The method of B\"acklund transformations is employed to derive in 2 + 1 and 3 + 1 dimensions exact solutions of the sine-Gordon equation [∇²-c^-2(∂²∂t²)]χ=sinχ. A formula is developed in 3 + 1 dimensions which permits us to generate without additional quadratures an infinite class of new time-dependent solutions.
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George Leibbrandt (1978) studied this question.
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