On the extension of the Painlevé property to difference equations

By Mark J. Ablowitz, R. Halburd, B. Herbst. 2000

235 citations recorded by the source corpus.

Subfields: Geometry and Topology, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/759161e7a65eba987e58d0db1b512532d1f55557

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. R. Korhonen
2. R. Halburd
3. Y. Chiang
4. Shaoji Feng
5. I. Laine
6. Ajweezi. aljali
7. K. Tohge
8. Zong-Xuan Chen

In Numerical Analysis

1. R. Korhonen
2. I. Laine
3. Y. Chiang
4. Shaoji Feng
5. R. Halburd
6. K. Tohge
7. D. Levi
8. P. Winternitz

In Statistical and Nonlinear Physics

1. R. Korhonen
2. Y. Chiang
3. Shaoji Feng
4. R. Halburd
5. P. Winternitz
6. D. Levi
7. K. Tohge
8. Zong-Xuan Chen

Citations in the corpus

• On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane (2006, methodology): “It is proposed in [1] that a difference equation admits a finite order meromorphic solution is a strong indication of integrability of the equation.”
• Existence of finite-order meromorphic solutions as a detector of integrability in difference equations (2006, background): “This helps to clarify a link between the complex analytic approach [7] and the singularity confinement test [9].”
• A Difference Picard Theorem for Meromorphic Functions of Several Variables (2009, methodology): “The purpose of this section is to extend some of the methods used in [1] to partial differences, and apply these generalized results to single out the difference Riccati”
• Continuous symmetries of difference equations (2005, methodology): “[2] based on the study of the asymptotic behaviour using the Nevanlinna theory of value distribution and growth of meromorphic functions.”
• Finite‐order meromorphic solutions and the discrete Painlevé equations (2005, methodology): “Ablowitz, Halburd and Herbst [2] considered discrete equations as delay equations in the complex plane which allowed them to analyze the equations with methods from complex analysis.”
• Clunie theorems for difference and q‐difference polynomials (2007, background): “On the other hand, the search for what could be called the ‘discrete Painlevé property’ has prompted us to determine whether a given complex difference equation permits meromorphic solutions of finite order; see [1, 16] for a detailed argument.”
• On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions (2006, background): “Abstract. A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen [ 2 ], [15] that a connection exists between discrete Painlev´e equations and (finite order) Nevanlinna theory is an estimate of the integrated average of log + |f(z + 1)/f(z)| on |z| = r. We obtained es”
• Holomorphic curves with shift-invariant hyperplane preimages (2009, methodology): “In the second-order case, it follows by the proof of [1, Theorem 3] that if (5.5) has a meromorphic solution of hyper-order less than one, then deg w ( R ) ≤ 2.”
• Zeros of differences of meromorphic functions (2005, background): “A number of papers (including [1, 5, 10, 11, 16, 18]) focus on the growth and zeros of solutions of such equations, investigating analogies and contrasts with the theory of linear differential equations in the complex plane.”