The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation

Abstract

We analyse the Dirichlet problem for the elliptic sine-Gordon equation in the upper half plane. We express the solution q(x, y) in terms of a Riemann–Hilbert problem whose jump matrix is uniquely defined by a certain function b(λ), , explicitly expressed in terms of the given Dirichlet data g0(x) = q(x, 0) and the unknown Neumann boundary value g1(x) = qy(x, 0), where g0(x) and g1(x) are related via the global relation {b(λ) = 0, λ ⩾ 0}. Furthermore, we show that the latter relation can be used to characterize the Dirichlet-to-Neumann map, i.e. to express g1(x) in terms of g0(x). It appears that this provides the first case that such a map is explicitly characterized for a nonlinear integrable elliptic PDE, as opposed to an evolution PDE.