Nonlinear Waves and Solitons on Contours and Closed Surfaces (eBook)

Nonlinear and solitary waves are historically related to quasi one-dimensional systems where the spatial extent in one direction is much bigger than in the other direction, such as channels and fibres. The present book treats the case of more compact systems and their nonlinear ascillations, which have only recently come into forms. Such systems include liquid drop models, Bose-Einstein condensates and even living cells. A general formalism is developed, based on the differential geometry of curved manifolds, and various applications are considered in the physical sciences and beyond.

Mathematical Prerequisites.- Mathematical Prerequisites.- The Importance of the Boundary.- Vector Fields, Differential Forms, and Derivatives.- Geometry of Curves.- Motion of Curves and Solitons.- Geometry of Surfaces.- Theory of Motion of Surfaces.- Solitons and Nonlinear Waves on Closed Curves and Surfaces.- Kinematics of Hydrodynamics.- Dynamics of Hydrodynamics.- Nonlinear Surface Waves in One Dimension.- Nonlinear Surface Waves in Two Dimensions.- Nonlinear Surface Waves in Three Dimensions.- Other Special Nonlinear Compact Systems.- Physical Nonlinear Systems at Different Scales.- Filaments, Chains, and Solitons.- Solitons on the Boundaries of Microscopic Systems.- Nonlinear Contour Dynamics in Macroscopic Systems.- Mathematical Annex.