zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Topological solitons. (English) Zbl 1100.37044
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 0-521-83836-3/hbk; 0-511-20783-2/ebook). xi, 493 p. £ 70.00; $ 115.00 (2004).

The authors of this very interesting book consider the main examples of topological solitons and survey in detail static and dynamical multi-soliton solutions. They discuss several topics of interest as kinks in one dimension, lumps and vortices in two dimensions, monopoles and skyrmions in three dimensions, instantons in four dimensions. In some field theories, there are no static forces between solitons, and there is a large class of static multi-soliton solutions satisfying equations of Bogomolny type. The manifold of solutions can be considered as a moduli space. Its dimension increases with the soliton number.

The authors survey a lot of results in this area. Of special interest here is the discussion on the skyrmion quantization as well as the unstable analogue of the solitons, known as sphalerons. A very interesting topic is the question for the geodesic dynamics on moduli space. It is an adiabatic theory of multi-soliton motion at modest speeds when the static forces vanish. Many numerical results concerning solitons and their properties are shown as well. The techniques of modern differential geometry and algebra, such as Lie groups and algebras, moduli space dynamics, homotopy theory, Chern-Simons forms are used wherever appropriately to illuminate the considered topics.

The book is self-contained and beautifully written. It should remain for a long period of time as a standard reference for anyone interested in soliton theory and its application in physics.


MSC:
37K40Soliton theory, asymptotic behavior of solutions
57-02Research monographs (manifolds)
81T13Yang-Mills and other gauge theories
57R17Symplectic and contact topology
57R22Topology of vector bundles and fiber bundles
81-02Research monographs (quantum theory)
35Q51Soliton-like equations