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SIGMA 3 (2007), 073, 6 pages arXiv:0706.2719
https://doi.org/10.3842/SIGMA.2007.073
Do All Integrable Evolution Equations Have the Painlevé Property?
K.M. Tamizhmani a, Basil Grammaticos b and Alfred Ramani c
a) Departement of Mathematics, Pondicherry University,
Kalapet, 605014 Puducherry, India
b) IMNC, Université Paris VII-Paris XI, CNRS, UMR
8165, Bât. 104, 91406 Orsay, France
c) Centre de Physique Théorique, Ecole Polytechnique,
CNRS, 91128 Palaiseau, France
Received June 12, 2007; Published online June 19, 2007
Abstract
We examine whether the Painlevé property is necessary for
the integrability of partial differential equations (PDEs). We show
that in analogy to what happens in the case of ordinary differential
equations (ODEs) there exists a class of PDEs, integrable through
linearisation, which do not possess the Painlevé property. The same
question is addressed in a discrete setting where we show that there
exist linearisable lattice equations which do not possess the
singularity confinement property (again in analogy to the
one-dimensional case).
Key words:
integrability; linearisability; Painlevé property; singularity confinement.
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