Spatially extended relativistic particles associated with multi-soliton solutions of the Sine-Gordon equation in more than one space dimension

TL;DR

This paper demonstrates that multi-soliton solutions of the Sine-Gordon equation in higher dimensions behave like extended relativistic particles, revealing new properties and violations of conservation laws in multi-soliton interactions.

Contribution

It shows that multi-soliton solutions in (1+2) and (1+3) dimensions act as spatially extended relativistic particles, a novel insight into higher-dimensional Sine-Gordon dynamics.

Findings

Multi-soliton solutions exist for all N in (1+n) dimensions.

Vertices form at collision regions, moving with the solution at constant velocity.

Vertices emulate free, extended relativistic particles and bound states.

Abstract

Contrary to the decades-old understanding, SGn, the Sine-Gordon equation in (1+n) dimensions, has N-soliton solutions for any N >= 1, not only for n = 1, but also for n = 2 and 3. While SG1 solitons are confined to a line, SG2- and SG3-solitons are confined to a plane. An SG2-soliton solution moves rigidly with a constant velocity in the plane, and an SG3-solution moves rigidly with a constant velocity in the plane, and along the normal to the plane. A conservation law for the current density, obeyed by the single-SGn-soliton solution, is violated by all multi-soliton solutions. The violation manifests itself by generating vertices, structures that are localized around the soliton collision regions and decay exponentially in all directions in the (1+n)-dimensional space. In (1+1) dimensions, vertices evolve and then decay. In (1+2) and (1+3) dimensions, they move with the whole solution at its constant velocity, preserving their profiles, thereby emulating free, spatially extended, relativistic particles and bound states of such particles.