Painlevé Conjecture Revisited

Abstract

The discovery of new integrable two-dimensional Hamiltonian systems is reported. The analytic structure of the solutions makes necessary the generalization of the Painleve conjecture, a widely used integrability criterion. Such a generalization is presented, which the authors believe should replace the usual conjecture for two-dimensional Hamiltonian systems. It is indeed compatible with all the systems already found and, in addition, leads to still new integrable cases.

Separation of variables in quasi-potential systems of bi-cofactor form

“…It is easy to show that for n = 2 it is the general solution of (25). For higher n the general solution of (25) depends on n(n + 1) 2 (n + 2)/12 parameters (its basis can be found e.g.…”

“…The equation (25) implies that the matrix A is a Killing tensor. We will restrict ourselves to a class of solutions of (25) that have the form…”

Untitled

“…Having obtained the equation (2.9) from (2.1) by using the reciprocal transformation (2.7), one may check that the standard WTC Painlevé test [22] is satisfied. In fact the original equation (2.1) admits expansions with square root branching, but such generalized (weak [18]) Painlevé expansions were specifically excluded from the classification of [10]. Further details of the Painlevé analysis will be given elsewhere [3].…”

“…In [10] a class of PDEs including all of the equations (1.7) was tested for integrability using Painlevé analysis. However, because both Camassa-Holm equation (1.1) and the new equation (1.6) are examples of integrable systems with algebraic branching in their solutions (the weak Painlevé property of [18]), they were explicitly excluded by the various Painlevé tests applied in [10], and in fact all of the equations in that class failed the combination of tests. The authors of [10] noted that the (strong) Painlevé property is destroyed by changes of variables, and thus a transformation may be required before applying the test.…”

Analytic integrability of Hamiltonian systems with a homogeneous polynomial potential of degree 4

“…We will say that a 2-degree of freedom Hamiltonian system (2) is completely or Liouville integrable if it has 2 functionally independent first integrals: H, and an additional one F , which are in involution. In the beginning of 80's all integrable Hamiltonian systems (1) with homogeneous polynomial potential of degree at most 5 and having a second polynomial first integral up to degree 4 in the variables p 1 and p 2 were found, see [11,4,2,5,1] and also [6] for the list of corresponding additional first integrals. We remark that all these first integrals are polynomials in the variables p 1 , p 2 , q 1 and q 2 .…”