### 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), {\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 language | English |
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Pages (from-to) | 1011-1031 |

Number of pages | 21 |

Journal | Nonlinearity |

Volume | 25 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 2012 |

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## Cite this

Fokas, A. S., & Pelloni, B. (2012). The Dirichlet-to-Neumann map for the elliptic sine-Gordon equation.

*Nonlinearity*,*25*(4), 1011-1031. https://doi.org/10.1088/0951-7715/25/4/1011