The Painlevé Integrability Test

The Painlevé test is a widely applied and quite successful technique to investigate the integrability [8] of nonlinear ODEs and PDEs by analyzing the singularity structure of the solutions. The test is named after the French mathematician Paul Painlevé (1863-1933) [18], who classified second order differential equations that are solvable in terms of known elementary functions or new transcendental functions [12]. The Painlevé test, allows one to verify whether or not a differential equation (perhaps after a change of variables) satisfies the necessary conditions for having the Painlevé property. If so, the equation is prime candidate for being completely inte-grable [1]. As originally formulated by Ablowitz et al. [2], the Painlevé conjecture asserts that all similarity reductions of a completely integrable PDE should have the Painlevé property (or be of Painlevé-type), i.e. their general solutions should have no movable singularities other than poles in the complex plane. A later version of the Painlevé test due to Weiss et al. [23] allows testing of PDEs directly, without recourse to the reduction(s) to ODEs. A PDE is said to have the Painlevé property if its solutions in the complex plane are single-valued in the neighborhood of all its movable singularities. In other words, the equation must have a solution without any branching around the singular points whose positions depend on the initial conditions. The traditional Painlevé test does not test for essential singu-larities and therefore cannot determine whether or not branching occurs about these. The algorithm The Painlevé test can be applied to nonlinear polynomial system of ODEs or PDEs with (real) polynomial terms. For brevity, we give the three steps of the test for a single PDE, F (x, t, u(x, t)) = 0, in two independent variables x and t. Following [23], the Laurent expansion of the solution u(x, t), u(x, t) = g α (x, t) ∞ k=0 u k (x, t) g k (x, t), (1) should be single-valued in the neighborhood of a non-characteristic, movable singular manifold g(x, t), which can be viewed as the surface of the movable poles in the complex plane. In (1), u 0 (x, t) = 0, α is a negative integer, and u k (x, t) are analytic functions in a neighborhood of g(x, t). Note that for ODEs the singular manifold is g(x, t) = x − x 0 , where x 0 is the initial value for x. For …