Breather surface

In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions[1] of the sine-Gordon equation, a differential equation appearing in theoretical physics.

There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation.

Due to the Lorentz invariance of sine-Gordon, a one-parameter family of Lorentz boosts can be applied to the static solution to obtain new solutions: on the pseudosphere side, these are known as Lie transformations, which deform the tractroid to the one-parameter family of surfaces known as Dini's surfaces.

Breather solutions are instead derived from the inverse scattering method for the sine-Gordon equation.

Locally the parameterization is well-behaved, but extended arbitrarily the resulting surface may have self-intersections and cusps.

Indeed, a theorem of Hilbert says that any pseudosphere cannot be embedded regularly (roughly, meaning without cusps) into