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

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The (1+1)-dimensional Sine-Gordon equation passes integrability tests commonly applied to nonlinear evolution equations. Its soliton solutions are obtained by a Hirota algorithm. In higher space-dimensions, the equation does not pass these tests. In this paper, using no more than the relativistic kinematics of the tachyonic momentum vectors, from which the soliton solutions are constructed through the Hirota algorithm, the existence and classification of N-soliton solutions of the (1+2)- and (1+3)-dimensional equations for all N greater than or equal to 1 are presented. In (1+2) dimensions, each multisoliton solution propagates rigidly at one velocity. The solutions are divided into two subsets: Solutions whose velocities are lower than the speed of light (c = 1), or are greater than or equal to c. In (1+3)-dimensions, multisoliton solutions are characterized by spatial structure 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 is lower than, equal to, or higher than c. A subset of the branes contains hybrids, in which different clusters of solitons propagate at different velocities. Some velocities may be lower than c but some must be equal to, or greater than c. Finally, the speed of light cannot be approached from within the subset of slower-than-light solutions in both (1+2) and (1+3) dimensions.
Original languageEnglish GB
StatePublished - 1 Apr 2013


  • Nonlinear Sciences - Exactly Solvable and Integrable Systems
  • 35Q51
  • 37K40
  • 35Q75


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