zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory. (English) Zbl 1002.58015
The $(2+1)$-dimensional relativistic Chern-Simons equations form a nonlinear system of partial differential equations for a gauge field $A_\mu$ and a Higgs field $\varphi$ defined on ${\Bbb R}^3$ with standard Lorentzian metric. The self-dual solutions absolutely minimize the energy. There are two possible boundary conditions $|\varphi(x)|\to 1$ or $|\varphi(x)|\to 0$ as ${\Bbb R}^2\ni x\to\infty$ consistent with finite energy. Solutions with $|\varphi(x)|\to 1$ have been dubbed `topological’ and were shown to exist by {\it R. Wang} [Commun. Math. Phys. 137, No. 3, 587-597 (1991; Zbl 0733.58009)]. In this article, the authors consider the existence of self-dual `non-topological’ solutions, i.e. with boundary condition $|\varphi(x)|\to 0$. They prove the existence of solutions with arbitrarily prescribed zeroes for the Higgs field and other good properties. In particular, these solutions are not in any way symmetric. The construction is obtained by perturbation about explicit solutions of the Liouville equation.

58E50Applications of variational methods in infinite-dimensional spaces
81T13Yang-Mills and other gauge theories
35J60Nonlinear elliptic equations
Full Text: DOI