The Bäcklund transformation and nonlinear superposition formula of solutions for the Liouville's equation in higher dimensions (Q1074760)

The aim of the note is to discuss a certain class of Bäcklund transformations for 3-D Liouville equation \(\Delta u=\exp (u)\). This class of transformations originally constructed by \textit{G. Leibbrandt}, \textit{S. S. Wang} and \textit{N. Zamani} [J. Math. Phys. 23, 1566--1572 (1982; Zbl 0493.35073) and \textit{G. Leibbrandt}, Lett. Math. Phys. 4, 317--321 (1980; Zbl 0451.35056)] is reconstructed in a more convenient form and in the general \(N\)-dimensional situation. It is printed out, however, that the subsequent nonlinear superposition technique exploiting the permutability of the Bäcklund transformations is based in fact on a severe error and leads to incorrect further results. The exact formulae derived in the note force to conjecture that the relevant permutability does not take place (except for trivial cases).