Introduction

Since this paper is one of several papers appearing recently concerning the Painlevé property for partial differential equations, we spare the reader a formal definition of the Painlevé property, Bäcklund transformations, etc. For this see [1]-[6]. Informally, when an equation possesses the Painlevé property, the solutions are meromorphic functions of the independent variables. For a reasonably self-contained presentation we review, in Sec. II, the calculation of the Painlevé property and Bäcklund transformation for the Boussinesq equation.

In this paper we propose an extension of the methods of [1], [2], [4], and [5] for calculating Bäcklund transformations and Lax pairs. That is, when an equation is found to possess the Painlevé property, a certain Bäcklund transformation is defined. This Bäcklund transformation, when formulated in terms of the Schwarzian derivative, leads to an equation invariant under the Moebius group. From this equation, by a specific change of dependent variables (Miura transformations), both the original and a form modified equation are obtained. The resulting Miura transformation from modified to original equation is then linearized to obtain the Lax pair.

Now, when the equation/modified equation both have a Hamiltonian structure a result of Fokas and Anderson [7] may be used to construct the recursion operators defining the sequences of higher-order equations. (See also [8].) For these equations we can recursively define Bäcklund transformations and, in certain cases, by observing the effect of the discrete symmetries of the modified equations acting on the singularities, prove that the entire sequence of equations possesses the Painlevé property [4].

In Sec. II the Bäcklund transformation, modified equations, Miura transformations, and Lax pair for the Boussinesq equation are calculated by the above method. The modified Boussinesq equations are also found to be invariant under a discrete group of symmetries.

In Sec. III the sequences of higher-order equations are investigated. The recursion operators are shown to preserve the discrete symmetries of the modified equations. These discrete symmetries, when interpreted in terms of the underlying equation for the singular manifold, and combined with the invariance of this equation under the Moebius group, allow the conclusion that the sequences of higher-order Boussinesq and modified Boussinesq equations identically possess the Painlevé property. We also define Bäcklund transformations for both sequences of equations.

With the view toward understanding the generality of the above procedures we consider in Appendix A the nonlinear Schrödingerequation. Insofar as obtaining the Bäcklund transformation, modified equations, and a (scalar) Lax pair the method proceeds as before. However, the modified nonlinear Schrödinger equations, while similar to the modified Boussinesq equations, do not allow a group of discrete symmetries. Therefore, the arguments used to conclude that the Boussinesq sequence is identically Painlevé do not apply to the nonlinear Schrödinger sequence.

Finally, in Appendix B certain rational solutions connected with the discrete group of symmetries are obtained.