characterize the existence and forms of finite order transcendental entire solutions for first order nonlinear partial differential equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>c</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>d</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:math> (af_z₁+bf_z₂)ᵐ(cf_z₁+df_z₂)ⁿ=1, and the systems of the partial differential equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="aligned"> <m:mtr> <m:mtd columnalign="right"> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:msub> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>c</m:mi> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>d</m:mi> <m:msub> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow>