Nonlinear Waves and Solitons on Contours and Closed Surfaces

1. Front Matter

   Pages i-xix

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2. Mathematical Prerequisites

   1. Front Matter

      Pages 1-3

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   2. Introduction

      Pages 5-10

   3. Mathematical Prerequisites

      Pages 11-22

   4. The Importance of the Boundary

      Pages 23-29

   5. Vector Fields, Differential Forms, and Derivatives

      Pages 31-67

   6. Geometry of Curves

      Pages 69-79

   7. Motion of Curves and Solitons

      Pages 81-102

   8. Geometry of Surfaces

      Pages 103-135

   9. Theory of Motion of Surfaces

      Pages 137-151

3. Solitons and Nonlinear Waves on Closed Curves and Surfaces

   1. Front Matter

      Pages 154-155

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   2. Kinematics of Hydrodynamics

      Pages 157-199

   3. Dynamics of Hydrodynamics

      Pages 201-235

   4. Nonlinear Surface Waves in One Dimension

      Pages 237-265

   5. Nonlinear Surface Waves in Two Dimensions

      Pages 267-287

   6. Nonlinear Surface Waves in Three Dimensions

      Pages 289-353

   7. Other Special Nonlinear Compact Systems

      Pages 355-362

4. Physical Nonlinear Systems at Different Scales

   1. Front Matter

      Pages 364-365

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   2. Filaments, Chains, and Solitons

      Pages 367-390

   3. Solitons on the Boundaries of Microscopic Systems

      Pages 391-422

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.

• Soliton
• closed curves
• contour dynamics
• differential geometry
• dynamische Systeme
• geometry
• manifold
• modeling
• nonlinear waves
• solitons
• surface waves
• fluid- and aerodynamics

From the reviews:

"The author succeeds in writing a monograph which introduces the physics of solitons on compact systems to readers who may not have any such prior knowledge. … The text is suitable for a graduate course on special topics or it can be used by readers with various backgrounds and interests who simply want to understand the connections between geometry and the phenomena of nonlinear waves." (Alina Stancu, Zentralblatt MATH, Vol. 1167, 2009)

• Department of Chemistry and Physics, Northwestern State University, Natchitoches, USA

  Andrei Ludu

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.

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