The Painlevé Property is proposed as a sufficient condition for integrability. In our definition of the Painlevé Property we require the solutions be single-valued as functionals of the arbitrary data in the neighborhood of a singularity manifold. Examples of this phenomenon are examined. Expansions about characteristic manifolds are required to be single-valued. Essential singularities are found to be determined by certain Psi series involving non-constant leading-orders and resonances.
Constrained Psi series expansions are applied to non-Painlevé systems. The constraints are expressed as nonlinear partial differential equations. We conjecture that these are integrable and provide the integrable reductions of the original (non-integrable) system.
The Singular Manifold Method finds Bäcklund transformations by truncating the functional Laurent series after the constant level term. This results in the formulation of modified equations in terms of the Schwarzian derivative. The Miura transformation between modified and given system can be used to determine the Lax pair and recursion operators for the gradients of conserved densities. The symmetries of the modified equations and the invariance under the Moebius group are a form of Bäcklund transformation for the modified equation. A simple construction, based on these Bäcklund transformations, produces the formal convergence of the expansions about the singular manifolds. The formal convergence is in the sense of and based on the Cauchy-Kovalevsky theorem.
The periodic fixed points of the Bäcklund transformations are finite dimensional invariant manifolds for the flow of the system. The dynamics occur as commuting hamiltonian flows on this finite dimensional manifold. We examine the flow of the KdV periodic fixed points in the neighborhood of steady states and reductions. These are analogous to a flow in the neighborhood of a sequence of heteroclinic points.
The periodic fixed points of Bäcklund transformations define a natural factorization of the two-dimensional Toda lattice as commuting hamiltonian flows. These have an interesting connection with caustic surfaces and the Laplace-Darboux transformation.