The Painlevé property and Bäcklund transformations for the sequence of Boussinesq equations

Abstract

We investigate the sequence of Boussinesq equations by the method of singular manifolds. For the Boussinesq equation, which is known to possess the Painlevé property, a Bäcklund transformation is defined. This Bäcklund transformation, which is formulated in terms of the Schwarzian derivative, obtains the system of modified Boussinesq equations and the resulting Miura-type transformation. The modified Boussinesq equations are found to be invariant under a discrete group of symmetries, acting on the dependent variables. By linearizing the Miura transformation (and modified equations) the Lax pair is readily obtained. Furthermore, by a result of Fokas and Anderson, the recursion operators defining the sequence of (higher-order) Boussinesq equations may be constructed from the Miura transformation. This allows the (recursive) definition of Bäcklund transformations for this sequence of equations. The recursion operator is found to preserve the discrete symmetries of the modified Boussinesq equations. This leads to the conclusion that the sequences of Boussinesq and modified Boussinesq equations identically possess the Painlevé property (are meromorphic). We also find that, by a simple reduction, the sequences of Caudrey–Dodd–Gibbon and Kuperschmidt equations are contained within the Boussinesq sequence. Rational solutions are iteratively constructed for the Boussinesq equation and a criterion is proposed for the existence of rational solutions of general integrable systems.