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A Study of the Continuous and Discrete Gambier Systems | | A. Ramani
; B. Grammaticos
; S. Lafortune
; | | Date: |
6 Apr 2001 | | Journal: | SIDE III - Symmetry and Integrability of Difference Equations (Decio Levi and Orlando Ragnisco, eds.) (Conference held in Italy, 1998), CRM Proc. Lectures Notes, vol.25, Amer. Math. Soc., Providence, RI (2000), 367-379 | | Subject: | Exactly Solvable and Integrable Systems | nlin.SI | | Abstract: | We present a systematic study of the Gambier system, which in the continuous case is given by two Riccati equations in cascade. We derive the condition for its integrability and show that the generic Gambier system contains one free function. We also derive the Schlesinger transformations for this system which allows in principle the systematic construction of the integrable cases. The above procedure is carried over to a discrete setting. We show thus how the discrete Gambier system can be expressed as a system of two homographic mappings in cascade. The integrable cases are obtained through the singularity confinement discrete integrability criterion. Finally the discrete Schlesinger transformations are also derived giving a handle to the construction of the integrable Gambier mapping. | | Source: | arXiv, nlin.SI/0104017 | | Services: | Forum | Review | PDF | Favorites | |
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