Nonlinear relationship between Liouville and sinh-Gordon field theories (Q1109985)

Recently Pogrebkov and Polivanov have suggested that there exist singular solutions of the sinh-Gordon (shG) equation, \(\phi_{xt}=\alpha \sinh \phi\) corresponding to the singular solutions of the Liouville field theory, \(\psi_{xt}=\alpha e^ 2\psi\). We prove the nonexistence of a Bäcklund transformation from the Liouville to the shG equation. Nevertheless we introduce a nonlinear map defined by \(\psi =1/2 \log (\cosh \phi)\) that connect particular solution classes of the two equations. A few conserved densities of the shG system are also derived.