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The (1+1)-dimensional Sine-Gordon equation passes integrability testscommonly applied to nonlinear evolution equations. Its soliton solutions areobtained by a Hirota algorithm. In higher space-dimensions, the equation doesnot pass these tests. In this paper, using no more than the relativistickinematics of the tachyonic momentum vectors, from which the soliton solutionsare constructed through the Hirota algorithm, the existence and classificationof N-soliton solutions of the (1+2)- and (1+3)-dimensional equations for all Ngreater than or equal to 1 are presented. In (1+2) dimensions, eachmultisoliton solution propagates rigidly at one velocity. The solutions aredivided into two subsets: Solutions whose velocities are lower than the speedof light (c = 1), or are greater than or equal to c.  In (1+3)-dimensions, multisoliton solutions are characterized by spatialstructure and velocity composition. The spatial structure is either planar(rotated (1+2)-dimensional solutions), or genuinely three-dimensional - branes.Some solutions, planar or branes, propagate rigidly at one velocity, which islower than, equal to, or higher than c. A subset of the branes containshybrids, in which different clusters of solitons propagate at differentvelocities. Some velocities may be lower than c but some must be equal to, orgreater than c. Finally, the speed of light cannot be approached from withinthe subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.

On multi soliton solutions of the Sine-Gordon equation in more than one space dimension