LEADER 03949nam a2200457 i 4500 001 991003634789707536 006 m o d 007 cr cnu---unuuu 008 190403s2018 sz o 000 0 eng d 020 $a303001276X$qelectronic book 020 $a9783030012762$qelectronic book 020 $z9783030012755$qprint 024 7 $a10.1007/978-3-030-01276-2$2doi 035 $ab14363653-39ule_inst 040 $aBibl. Dip.le Aggr. Matematica e Fisica - Sez. Matematica$beng 082 0 $a515.353 084 $aAMS 35-02 084 $aLC QA3.L28 100 1 $aKlein, Sebastian$0760812 245 12$aA spectral theory for simply periodic solutions of the Sinh-Gordon equation$h[e-book] /$cSebastian Klein 264 1$aCham, Switzerland :$bSpringer,$c[2018] 300 $a1 online resource 336 $atext$btxt$2rdacontent 337 $acomputer$bc$2rdamedia 338 $aonline resource$bcr$2rdacarrier 490 1 $aLecture notes in mathematics,$x0075-8434 ;$v2229 504 $aIncludes bibliographical references and index 505 0 $aPart I Spectral Data. -- Introduction. -- Minimal Immersions into the 3-Sphere and the Sinh-Gordon Equation. -- Spectral Data for Simply Periodic Solutions of the Sinh-Gordon Equation. -- Part II The Asymptotic Behavior of the Spectral Data. -- The Vacuum Solution. -- The Basic Asymptotic of the Monodromy. -- Basic Behavior of the Spectral Data. -- The Fourier Asymptotic of the Monodromy. -- The Consequences of the Fourier Asymptotic for the Spectral Data. -- Part III The Inverse Problem for the Monodromy. -- Asymptotic Spaces of Holomorphic Functions. -- Interpolating Holomorphic Functions. -- Final Description of the Asymptotic of the Monodromy. -- Non-special Divisors and the Inverse Problem for the Monodromy. -- Part IV The Inverse Problem for Periodic Potentials (Cauchy Data). -- Divisors of Finite Type. -- Darboux Coordinates for the Space of Potentials. -- The Inverse Problem for Cauchy Data Along the Real Line. -- Part V The Jacobi Variety of the Spectral Curve. -- Estimate of Certain Integrals. -- Asymptotic Behavior of 1-Forms on the Spectral Curve. -- Construction of the Jacobi Variety for the Spectral Curve. -- The Jacobi Variety and Translations of the Potential. -- Asymptotics of Spectral Data for Potentials on a Horizontal Strip. -- Perspectives 520 $aThis book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces 650 0$aDifferential equations, Partial 650 0$aDifferential equations 650 0$aFunctional analysis 650 0$aFunctions of complex variables 650 0$aGlobal differential geometry 776 08$iPrinted edition:$z9783030012779 856 40$zA book accessible through the World Wide Web$uhttp://link.springer.com/10.1007/978-3-030-01276-2 907 $a.b14363653$b03-03-22$c03-04-19 912 $a991003634789707536 996 $aSpectral theory for simply periodic solutions of the Sinh-Gordon equation$91539995 997 $aUNISALENTO 998 $ale013$b03-04-19$cm$d@ $e-$feng$gsz $h2$i0