Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II. Tzitzéica equation

From the existing methods of singularity analysis only, we derive the two equations which define the Bäcklund transformation of the Tzitzéica equation. This is achieved by defining a truncation in the spirit of the approach of Weiss et al., so as to preserve the Lorentz invariance of the Tzitzéica equation. If one assumes a third-order scattering problem, this truncation admits a unique solution, thus leading to a matrix Lax pair and a Darboux transformation. In order to obtain the Bäcklund transformation (BT), which is the main new result of this paper, one represents the Lax pair by an equivalent two-component Riccati pseudopotential. This yields two different BTs; the first one is a BT for the Hirota–Satsuma equation, while the second one is a BT for the Tzitzéica equation. One of the two equations defining the BT is the fifth ordinary differential equation of Gambier.