On Bäcklund transformations and solutions to the 2+1 and 3+1 - dimensional sine — Gordon equation

Abstract

A Bäcklund transformation for the 3+1 - dimensional sine-Gordon equation is applied successively with different Bäcklund parameters two, three, and four times. The resulting matrix Bianchi relations are useful for generation of scalar Bianchi relations from which solutions to the sine-Gordon equation can be obtained. The Bianchi-Lamb parallelogram is generalized to a new Bianchi-Lamb parallelepiped for three successive Bäcklund transformations, and to a hyperparallelepiped for four successive Bäcklund transformations. Constraints on the Bäcklund parameters relevant for soliton wave solutions are interpreted geometrically in connection with these generalized Bianchi-Lamb diagrams. It is also shown that the constraints lead to conservation in time of the area between three line solitons moving in the XY-plane and of the volume between four plane solitons moving in the XYZ-space. The latter result which is a necessary condition for plane solitons is believed to be new.