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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!

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2 Answers 2

Google these if not already done: Breather, Solilton

And also check out Richard Palais's home page:

And in math overflow..

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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")
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