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

A. S. Fokas, Beatrice Pelloni

Research output: Contribution to journalArticlepeer-review

22 Citations (Scopus)


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), {\lambda}\in{\mathbb R} , 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.
Original languageEnglish
Pages (from-to)1011-1031
Number of pages21
Issue number4
Publication statusPublished - Apr 2012


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