Nonlinear Waves and Solitons on Contours and Closed Surfaces

Andrei Ludu

Sep 2007 · Springer Science & Business Media

About this ebook

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 (1863–1950) 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.

About the author

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.