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

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 g 0 ( x ) = q ( x , 0) and the unknown Neumann boundary value g 1 ( x ) = q y ( x , 0), where g 0 ( x ) and g 1 ( 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 g 1 ( x ) in terms of g 0 ( 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.