2009/11/10 by Jonatan Lenells, J. Lenells, A. S. Fokas +2
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Black Holes and Theoretical Physics #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #nlin.SI
paper · pdf · doi:10.48550/arxiv.0911.1898
36 pages
arxiv created 2009/11/10 · openalex publication_date 2009/11/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The stationary, axisymmetric reduction of the vacuum Einstein equations, the so-called Ernst equation, is an integrable nonlinear PDE in two dimensions. There now exists a general method for analyzing boundary value problems for integrable PDEs, and this method consists of two steps: (a) Construct an integral representation of the solution characterized via a matrix Riemann-Hilbert (RH) problem formulated in the complex k-plane, where k denotes the spectral parameter of the associated Lax pair. This representation involves, in general, some unknown boundary values, thus the solution formula is \it not yet effective. (b) Characterize the unknown boundary values by analyzing a certain equation called the \it global relation. This analysis involves, in general, the solution of a nonlinear problem; however, for certain boundary value problems called linearizable, it is possible to determine the unknown boundary values using only linear operations. Here, we employ the above methodology for the investigation of certain boundary value problems for the elliptic version of the Ernst equation. For this problem, the main novelty is the occurence of the spectral parameter in the form of a square root and this necessitates the introduction of a two-sheeted Riemann surface for the formulation of the relevant RH problem. As a concrete application of the general formalism, it is shown that the particular boundary value problem corresponding to the physically significant case of a rotating disk is a linearizable boundary value problem. In this way the remarkable results of Neugebauer and Meinel are recovered.