Editing Pseudosphere (section)

== Relation to solutions to the sine-Gordon equation ==
Pseudospherical surfaces can be constructed from solutions to the [[sine-Gordon equation]].<ref name="wheeler">{{cite web |last1=Wheeler |first1=Nicholas |title=From Pseudosphere to sine-Gordon equation |url=https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Geometric%20Origin%20of%20Sine-Gordon/Pseudosphere%20to%20Sine-Gordon.pdf |access-date=24 November 2022 }}</ref> A sketch proof starts with reparametrizing the tractroid with coordinates in which the [[Gauss–Codazzi equations]] can be rewritten as the sine-Gordon equation. 

In particular, for the tractroid the Gauss–Codazzi equations are the sine-Gordon equation applied to the static soliton solution, so the Gauss–Codazzi equations are satisfied. In these coordinates the [[first fundamental form|first]] and [[second fundamental form]]s are written in a way that makes clear the [[Gaussian curvature]] is −1 for any solution of the sine-Gordon equations.

Then any solution to the sine-Gordon equation can be used to specify a first and second fundamental form which satisfy the Gauss–Codazzi equations. There is then a theorem that any such set of initial data can be used to at least locally specify an immersed surface in <math>\mathbb{R}^3</math>.

A few examples of sine-Gordon solutions and their corresponding surface are given as follows:
* Static 1-soliton: pseudosphere
* Moving 1-soliton: [[Dini's surface]]
* Breather solution: [[Breather surface]]
* 2-soliton: [[Kuen surface]]