Introduction -- 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 -- 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 -- Filaments, Chains and Solitons -- Solitons on the Boundaries of Microscopic Systems -- Nonlinear Contour Dynamics in Macroscopic Systems -- Mathematical Annex -- References -- Index.

This volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such a closed curves and surfaces and other domain contours. It assumes familiarity with basic soliton theory and nonlinear dynamical systems. The first part of the book introduces the mathematical concept required for treating the manifolds considered, providing relevant notions from topology and differential geometry. An introduction to the theory of motion of curves and surfaces - as part of the emerging field of contour dynamics - is given. The second and third parts discuss the modeling of various physical solitons on compact systems, such as filaments, loops and drops made of almost incompressible materials thereby intersecting with a large number of physical disciplines from hydrodynamics to compact object astrophysics. This book is intended for graduate students and researchers in mathematics, physics and engineering. This new edition has been thoroughly revised, expanded and updated.