Initial value problems of the Sine-Gordon equation and geometric solutions

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15 Scopus citations


Recent results using inverse scattering techniques interpret every solution φ(x, y) of the sine-Gordon equation as a nonlinear superposition of solutions along the axes x=0 and y=0. This has a well-known geometric interpretation, namely that every weakly regular surface of Gauss curvature K=-1, in arc length asymptotic line parametrization, is uniquely determined by the values φ(x, 0) and φ(0, y) of its coordinate angle along the axes. We introduce a generalized Weierstrass representation of pseudospherical surfaces that depends only on these values, and we explicitely construct the associated family of pseudospherical immersions corresponding to it.

Original languageEnglish
Pages (from-to)257-271
Number of pages15
JournalAnnals of Global Analysis and Geometry
Issue number3
StatePublished - May 2005


  • Generalized Weierstrass representation
  • Loop algebra
  • Loop group
  • Pseudospherical surface


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