New Exact Solutions of the Classical Sine-Gordon Equation in 2 + 1 and 3 + 1 Dimensions

Abstract

The method of B\"acklund transformations is employed to derive in 2 + 1 and 3 + 1 dimensions exact solutions of the sine-Gordon equation $[{\ensuremath{\nabla}}^{2}\ensuremath{-}{c}^{\ensuremath{-}2}(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}})]\ensuremath{\chi}=sin\ensuremath{\chi}$. A formula is developed in 3 + 1 dimensions which permits us to generate without additional quadratures an infinite class of new time-dependent solutions.