The Sine-Gordon Equation (SGE), q_{xt}= sin q, is a well-known soliton equation. It is also the Gauss-Codazzi equation for surfaces in R^3 with Gaussian curvature -1. For a survey see Geometry of solitons by Chuu-Lian Terng and Karen Uhlenbeck. The SGE is written in light cone coordinate, and the space time coordinate is X= x+t, T= x-t. Below the wave for the SEG soliton solution q is shown as the graph of the derivative of q with respect to X for a sequence of increasing time. All waves and K=-1 surfaces shown here were made by 3D-XplorMath program.
(1) 1-soliton solutions: q(x,t)= 4 arctan (e^{sx+ s^{-1}t}). The corresponding K=-1 surfaces are the Dini surfaces. When s=1, the 1-soliton is stationary (in space-time coordinates) and the corresponding surface is the pseudo-sphere. You can see a picture of the stationary 1-soliton wave and a movie of a family of Dini surfaces
(2) Bianchi permutability formula gives explicit 2-soliton solutons of the SGE:
q(x,t)= 4 arctan (A/B), where A=(s_1+s_2) (p_1-p_2), B= (s_1-s_2)(1+ p_1p_2), where p_1= e^{s_1 x+ s_1^{-1} t}, p_2= e^{s_2 x+ s_2^{-1}t}
You can see a movie of 2-soliton waves and a 2-soliton K=-1surface
(3) There are infinitely many 2-solitons that are periodic in time, and are called breather. You can see a movie of breather waves and two breather K=-1 surfaces: Breather surface 1, Breather surface 2.
(4) a movie of 3-soliton wave and a 3-soliton K=-1surface
(5) a K=-1 surface corresponds to a 3-soliton solution of the SGE which is an "algebraic sum" of a 1-soliton and a breather: Pseudo-sphere plus breather surface
