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Extensively revised, expanded and updated for the new edition, this book offers a comprehensive introduction to nonlinear waves and soliton theory in the special environment of compact spaces such as closed curves and surfaces and other domain contours.

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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.

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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.

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From the reviews of the second edition:

"This book is devoted to the detailed exposition of the present state of research in the field of nonlinear dynamics of boundaries of physical systems which in the two-dimensional case reduces to the so-called contour dynamics. ... The text is intended to be an introduction to the physics and mathematics of solitons on compact systems ... . In general, this book can serve as a source of information for students and researchers in nonlinear physics on nonlinear dynamics of compact systems." (Anatoly M. Kamchatnov, Mathematical Reviews, September, 2013)

"This book is an update of the first edition ... and deals with models of nonlinear media filling closed (compact) curves and surfaces. The main subject is a systematic construction and study of solitons states in such systems. ... the book may be used as a basis for a graduate course on the theory of nonlinear waves and solitons." (Boris A. Malomed, Zentralblatt MATH, Vol. 1253, 2013)

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