Meromorphic solutions of certain types of differential equations

In this paper, we prove the uniqueness of finite-order transcendental meromorphic solutions of differential-difference Painlevé III and V equations: $$\omega(z+1)\omega(z-1)+a(z)\frac{\omega^{\prime}(z)}{\omega(z)}=\frac{\sum_{m=0}^{3}a_{m}\omega^{m}(z)}{\sum_{n=0}^{2}b_{n}\omega^{n}(z)},$$ and $$(\omega(z)\omega(z+1)-1)(\omega(z)\omega(z-1)-1)+a(z)\frac{\omega^{\prime}(z)}{\omega(z)}=\frac{\sum_{m=0}^{6}a_{m}\omega^{m}(z)}{\omega(z)-b_{1}(z)},$$ where $$a_{m}$$ , $$b_{n}$$ and $$a(z)$$ are small functions of solution $$f(z)$$ . We show that if the solution $$f(z)$$ shares $$e_{1}$$ , $$e_{2}$$ , and $$\infty$$ CM with another meromorphic function $$g(z)$$ , then $$f(z)\equiv g(z)$$ . Moreover, for Painlevé V equation, if $$g(z)$$ is replaced by $$f(z+c)$$ , it is sufficient for $$f(z)$$ and $$f(z+c)$$ to share the values $$e_{1}$$ and $$e_{2}$$ CM. MSC2020 numbers:30D35.

Abstract In this article, by utilizing the properties of elliptic functions, we characterize the meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb {C}$ , where $L(z)$ is a nonconstant entire function, and m and n are two positive integers. As applications, we also investigate the meromorphic solutions of Fermat-type difference and q -difference equations.

Meromorphic Solutions of Logistic Delay Differential Equations of the Lotka-Volterra Type and Beyond

On the exact structure of meromorphic solutions to non-linear difference equations with exponential terms

Abstract In this paper, we investigate a special autonomous Schwarzian differential equation This equation was overlooked by K. Ishizaki during his classification of autonomous differential equations with transcendental meromorphic solutions. It was discovered by Zhang–Liao–Wu–Zhao when they were exploring the rational solutions of autonomous differential equations. In fact, we prove this equation admits no meromorphic solutions, thus completing the classification of autonomous Schwarzian differential equations.

In this paper, we study the transcendental meromorphic solutions to a certain type of nonlinear complex differential equation $$f^{n}(z)+P(z,f,f^{\prime},f^{\prime\prime},\cdots,f^{(t)})=H_{0}(z)+H_{1}(z)e^{\alpha_{1}z^{q}}+\cdots+H_{m}(z)e^{\alpha_{m}z^{q}},$$ where $$m\geq 1$$ , $$n\geq m+2$$ , $$t\geq 0$$ , $$q\geq 1$$ are integers, $$P(z,f,f^{\prime},f^{\prime\prime},\cdots,f^{(t)})$$ is a differential polynomial in $$f(z)$$ of degree $$d\leq n-m-1$$ with small functions of $$f(z)$$ as its coefficients, $$\alpha_{i}$$ $$(i=1,2,\cdots,m)$$ are nonzero complex constants such that $$|\alpha_{1}|>|\alpha_{2}|>\cdots>|\alpha_{m}|$$ , $$H_{i}(z)$$ are entire functions of order less than $$q$$ such $$H_{i}(z)\not\equiv 0$$ for $$1\leq i\leq m$$ . In fact, we give the exact forms of all possible meromorphic solutions satisfying $$N(r,f)=S(r,f)$$ of the above equation. Particularly, we weaken the condition and obtain other properties of the meromorphic solutions when $$m=3$$ , $$q=1$$ . Some examples are given to illustrate our results.

Abstract In this paper, we discuss the existence of meromorphic solutions to the following system of difference equations $$\begin{cases}f(z+c)=e^{P}f-ae^{P}+a,\\ f(z+c)=e^{Q}f-be^{Q}+b,\end{cases}$$ where $$P$$ and $$Q$$ are entire functions, $$a$$ and $$b$$ are distinct complex numbers. Additionally, this work elucidates the application of these findings, thereby enhancing uniqueness results associated with meromorphic functions $$f(z)$$ and their shifts $$f(z+c)$$ .

On Meromorphic Solutions of Non-linear Differential Equations and Their Applications

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and transcendental entire function solutions of infinite order in the complex plane. The exact solutions contribute to understanding the predator–prey model from the perspective of complex differential equations. In fact, the presented synthesis method provides a new technology for studying some systems of partial differential equations.

In this paper, we study the transcendental meromorphic solutions of the higher order differential-difference equations $$w(z+1)w(z-1)+a(z)\frac{w^{(k)}(z)}{w(z)}=R(z,w(z))=\frac{P(z,w(z))}{Q(z,w(z))},$$ where $$k$$ is a positive integer, $$a(z)$$ is a rational function, $$P(z,w(z))$$ and $$Q(z,w(z))$$ are polynomials with rational coefficients in $$w(z)$$ . We obtain an upper bound on the degree of $$R$$ , and the degrees of $$P$$ and $$Q$$ satisfy certain conditions. These results generalize some of the previous results by Liu and Song to higher order case. Finally, we provide some examples to illustrate that all cases occur.

In this paper, we study a Malmquist-Yosida type theorem for Schwarzian differential equations S ( f , z ) m = R ( z , f ) = P ( z , f ) Q ( z , f ) , \begin{equation*} S(f,z)^{m} = R(z,f) = \frac {P(z,f)}{Q(z,f)}, \end{equation*} where m ∈ N + m \in \mathbb {N}^{+} , P ( z , f ) P(z,f) and Q ( z , f ) Q(z,f) are irreducible polynomials in f f with rational coefficients. If the above equation admits a transcendental meromorphic solution f f , then by a suitable Möbius transformation f → u f \to u , u u satisfies a Riccati differential equation with small meromorphic coefficients, or one of the six types of first-order differential equations \eqrefRange{1.2}Thm.1.E.6, or u u satisfies one of types \eqrefRange{Thm.1.E.7}E.14. In addition, we improve the result of Ishizaki [Kodai Math. J. 20 (1997), pp. 67–78; Theorem 1.1] on Schwarzian differential equations with small meromorphic coefficients when m = 1 m=1 .

Abstract We study the higher-order delay differential equation $$w(z+1)-w(z-1)+a(z)\frac{w^{(k)}(z)}{w^2(z)}=R(z,w(z)),$$ w ( z + 1 ) - w ( z - 1 ) + a ( z ) w ( k ) ( z ) w 2 ( z ) = R ( z , w ( z ) ) , where $$k$$ k is a positive integer, $$a(z)$$ a ( z ) is a rational function and $$R(z,w)$$ R ( z , w ) is rational in $$w$$ w with rational coefficients. We obtain necessary conditions on the degree of $$R(z, w)$$ R ( z , w ) for this delay differential equation to admit subnormal transcendental meromorphic solutions. On the other hand, we obtain a reduced form of the equation above when $$R(z,w)$$ R ( z , w ) becomes a rational function.

Abstract In this paper, we consider the equation $$\overline{y}\underline{y}+\alpha(x)\frac{y^{(d)}}{y^{2}}=R(x,y)=\frac{H(x, y)}{G(x, y)},$$ y ¯ y ̲ + α ( x ) y ( d ) y 2 = R ( x , y ) = H ( x , y ) G ( x , y ) , where $$\alpha(x)$$ α ( x ) is a nonzero rational, $$H(x,y)$$ H ( x , y ) and $$G(x,y)$$ G ( x , y ) are co-prime polynomials of $$y$$ y with rational coefficients. If there exists a non-rational meromorphic solution with $$\rho_{2}(y)<1 $$ ρ 2 ( y ) < 1 , then $$\deg_{y}(H)$$ deg y ( H ) and $$\deg_{y}(G)$$ deg y ( G ) must satisfy certain conditions.

Correction to: Properties of meromorphic solutions of some delay differential equations

Abstract The paper concerns the algebraic differential independence of meromorphic solutions of difference equation , where is a rational function with a certain condition. Based on ideas by Bank and Kaufman [Funkcial. Ekvac. 19 (1976), 53–63; Math Ann. 232 (1978), 115–120] and Wang and Yao [J. Differ. Equations. 412 (2024), 797–807], we confirm that Markus's conjecture holds in a more general form. Meanwhile, an example is given to show that the condition on is necessary.

Properties of meromorphic solutions of some delay differential equations

ABSTRACTThe coupled Whitham–Broer–Kaup system can be used to describe the propagation of shallow water waves in fluid dynamics. In this paper, we investigate the nonlinear conformable fractional‐order Whitham–Broer–Kaup system by complex analytic method and obtain plentiful new closed‐form meromorphic solutions. These derived meromorphic solutions include rational solutions, simply periodic solutions, and elliptic function solutions. At the same time, we give the real‐valued characterizations of such meromorphic solutions and illustrate the dynamic behaviors of these solutions with some graphs.

In this paper, we slightly refine the Nevanlinna theory approach, incorporating singularity confinement analysis, to study a class of delay-differential equations. Some necessary conditions are obtained for the existence of non-rational meromorphic solutions of hyper-order less than one.

Let K be a complete, algebraically closed ultrametric field, and let ℳ(K) denote the field of meromorphic functions over K. In this article, we investigate the properties of solutions to certain ultrametric q-difference equations. Specifically, we examine the growth behavior of ultrametric meromorphic functions f that satisfy these equations. Furthermore, we establish necessary conditions on the coefficients {for} which a difference equation of the form: ∑si=0 Ai(x)f(x+i) = B(x) where B(x), A0(x), …, As(x) (with s ≥ 1) {is a polynomial, admits a meromorphic solution.