Results on Solutions for Several q-Painlevé Difference Equations concerning Rational Solutions, Zeros, and Poles

Bu Sheng Li,Xiu Min Zheng,Rui Ying,Hong Yan Xu

doi:10.1155/2020/3781942

Abstract

In this article, we discuss the problem about the properties on solutions for several types of q-difference equations and obtain some results on the exceptional values of transcendental meromorphic solutions fz with zero order, their q-differences Δqfz=fqz−fz, and divided differences Δqfz/fz. In addition, we also investigated the condition on the existence of rational solution for a class of q-difference equations. Our theorems are some extensions and supplement to those results given by Liu and Zhang and Qi and Yang.

Highlights

Painleveequations have attracted much interest due to the reduction of solution equations, which are solvable by inverse scattering transformations, and they often occur in many physical situations: plasma physics, statistical mechanics, and nonlinear waves. e study of Painleveequations has spanned more than one hundred years
Where R(z, f) is rational in f and meromorphic in z, respectively, and they singled out the following difference equations: az + b f(z + 1) + f(z − 1)
Motivated by the idea [27, 28], a natural question is what is the result if we give q-difference analogues of (9)

Summary

Introduction and Main Results

Painleveequations have attracted much interest due to the reduction of solution equations, which are solvable by inverse scattering transformations, and they often occur in many physical situations: plasma physics, statistical mechanics, and nonlinear waves. e study of Painleveequations has spanned more than one hundred years (see [1,2,3]). In 2013 and 2018, Zhang and Yi [11] and Du et al [13] studied the difference Painleve III equations with the constant coefficients and obtained the result as follows. Let f(z) be a transcendental meromorphic solution with zero order of equation (10) and a, b, c be three constants such that a, b cannot vanish simultaneously. Let Y(z) be a transcendental meromorphic solution with zero order of (11) and ξ, o, ] be three constants such that ξ, o cannot vanish simultaneously. Motivated by the idea [27, 28], a natural question is what is the result if we give q-difference analogues of (9) For this question, our main aim of this article is further to investigate some properties of meromorphic solutions for some q-Painlevedifference IV equations. (iii) Δqf/f has infinitely many fixed points and τ(Δqf/f) σ(f)

Proof of Theorem 6

Proof of Theorem 7

Proof of Theorem 8