Entire solutions of certain type of non-linear differential-difference equations

Abstract

The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate the nonlinear differential-difference equations of form \begin{equation*} f(z)^{n}+L(z,f)=q(z)e^{p(z)},\eqno(*) \end{equation*} where $n\geq 2, L(z,f)(\not\equiv 0)$ is a linear differential-difference polynomial in $f(z)$, with small functions as its coefficients, $p(z)$ and $q(z)$ are non-vanishing polynomials. We first obtain that $n=2$ and $f(z)$ satisfies $\overline{\lambda}(f)=\sigma(f)=\deg p(z)$ under the assumption that the equation (*) possesses a transcendental entire solution of hyper order $\sigma_{2}(f)<1$. Furthermore, we give the exact form of the solutions of equation (*) when $p(z)=a, q(z)=b$, $\eta$ are constants and $L(z,f)=g(z)f(z+\eta)+h(z)f^{'}(z)+u(z)f(z)+v(z)$ is a linear differential-difference polynomial in $f(z)$ with polynomial coefficients $g(z), h(z), u(z)$ and $v(z)$ such that $L(z,f)\not\equiv 0$ and $a b \eta\neq 0$.