Do integrable mappings have the Painlevé property?
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Publication:4491892
DOI10.1103/PhysRevLett.67.1825zbMath0990.37518WikidataQ74510030 ScholiaQ74510030MaRDI QIDQ4491892
V. G. Papageorgiou, Alfred Ramani, Basile Grammaticos
Publication date: 16 July 2000
Published in: Physical Review Letters (Search for Journal in Brave)
Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) (37K10) Discrete version of topics in analysis (39A12)
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Cites Work
- Unnamed Item
- Integrable discrete-time systems and difference operators
- Linear integral equations and nonlinear difference-difference equations
- Integrable mappings and soliton equations
- Nonlinear Partial Difference Equations. IV. Bäcklund Transformation for the Discrete-Time Toda Equation
- Method for Generating Discrete Soliton Equation. III
- A Nonlinear Difference Scheme and Inverse Scattering
- On the Solution of a Class of Nonlinear Partial Difference Equations
- Generalized affine lie algebras and the solution of a class of flows associated with the QR eigenvalue algorithm
- Numerical study of quadratic area-preserving mappings