TY - JOUR
T1 - Rational solutions of difference painlevé equations
AU - Shimomura, Shun
N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.
PY - 2012/6
Y1 - 2012/6
N2 - We capture all the rational solutions of some difference Painlevé equations of PI and PII types. For non-autonomous cases, it is shown that all the rational solutions of the difference PII are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of PI type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.
AB - We capture all the rational solutions of some difference Painlevé equations of PI and PII types. For non-autonomous cases, it is shown that all the rational solutions of the difference PII are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of PI type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.
KW - Difference Painlevé equation
KW - Rational solution
UR - http://www.scopus.com/inward/record.url?scp=84866927546&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84866927546&partnerID=8YFLogxK
U2 - 10.3836/tjm/1342701346
DO - 10.3836/tjm/1342701346
M3 - Article
AN - SCOPUS:84866927546
SN - 0387-3870
VL - 35
SP - 85
EP - 95
JO - Tokyo Journal of Mathematics
JF - Tokyo Journal of Mathematics
IS - 1
ER -