Exact solutions are derived for the three-dimensional and four-dimensional sine-Gordon equation [∇2−∂2/(c2∂t2)]χ =sinχ. The principal tools in the derivation are a new Bäcklund transformation and the appropriate generating formulas which allow us to generate an infinite number of real classical solutions. Three-dimensional computer plots depicting the time evolution are presented (a) for the two-soliton and four-soliton solutions of the sine-Gordon equation, and (b) for the three-wave interaction of the associated sine-Gordon equation.
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Leibbrandt et al. (1980) studied this question.
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