Sine-Gordon equation in higher dimensions: A fresh look at integrability

TL;DR

This paper explores the integrability and soliton solutions of the Sine-Gordon equation in higher dimensions, revealing connections between non-integrable cases and lower-dimensional integrable equations through transformations.

Contribution

It demonstrates that higher-dimensional Sine-Gordon equations, despite lacking known integrability features, still admit soliton solutions connected to lower-dimensional integrable cases via transformations.

Findings

N soliton solutions exist in higher dimensions.

Solutions split into subspaces based on velocity relative to light speed.

Transformations relate higher-dimensional solutions to lower-dimensional integrable equations.

Abstract

The Sine-Gordon equation is integrable in (1+1)-dimensional Minkowski and in 2-dimensional Euclidean spaces. In each case, it has a Lax pair, and a Hirota algorithm generates its N soliton solutions for all N greater than or equal to 1. The (1+2)-dimensional equation does not pass known integrability tests and does not have a Lax pair. Still, the Hirota algorithm generates N soliton solutions of that equation for all N greater than or equal to 1. Each multi-soliton solution propagates rigidly at a constant velocity, v. The solutions are divided into two unconnected subspaces: Solutions with v greater than or equal to c =1, and v smaller than c. Each subspace is connected by an invertible transformation (rotation plus dilation) to the space of soliton solutions of an integrable Sine-Gordon equation in two dimensions. The faster-than-light solutions are connected to the solutions in (1+1)-dimensional Minkowski space. The slower-than-light solutions are connected to the solutions in 2-dimensional Euclidean space by this transformation and also by Lorentz transformations. The Sine-Gordon equation in (1+3)-dimensional Minkowski space has a richer variety of solutions. Its slower-than-light solutions are connected to the solutions of the integrable equation in 2-dimensional Euclidean space. However, only a subset of its faster-than-light solutions is connected to the solutions of the integrable equation in (1+1)-dimensional Minkowski space.