Introduction

In [1] we have formulated a procedure for calculating the Lax pair and rational solutions of partial differential equations that possess the Painlevé property. That is, for an equation with the Painlevé property, a Bäcklund transformation is defined in terms of an expansion about the ``singular manifold". This Bäcklund transformation obtains (1) a type of ``modified equation" that can be expressed in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the Ricati-type Miura transformation (and the modified equations), the Lax pair is found. Then, further consideration of the Bäcklund transformations for the modified equations provides a method for the iterative construction of ``rational" solutions, and finds the Lax pair for the modified equations as well.

We recall that the partial differential equation is said to possess the Painlevé property [2]-[7] when the solutions of the partial differential equation (pde) are ``single valued" about the movable, singularity manifold and the singularity manifold is ``noncharacteristic." To be precise, if the singularity manifold is determined by

$$\varphi(z_1,z_2,\ldots,z_n)=0\ ,\eqno (1.1)$$

and $u=u(z_1,\ldots,z_n)$ is a solution of the pde, then we require that

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

where $u_0\ne 0$, $\varphi=\varphi(z_1,\ldots,z_n)$, and $u_j=u_j(z_1, \ldots,z_n)$ are analytic functions of $(z_j)$ in a neighborhood of the manifold (1.1) and $\alpha$ (the leading-order exponent) is a (negative) rational number. The requirement that the manifold (1.1) be noncharacteristic (for the pde) insures that the expansion (1.2) will be well defined, in the sense of the Cauchy-Kovalevskaya theorem [8]. Substitution of(1.2) into the pde determines that value(s) of $\alpha$, and defines the recursion relations for $u_j$, $j=0,1,2,\ldots$ . When the expansion (1.2) is well defined and contains the maximum number of arbitrary functions allowed at the ``resonances" [2], [9], [10], the pde is said to possess the Painlevé property and is conjectured to be integrable. Informally, the resonances are the values of $j$ for which the $u_j$ are not ``fixed" by the recursion relations (i.e., are arbitrary).

The Bäcklund transformation is defined by truncating the expansion (1.2) at the constant level term. That is, we set

$$u=u_0\varphi^{-n}+u_1\varphi^{-n+1}+\cdots+u_n\ ,\eqno (1.3)$$

and find, from the recursion relations for $u_j$ and the condition that $u_j$ vanish for $j>n$, a system of equations for $(\varphi,\ u_j,\ j=j0,1,\ldots,n)$), where $u_n$ will satisfy the (original) pde. This system of equations will, in general (depending on the values of the resonances), be overdetermined. Upon solving this system, it is found, for those equations considered, the $\varphi$ satisfies an equation formulated in terms of Schwarzian derivatives [3]:

$$\{\varphi;x\}={\partial\over\partial x}\left({\varphi_{xx}\ov... ...r 2} \left({\varphi_{xx}\over\varphi_x}\right)^2\ .\eqno (1.4)$$

This equation, or system of equations, we regard as a type of modified equation. By the invariance of (1.4) under the Moebius group,

$$\varphi=(a\psi+b)/(c\psi+d)\ ,\qquad\{\varphi;x\}=\{\psi;x\}\ ,\eqno (1.5)$$

the ``modified" equations allow the Bäcklund transformation (1.5).

The above procedure may now be reapplied to the ``modified" (or equivalent) equations to find different forms of Bäcklund transformations. These Bäcklund transformations may take the form of discrete symmetries [1],[5], [6], reductions [1], or, as we shall see, more complicated structures. The group of Bäcklund transformations for the modified equations may be conveniently employed to iteratively construct sequences of rational solutions. Also, by linearizing the Miura transformation from modified to original equation we propose to calculate the Lax pair [1], [6].

In this paper we consider the Kadomtsev-Petviashvili (KP) equation and the Hirota-Satsuma equations. The modified equations are derived, their (modified) Bäcklund transformations are calculated, and the sequences of rational solutions are found.