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

Abstract

Recent results using inverse scattering techniques interpret every solution $ϕ(x,y)$ of the sine-Gordon equation as a non-linear superposition of solutions along the axes $x=0$ and $y=0$. Here we provide a geometric method of integration, as well as a geometric interpretation. Specifically, 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. Based on a generalized Weierstrass pair that depends only on these values, we prove that to each such unconstrained pair of differentiable functions, there corresponds uniquely an associated family of pseudospherical immersions; we construct these immersions explicitely.