Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Due to my not being very skilled in differential geometry, I want to ask if there is a reference (book, paper, etc.) that explicitly works out how one constructs the parametric equations of a pseudospherical surface from a given solution of the sine-Gordon equation (or Bäcklund transformations of those solutions). My searches are only turning up links between sine-Gordon solutions and pseudospherical surfaces, but I have not been able to find an explicit demonstration of how one derives the parametric equations from a soliton solution (or more likely it is my searching skills that are as sub-par as my differential geometry aptitude).

As an example of what I want to see, consider the following form of SGE:

$$w_{tu}=\sin w$$

One of the simplest solutions to this form of the SGE is


I am told that from this solution, one could obtain the parametric equations of either of the pseudosphere proper, or Dini's surface, depending on the value of the arbitrary constant $a$. I gather that the particular parametric equations for the Dini surface (formulas 1-3 here) and the pseudosphere (formulas 5-7 here) were obtained from the expression for $w$, but it is not immediately obvious to me how to go about the derivation.

I will appreciate either being pointed to references, or if somebody might want to write out the explicit derivation for a non-expert like me. Thank you!

share|cite|improve this question

Google these if not already done: Breather, Solilton

And also check out Richard Palais's home page:

And in math overflow..

share|cite|improve this answer

I wrote a thesis on it long time ago. For what can help you I can give you my code for Mathematica. For the constructions of Dini and Ulisse surfaces as Backlund Trasnformation you can have a look at Luigi Bianchi, Opere, volume V, Edizioni Cremonese, Roma, 1957

breather[d_][x_,y_]:={2 d/Sqrt[1-d^2] *
Cosh[Sqrt[1-d^2] x]/
(d^2 Cosh[Sqrt[1-d^2] x]^2 + (1-d^2) Sin[d y]^2) *
(Sin[d y] Sin[ y] + d Cos[y] Cos[d y]),
2 d/Sqrt[1-d^2] * Cosh[Sqrt[1-d^2] x]/
(d^2 Cosh[Sqrt[1-d^2] x]^2 + (1-d^2) Sin[d y]^2) *
(-Sin[d y] Cos[y] + d Sin[y] Cos[d y]),
x - 2 d^2/Sqrt[1-d^2] * Cosh[Sqrt[1-d^2] x]/
(d^2 Cosh[Sqrt[1-d^2] x]^2 + (1-d^2) Sin[d y]^2) * 
Sinh[Sqrt[1-d^2] x]}

If you want to plot you just need to

ParametricPlot3D[breather[d=1/3][x, y], {x, -1.3 Pi, 1.3 Pi}, {y, 0, 3   Pi}, 
 Boxed -> False, PlotPoints -> 60, Axes -> None, PlotRange -> All]

Changing the parameter d you can see differents breathers.

Finally if you are into Open source software too you can use the following code for sagemath

u, v = var(’u,v’)
fx = 2*d/sqrt(1-d^2)*cosh(sqrt(1-d^2)*x)/(d^2 *cosh(sqrt(1-d^2)*x)^2+(1-d^2)*sin(d*y)^2)*(sin(d*y)*sin(y)+d*cos(y)*cos(d*y))

fy = 2*d/sqrt(1-d^2)*cosh(sqrt(1-d^2)*x)/(d^2*cosh(sqrt(1-d^2)*x)^2+(1-d^2)*sin(d*y)^2)*(-sin(d*y)*cos(y)+d*sin(y)*cos(d*y))

fz = x-2*d^2/sqrt(1-d^2)*cosh(sqrt(1-d^2)*x)/(d^2*cosh(sqrt(1-d^2)*x)^2+(1-d^2)*sin(d*y)^2)*sinh(sqrt(1-d^2)*x)

parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*(1-d)^(-1)*pi),frame=False,color="red")
share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.