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

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.