Backlund transformation of partial differential equations from the Painleve-Gambier classification. II. Tzitzeica equation

From the existing methods of singularity analysis only, we derive the two e quations which define the Backlund transformation of the Tzitzeica equation . This is achieved by defining a truncation in the spirit of the approach o f Weiss et al., so as to preserve the Lorentz invariance of the Tzitzeica e quation. If one assumes a third-order scattering problem, this truncation a dmits a unique solution, thus leading to a matrix Lax pair and a Darboux tr ansformation. In order to obtain the Backlund transformation (BT), which is the main new result of this paper, one represents the Lax pair by an equiv alent two-component Riccati pseudopotential. This yields two different BTs; the first one is a BT for the Hirota-Satsuma equation, while the second on e is a BT for the Tzitzeica equation. One of the two equations defining the BT is the fifth ordinary differential equation of Gambier. (C) 1999 Americ an Institute of Physics. [S0022-2488(99)01503-0].