Breather surface

Surface of constant negative curvature In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature − 1 {\displaystyle -1} , where the curvature is well-defined. This makes them examples of generalized pseudospheres. Mathematical background Sine-Gordon standing breather is a swinging in time coupled kink-antikink 2-soliton solution. There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation. This correspondence can be built starting with the simplest example of a pseudosphere, the tractroid. In a special set of coordinates, known as asymptotic coordinates, the Gauss–Codazzi equations, which are consistency equations dictating when a surface of prescribed first and second fundamental form can be embedded into three-dimensional space with the flat metric, reduce to the sine-Gordon equation. In the correspondence, the tractroid corresponds to the static 1-soliton s