### 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 π_{3}(S^{2}) = 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 language | English (US) |
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Pages (from-to) | 273-303 |

Number of pages | 31 |

Journal | Communications In Mathematical Physics |

Volume | 249 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2004 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics