Finite-order Meromorphic Solutions and the Discrete Painlevé Equations
نویسنده
چکیده
Let w(z) be a finite-order meromorphic solution of the second-order difference equation w(z + 1) + w(z − 1) = R(z, w(z)) (†) where R(z, w(z)) is rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else equation (†) can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painleve equation of the form (†), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.
منابع مشابه
Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations
The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the indep...
متن کاملGrowth of meromorphic solutions for complex difference equations of Malmquist type
In this paper, we give some necessary conditions for a complex difference equation of Malmquist type $$sum^n_{j=1}f(z+c_j)=frac{P(f(z))}{Q(f(z))},$$ where $n(in{mathbb{N}})geq{2}$, and $P(f(z))$ and $Q(f(z))$ are relatively prime polynomials in $f(z)$ with small functions as coefficients, admitting a meromorphic function of finite order. Moreover, the properties of finite o...
متن کاملOn meromorphic solutions of certain type of difference equations
We mainly discuss the existence of meromorphic (entire) solutions of certain type of non-linear difference equation of the form: $f(z)^m+P(z)f(z+c)^n=Q(z)$, which is a supplement of previous results in [K. Liu, L. Z. Yang and X. L. Liu, Existence of entire solutions of nonlinear difference equations, Czechoslovak Math. J. 61 (2011), no. 2, 565--576, and X. G. Qi...
متن کاملA new positive definite semi-discrete mixed finite element solution for parabolic equations
In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations. Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...
متن کاملON THE NEVANLINNA CHARACTERISTIC OF f(z + η) AND DIFFERENCE EQUATIONS IN THE COMPLEX PLANE
and T (r, f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an...
متن کامل