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:
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!