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Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations | J. Lenells
; A. S. Fokas
; | Date: |
12 Feb 2020 | Abstract: | By employing a novel generalization of the inverse scattering transform
method known as the unified transform or Fokas method, it can be shown that the
solution of certain physically significant boundary value problems for the
elliptic sine-Gordon equation, as well as for the elliptic version of the Ernst
equation, can be expressed in terms of the solution of appropriate $2 imes
2$-matrix Riemann--Hilbert (RH) problems. These RH problems are defined in
terms of certain functions, called spectral functions, which involve the given
boundary conditions, but also unknown boundary values. For arbitrary boundary
conditions, the determination of these unknown boundary values requires the
analysis of a nonlinear Fredholm integral equation. However, there exist
particular boundary conditions, called linearizable, for which it is possible
to bypass this nonlinear step and to characterize the spectral functions
directly in terms of the given boundary conditions. Here, we review the
implementation of this effective procedure for the following linearizable
boundary value problems: (a) the elliptic sine-Gordon equation in a semi-strip
with zero Dirichlet boundary values on the unbounded sides and with constant
Dirichlet boundary value on the bounded side; (b) the elliptic Ernst equation
with boundary conditions corresponding to a uniformly rotating disk of dust;
(c) the elliptic Ernst equation with boundary conditions corresponding to a
disk rotating uniformly around a central black hole; (d) the elliptic Ernst
equation with vanishing Neumann boundary values on a rotating disk. | Source: | arXiv, 2002.5244 | Services: | Forum | Review | PDF | Favorites |
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