Transformations of Manifolds and Applications to Differential Equations

Transformations of Manifolds and Applications to Differential Equations

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Description

The interaction between differential geometry and differential equations is the general theme of this analysis. The author focuses on the role played by differential geometry in the study of differential equations, combining the geometric and analytic aspects of the theory, not only in the classical examples, but also in results obtained since 1980, on integrable systems with an arbitrary number of independent variables. The discussion applies to problems in evolution equations, strongly hyperbolic systems of the hydrodynamic type, linear Weingarten surfaces, and submanifolds of constant curvature.show more

Product details

  • Hardback | 224 pages
  • 159 x 235 x 19.05mm | 476g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • 1584880341
  • 9781584880349

Table of contents

Transformations of Surfaces and ApplicationsThe Structure EquationsDifferential Equations Associated to Linear Weingarten SurfacesGeodesic Congruences and Parallel SurfacesPseudo-Spherical Geodesic CongruencesBäcklund Transformation for the sine-Gordon and the Elliptic sinh-Gordon Equations. Superposition FormulaThe Laplace Transformation for Second-Order Hyperbolic Equations and its Geometric InterpretationDifferential Equations which Describe Pseudo-Spherical SurfacesSubmanifolds of Constant Sectional CurvatureThe Structure Equations in a Pseudo-Riemannian Space FormSubmanifolds of Constant Sectional Curvature. The Generating EquationPseudo-Spherical Geodesic Congruences and ApplicationsPseudo-Spherical Geodesic Congruences. A Generalization of Bäcklund's TheoremPermutability TheoremBäcklund Transformation and Superposition Formula for the Generalized Wave Equation and the Generalized sine-Gordon EquationLinearization of the Bäcklund TransformationThe Inverse Scattering Method for the Generalized Wave EquationThe Inverse Scattering Method for the Generalized sine-Gordon EquationThe Bäcklund Transformation in Terms of Scattering Data. Soliton SolutionsThe Generating EquationThe Generating EquationBäcklund Transformation for the Generating Equation and its LinearizationSuperposition FormulaThe Generating Intrinsic EquationThe Generating Intrinsic Equation. Submanifolds of Constant Curvature Characterized by the MetricBäcklund Transformation for the Generating Intrinsic Equation. Symmetry GroupHyperbolic Toroidal Submanifolds of Euclidean SpaceFlat Toroidal Submanifolds of the Unit SphereGeometric Properties of Submanifolds Associated to Special SolutionsLaplace Transformation in Higher DimensionsLaplace Transformations of Cartan ManifoldsThe Higher-Dimensional Laplace Invariants for Systems of Second Order PDEsThe Generalized Method of Laplace for Systems of Second Order PDEsApplications of the Laplace Transformation to Hydrodynamic Systems Rich in Conservation LawsReferencesIndexshow more