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.