Finite‐order meromorphic solutions and the discrete Painlevé equations
By R. Halburd, R. Korhonen. 2005
204 citations recorded by the source corpus.
Let w(z) be an admissible 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) with coefficients that are meromorphic in z. Then either w(z) satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painlevé equations of the above form, together with their autonomous versions. This suggests that the existence of finite‐order meromorphic solutions is a good detector of integrable difference equations.
Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/7cfc34faf833e5ec7b1bc978d83218c89f8e56b3
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In Applied Mathematics
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Citations in the corpus
- Some properties of meromorphic solutions of q-difference equations☆ (2010, methodology): “[27] J. L. Zhang, R. Korhonen; On the Nevanlinna characteristic of f(qz) and its applications, J. Math.”
- Nevanlinna theory for the difference operator (2005, background): “16) is a subcase of a more general equation studied in [3] where it was shown that the existence of one finite-order meromorphic solution is sufficient to reduce a large class of difference equations into a difference Painlevé equation or into a linear difference equation, provided that the solution”
- Uniqueness of meromorphic functions sharing values with their shifts (2011, unknown): “4 [19] Let T : (0, þ1)! (0, þ1) be a non-decreasing continuous function, s4 0, 5 1, and let F Rþ be the set of all r such that TðrÞ Tðrþ sÞ:”