### Abstract

The problem of existence of knot-like solitons as the energy-minimizing configurations in the Faddeev model, topologically characterized by Hopf invariant, Q, is considered. It is proved that, in the full space situation, there exists an infinite set S of integers so that for any m ∈ S, the Faddeev energy, E, has a minimizer among the class Q = m; in the bounded domain situation, the same existence theorem holds when S is the set of all integers. One of the important technical results is that E and Q satisfy the sublinear inequality E ≤ C|Q|^{3/4}, where C > 0 is a universal constant, which 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) | 187-197 |

Number of pages | 11 |

Journal | Science in China, Series A: Mathematics |

Volume | 47 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2004 |

### Keywords

- Hopf invariant
- Knots
- Minimization
- Solitons

### ASJC Scopus subject areas

- Mathematics(all)