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

TL;DR

This paper explores the existence, classification, and properties of multi-soliton solutions to the higher-dimensional Sine-Gordon equation, revealing diverse structures and velocity behaviors, including superluminal propagation and hybrid solutions.

Contribution

It introduces a novel classification of N-soliton solutions in (1+2) and (1+3) dimensions using relativistic kinematics, extending soliton theory beyond integrability constraints.

Findings

Multi-soliton solutions exist in higher dimensions with diverse velocities.

Solutions include subluminal, superluminal, and hybrid propagations.

Spatial structures include planar and brane-like configurations.

Abstract

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.