Introduction to the Painlevé property, test and analysis

Abstract

A new approach for Painlevé analysis of the generalized Kawahara equation

Fourth order superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials

“…We shall use the Painlevé test in the form introduced in [1]. For a review and further developments see Conte, Fordy, and Pickering [13], Conte [14], Conte and Musette [15,16], Grammaticos and Ramani [33], Hone [38], Kruskal and Clarkson [43]. Passing the test is a necessary condition for having the Painlevé property.…”

Vortices, Painlevé integrability and projective geometry

“…This definition is motivated by the possibility of defining meromorphic functions from ODE's on the complex plane (c.f. [14] for a short survey on the subject). Non-linear second order ODE's of the form y = R(x, y, y ), where R(x, y, y ) is a rational function of (y, y ) with coefficients analytic in x, that possess the Painlevé property were classified [55,54,29,33] and there are 50 of them up to the change of variables…”

“…: we have not studied hydrodynamic-type systems coming from the two last Frobenius manifolds of Theorem C.5 because the flows constructed from the fifth one have two identical characteristic velocities while the last one has vanishing A and B for every flow. These cases are not taken into account in the present work.C.4 The third Hamiltonian structureThe metric corresponding toC 1 = C 2 = 0 in (C 14…”