Geometrical interpretation of the solutions of the sine–Gordon equation

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Abstract
The sine–Gordon equation is known to possess solutions that correspond to solitons, that is, localized entities that maintain their shape after collisions, and have certain properties characteristic of elementary particles. Although the algebraic structure of these solutions is well known, their geometric interpretation as surfaces of constant negative curvature has not been previously illuminated. We discuss these surfaces herein. Curves drawn on these surfaces along the asymptotic directions at each point simulate solutions of the nonlinear wave equation φxx−φtt=sin φ.