On the Lax pairs of the continuous and discrete sixth Painlevé equations
- 10.2991/jnmp.2003.10.s2.10How to use a DOI?
Among the recently found discretizations of the sixth Painlevé equation P6, only the one of Jimbo and Sakai admits a discrete Lax pair, which does establish its integrabiity. However, a subtle restriction in this Lax pair prevents the possibility to generalize it in order to find the other missing Lax pairs. It happens that the same restriction already exists in the matrix Lax pair of Jimbo and Miwa for the continuous P6. In this preliminary article, we remove this last restriction and give a matrix Lax pair for P6 which is traceless, rational in the dependent and independent variables, holomorphic in the monodromy exponents, with four Fuchsian singularities in the complex plane of the spectral parameter. Its only minor drawback is the presence of the apparent singularity which always exists in the scalar Lax pair. Contents
- © 2006, the Authors. Published by Atlantis Press.
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Cite this article
TY - JOUR AU - Robert Conte AU - Runliang Lin AU - Micheline Musette PY - 2003 DA - 2003/12/01 TI - On the Lax pairs of the continuous and discrete sixth Painlevé equations JO - Journal of Nonlinear Mathematical Physics SP - 107 EP - 118 VL - 10 IS - Supplement 2 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2003.10.s2.10 DO - 10.2991/jnmp.2003.10.s2.10 ID - Conte2003 ER -