Bäcklund generated solutions of Liouville’s equation and their graphical representations in three spatial dimensions

Source abstract

The Bäcklund–Bianchi method is employed to generate, in three spatial dimensions, the following multiple solutions of Liouville’s equation ∇2α = exp α: The three-wave interaction function α3 and the five-wave interaction function α5. It is verified numerically that α3 satisfies Liouville’s equation to an accuracy of one part in 1014, while α5 satisfies it to one part in 106. The construction of α5 is conditional upon solving ten nonlinear constraint equations. We analyze the complicated structures of α3 and α5 with the help of a three-dimensional plotting routine. It is found that α3 is, surprisingly enough, only characterized by a single ring singularity, while α5 exhibits three ring singularities. It is speculated that the function tanh α3 represents a ring soliton whose shape appears to be preserved in the nonlinear superposition of similar ring solitons. The derivation of Liouville’s solutions α3 and α5 is intimately connected with the auxiliary functions β2 and β4 which solve Laplace’s equation. The latter are also derived and plotted in the paper.