The Study of Solutions of Several Systems of Nonlinear Partial Differential Difference Equations. (12th July 2022) - Xu, Hong Yan; Yu, Meiying; Zhang, Keyu
Abstract : Our main aim is to describe the entire solutions of several systems of α 1 f 1 z 2 + α 2 f 2 z + c 2 = 1, β 1 f 2 z 2 + β 2 f 1 z + c 2 = 1, α 1 ∂ f 1 / ∂ z 1 n 1 + α 2 f 2 z + c m 1 = 1, β 1 ∂ f 2 / ∂ z 1 n 2 + β 2 f 1 z + c m 2 = 1, and α 1 ∂ f 1 / ∂ z 1 2 + α 2 f 2 z + c 2 = 1, β 1 ∂ f 2 / ∂ z 1 2 + β 2 f 1 z + c 2 = 1, α 1 ∂ f 1 / ∂ z 1 2 + α 2 f 2 z + c + α 3 f 1 z 2 = 1, β 1 ∂ f 2 / ∂ z 1 2 + β 2 f 1 z + c + β 3 f 2 z 2 = 1, where α j, β j j = 1, 2, 3 are nonzero constants in ℂ and m j, n j j = 1, 2 are positive integers. We obtain several theorems on the existence and the forms of solutions for these systems, which are some improvements and supplements of the previous theorems given by Xu and Cao, Gao, and Liu and Yang. Moreover, we give some examples to explain the existence of solutions for such systems.