1. For the narrow definition, and also for much information about a system we shall encounter shortly, the sine-Gordon equation (and about many similar systems), see A. Scott, F. Chu, and D. McLaughlin, Proc. IEEE 61, 1443 (1973).
2. Much of the material in this section is plagiarized from two long papers on the quantum problem: J. Goldstone and R. Jackiw, Phys. Rev. D11, 1486 (1975); N. Christ and T. D. Lee, Columbia Univ. preprint.
3. This equation has an interesting history. It was studied extensively by differential geometers in the last quarter of the nineteenth century, because of its role in the theory of surfaces of constant negative curvature. Bianchi called it, “l’equazione fondamentale di tutta la teoria delle superficie pseudosferiche”. (Lezioni di Geometria Differenziale, 3rd ed., I, 658.) The equation enters geometry in the following way: On any two-dimensional Riemannian manifold, it is possible to choose coordinates in some neighborhood of any point such that
$$
d{s^2}\;{\rm{ = d}}{{\rm{u}}^2}\;{\rm{ + d}}{{\rm{v}}^2}\;{\rm{2dudv cos }}\theta (u, v).
$$
In terms of such coordinates, a simple computation shows that the statement that the manifold has constant negative curvature is equivalent to
$$
{\partial ^2}\;\theta /\partial u\partial v\;{\rm{ = }}\alpha \;{\rm{sin }}\theta ,
$$
where α is a constant related to the magnitude of the curvature. This is the sine-Gordon equation, in lightcone coordinates. The equation entered particle physics through the work of Skyrme [Proc. Roy. Soc. A247, 260 (1958); A262, 237 (1961)] who studied it as a simple model of a non-linear field theory. Skyrme’s work prefigured many of the results we shall obtain in Sec. 5. The silly name is of recent origin. From a letter from David Finkelstein: “I am sorry that I ever called it the sine-Gordon equation. It was a private joke between me and Julio Rubinstein, and I never used it in print. By the time he used it as the title of a paper he had earned his Ph.D. and was beyond the reach of justice.”
4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, I, 721.
5. G. H. Derrick, J. Math. Phys. 5, 1252 (1964).