Existence of energy minimizers as stable knotted solitons in the Faddeev model

Research output: Contribution to journalArticle

Abstract

In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S2) = ℤ. We prove in the full space situation that there exists an infinite subset script S sign of ℤ such that for any m ∈ script S sign, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for script S sign = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C C|Q|3/4, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

Original languageEnglish (US)
Pages (from-to)273-303
Number of pages31
JournalCommunications in Mathematical Physics
Volume249
Issue number2
StatePublished - Aug 2004

Fingerprint

Minimizer
Solitons
solitary waves
Hopf Invariant
Energy
Configuration
configurations
Homotopy Groups
energy
Quantum Field Theory
Model
Knot
set theory
Bounded Domain
Subset
Class

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

Existence of energy minimizers as stable knotted solitons in the Faddeev model. / Lin, Fanghua; Yang, Yisong.

In: Communications in Mathematical Physics, Vol. 249, No. 2, 08.2004, p. 273-303.

Research output: Contribution to journalArticle

@article{ea9343895d384353ab5b1790dcddd176,
title = "Existence of energy minimizers as stable knotted solitons in the Faddeev model",
abstract = "In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S2) = ℤ. We prove in the full space situation that there exists an infinite subset script S sign of ℤ such that for any m ∈ script S sign, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for script S sign = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C C|Q|3/4, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.",
author = "Fanghua Lin and Yisong Yang",
year = "2004",
month = "8",
language = "English (US)",
volume = "249",
pages = "273--303",
journal = "Communications in Mathematical Physics",
issn = "0010-3616",
publisher = "Springer New York",
number = "2",

}

TY - JOUR

T1 - Existence of energy minimizers as stable knotted solitons in the Faddeev model

AU - Lin, Fanghua

AU - Yang, Yisong

PY - 2004/8

Y1 - 2004/8

N2 - In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S2) = ℤ. We prove in the full space situation that there exists an infinite subset script S sign of ℤ such that for any m ∈ script S sign, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for script S sign = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C C|Q|3/4, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

AB - In this paper, we study the existence of knot-like solitons realized as the energy-minimizing configurations in the Faddeev quantum field theory model. Topologically, these solitons are characterized by an Hopf invariant, Q, which is an integral class in the homotopy group π3(S2) = ℤ. We prove in the full space situation that there exists an infinite subset script S sign of ℤ such that for any m ∈ script S sign, the Faddeev energy, E, has a minimizer among the topological class Q = m. Besides, we show that there always exists a least-positive-energy Faddeev soliton of non-zero Hopf invariant. In the bounded domain situation, we show that the existence of an energy minimizer holds for script S sign = ℤ. As a by-product, we obtain an important technical result which says that E and Q satisfy the sublinear inequality E ≤ C C|Q|3/4, where C > 0 is a universal constant. Such a fact explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.

UR - http://www.scopus.com/inward/record.url?scp=4344717357&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344717357&partnerID=8YFLogxK

M3 - Article

VL - 249

SP - 273

EP - 303

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -