The elliptic sine-Gordon equation in a half plane

Beatrice Pelloni, Dimitrios Pinotsis

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We consider boundary value problems for the elliptic sine-Gordon equation posed in the half plane y > 0. This problem was considered in Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) using the classical inverse scattering transform approach. Given the limitations of this approach, the results obtained rely on a nonlinear constraint on the spectral data derived heuristically by analogy with the linearized case.We revisit the analysis of such problems using a recent generalization of the inverse scattering transform known as the Fokas method, and show that the nonlinear constraint of Gutshabash and Lipovskii (1994 J. Math. Sci. 68 197–201) is a consequence of the so-called global relation. We also show that this relation implies a stronger constraint on the spectral data, and in particular that no choice of boundary conditions can be associated with a decaying (possibly mod 2π) solution analogous to the pure soliton solutions of the usual, time-dependent sine-Gordon equation.We also briefly indicate how, in contrast to the evolutionary case, the elliptic sine-Gordon equation posed in the half plane does not admit linearisable boundary conditions.
Original languageEnglish
Pages (from-to)77-88
Number of pages12
JournalNonlinearity
Volume23
Issue number1
DOIs
Publication statusPublished - Jan 2010

Keywords

  • Partial differential equations; Tunneling phenomena; point contacts
  • weak links
  • Josephson effects
  • Inverse problems

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