Static solitons, Lorentz invariance, and a new perspective on the integrability of the Sine Gordon equation in (1+2) dimensions

TL;DR

This paper demonstrates that the (1+2)-dimensional Sine-Gordon equation admits N-soliton solutions for any N, using the Hirota algorithm and Lorentz transformations, offering a new perspective on its integrability.

Contribution

It reveals that static N-soliton solutions exist for the (1+2)D Sine-Gordon equation and can be Lorentz transformed, challenging previous assumptions about its integrability.

Findings

N-soliton solutions exist for any N in (1+2)D Sine-Gordon equation

Static solutions can be Lorentz transformed to moving frames

Momentum vectors of solitons are linearly related in solutions

Abstract

Contrary to the common understanding, the Sine-Gordon equation in (1+2) dimensions does have N-soliton solutions for any N. The Hirota algorithm allows for the construction of static N-soliton solutions (i.e., solutions that do not depend on time) of that equation for any N. Lorentz transforming the static solutions yields N-soliton solutions in any moving frame. They are scalar functions under Lorentz transformations. In an N-soliton solution in a moving frame, (N-2) of the (1+2)-dimensional momentum vectors of the solitons are linear combinations of the two remaining vectors.