Initial Value Problems of the Sine-Gordon Equation and Geometric Solutions

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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
JournalAnnals of Global Analysis and Geometry (Springer Netherlands)
StatePublished - 2005


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