Introduction

We say [1] that a partial differential equation has the Painlevé property when the solutions of the pde are "single-valued" about the movable, singularity manifold. To be precise, if the singularity manifold is determined by

$$\phi(z_1,\ldots,z_n)=0\eqno (1.1)$$

and $u=u(z_1,\ldots,z_n)$ is a solution of the partial differential equation, then we assume that

$$u=\phi^{\alpha}\sum_{j=0}^{\infty} u_j\phi^j ,\eqno (1.2)$$

where

$$\phi=\phi(z_1,\ldots,z_n) ,\qquad u_j=u_j(z_1,\ldots,z_n) ,\qquad u_0\ne 0 ,$$

are analytic functions of $(z_j)$ in a neighborhood of the manifold (1.1) and $\alpha$ is an integer. Substitution of (1.2) into the partial differential equation determines the value(s) of $\alpha$ and defines the recursion relations for $u_j$, $j\ne 0,1,2,\ldots$ . When the ansatz (1.2) is correct, the pde is said to possess the Painlevé property and is conjectured to be integrable.

In this paper we demonstrate a connection between the Painlevé property and the occurrence of the Lax pairs (and Bäcklund transforms) for several well-known nonlinear pde's. These include the Burgers', KdV, MKdV, Bousinesq, higher-order KdV, and Kadomtsev-Petviashvili equations. This connection is most clearly formulated in terms of the "Schwarzian derivative" [2], [3] of the function defining the singular manifold, (1.1). Consideration of the Painlevé property associated with the Schwarzian derivative leads to the identification of a wide class of nonlinear pde's that possess, in part, the Painlevé property. With some care, it appears possible to extend this identification to equations in any (finite) number of independent variables. Study of this class of equations is currently in progress.

In Sec. 2 we develop, at some length, the above procedures for the Korteweg-de Vries equation. Then in the succeeding sections the results for a number of equations are stated.

In Appendix A it is shown how an infinite sequence of "conserved densities", can be defined by the expansion of the solution (of the KdV equation) about the "singularity manifold".

In Appendix B the Harry Dym equation is obtained, by a charge of variables, from the KdV equation.