A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation
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Beschreibung
This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.
This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.
Features a self-contained elaboration of the spectral theory for a specific integrable system Provides detailed proofs of all necessary asymptotic estimates Includes a complete treatment of singular spectral curves
Autor*in
Sebastian Klein
Themen in »A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation«
37K10,53A10,58E12,35P05,35R30,37K25,40A05,42B05,46C05,46E15 Asymptotic Estimates Constant Mean Curvature Surfaces Integrable Systems Sinh-Gordon Equation Spectral Theory partial differential equations ordinary differential equations
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“The book is useful for specialists studying periodic solutions to integrable nonlinear partial differential equations.” (Dmitry E. Pelinovsky, Mathematical Reviews, October, 2019)
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