Painleve Equations in the Differential Geometry of Surfaces

1. Auflage 2000
ISBN: 978-3-540-41414-8
Verlag: Springer

Buch, Englisch, Band 1753, 120 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 201 g

Reihe: Lecture Notes in Mathematics

Since the time of surfaces -+ in differential Gauss, parametrized (x, y) P(x, y) have been described a frame attached to the moving geometry through TI(x, y) surface. One introduces the Gauss- which linear dif- Weingarten equations are, ferential equations = U = TIX T1, VT', !PY (1. for the and their condition frame, compatibility - = V + [U, V] 0, UY (1.2) which the Gauss-Codazzi For surfaces in three-dim- represents equations. a sional Euclidean the frame T1 lies in the usually or space, group SO(3) SU(2). On the other a of a non-linear in the form hand, representation equation (1.2) is the of the of of starting point theory integrable equations (theory solitons), which in mathematical in the 1960's appeared physics [NMPZ, AbS, CD, FT, More the differential for the coefficients of AbC]. exactly, partial equation (1.2) the matrices U and V is considered to be if these matrices can be integrable, extended to U V non-trivially a one-parameter family (x, y, A), (x, y, A) satisfying - = + U(A)y V(A). [U(A), V(A)] 0, (1-3) so that the differential is and original partial equation preserved.'. Usually U(A) V are rational functions of the which is called the (A) parameter A, spectral param- In soliton the eter is called the Lax. theory, representation (1.3) representation the Zakharov-Shabat or representation [ZS].

Zielgruppe

Research

Autoren/Hrsg.

TU Berlin, Alexander I. Bobenko

Eitner, Ulrich

Fachgebiete

• Mathematik | Informatik Mathematik Geometrie Nicht-Euklidische Geometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionentheorie, Komplexe Analysis
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

1. Introduction.- 2. Basics on Painlevé Equations and Quaternionic Description of Surfaces.- 3. Bonnet Surfaces in Euclidean Three-space.- 4. Bonnet Surfaces in S3 and H3 and Surfaces with Harmonic Inverse Mean Curvature.- 5. Surfaces with Constant Curvature.- 6. Appendices.