Nonlinear Waves and Solitons on Contours and Closed Surfaces

Everything the Power of the World does is done in a circle. The sky is round and I have heard that the earth is round like a ball and so are all the stars. The wind, in its greatest power, whirls. Birds make their nests in circles, for theirs is the same religion as ours. The sun comes forth and goes down again in a circle. The moon does the same and both are round. Even the seasons form a great circle in their changing and always come back again to where they were. The life of a man is a circle from childhood to childhood. And so it is everything where power moves. Black Elk (18631950) Nonlinearity is a captivating manifestation of the observable Universe, whose importance has increased over the decades, and has found more and more ?elds of application ranging from elementary particles, nuclear physics, biology, wave dynamics at any scale, ?uids, plasmas to astrophysics. The central character of this 172-year-old story is the soliton. Namely, a localized pulse traveling without spreading and having particle-like properties plus an in?nite number of conservation laws associated to its dynamics. In general, solitons arise as exact solutions of approximative models. There are di?- ent explanation, at di?erent levels, for the existence of solitons. From the experimentalist point of view, solitons can be created if the propagation c- ?gurationislongenough,narrowenough(likelongandshallowchannels,?ber optics, electric lines, etc.

Includes supplementary material: sn.pub/extras

Autorentext

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.

Klappentext

The present 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. Emphasis on the 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.

Nonlinear Waves and Solitons on Contours and Closed Surfaces provides graduate students and researchers in mathematics, physics and engineering with a ready tutorial and reference

Inhalt
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.