Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 073, 6 pages      arXiv:0706.2719

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

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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