Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms
Abstract
Let n≥ 2 and k≥ 1 be integers. We study finite order meromorphic solutions of the nonlinear differential-difference equation fn(z)f'(z)+Pd(z,f)=p1(z)eα1zk+p2(z)eα2zk, where Pd(z,f)not≡ 0 is a difference polynomial in f of degree d≤ n−1 whose coefficients are small meromorphic functions with respect to f, p1, p2 are nonzero meromorphic functions with T(r,pj)=o(rk), j=1,2, and α1≠α2 are nonzero constants. We prove that every such solution has order exactly k, and obtain sufficient conditions on n, d, k for the equation to admit no finite order meromorphic solution with N(r,1/f)=S(r,f). Our results extend the known ones from linear exponents z to polynomial exponents zk, and replace the linear difference polynomials with the degree restriction d≤ n−2 assumed in the literature by general ones of degree d≤ n−1. Examples show that the restrictions on n and d are sharp.
Related
- Explicit meromorphic solutions of certain types of nonlinear differential-difference equations
- Meromorphic Solutions of Certain Kind of Non-Linear Differential equation with finite sums of Exponential Functions
- Meromorphic solutions of first-order differential equations with rational exponential coefficients
- Existence and Nonexistence of Solutions of Certain type of Nonlinear Differential and Differential-Difference Equations
- Stability and Periodicity Performance of Solutions of Nonlinear Difference Equations
BibTeX
@misc{chen2026meromorphic,
title = {{Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms}},
author = {Minfeng Chen and Zongsheng Gao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38247},
eprint = {2610.38247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38247},
}
RIS
TY - UNPB
TI - Meromorphic solutions of a nonlinear differential-difference equation with two exponential terms
AU - Minfeng Chen
AU - Zongsheng Gao
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38247
UR - https://arxiv.org/abs/2610.38247
AB - Let n≥ 2 and k≥ 1 be integers. We study finite order meromorphic solutions of the nonlinear differential-difference equation f^n(z)f'(z)+P_d(z,f)=p_1(z)e^α_1z^k+p_2(z)e^α_2z^k, where P_d(z,f)not≡ 0 is a difference polynomial in f of degree d≤ n−1 whose coefficients are small meromorphic functions with respect to f, p_1, p_2 are nonzero meromorphic functions with T(r,p_j)=o(r^k), j=1,2, and α_1≠α_2 are nonzero constants. We prove that every such solution has order exactly k, and obtain sufficient conditions on n, d, k for the equation to admit no finite order meromorphic solution with N(r,1/f)=S(r,f). Our results extend the known ones from linear exponents z to polynomial exponents z^k, and replace the linear difference polynomials with the degree restriction d≤ n−2 assumed in the literature by general ones of degree d≤ n−1. Examples show that the restrictions on n and d are sharp.
ER -