Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

Lin, Fanghua; Yang, Yisong

doi:10.1007/s00220-004-1110-y

Cited by 68 publications

(99 citation statements)

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“…Sublinear growth laws of the form (4.20) have profound implications for the formation of knotted/tangled soliton structures. See Lin & Yang (2004, 2007 for details in the classical situation (Faddeev 1979(Faddeev , 2002Faddeev & Niemi 1997) of three spatial dimensions (nZ1). …”

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confidence: 99%

Universal growth law for knot energy of Faddeev type in general dimensions

Lin

Yang²

2008

Proc. R. Soc. A.

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The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of 'ideal' knots. In this paper, we show that the energy infimum E N stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S 4nK1 / S 2n (n R1) in general dimensions obeys the sharp fractional-exponent growth law E N wjN j p , where the exponent p is universally rendered as pZ ð4nK1Þ=4n, which is independent of the detailed fine structure of the knot energy but determined completely by the dimensions of the domain and range spaces of the field configuration maps.

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confidence: 99%

Universal growth law for knot energy of Faddeev type in general dimensions

Lin

Yang²

2008

Proc. R. Soc. A.

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“…The corresponding topological index classifying a finite energy field configuration is a linking number (the Hopf index), whereas it is a winding number for the Skyrme model. The existence of soliton solutions in the Faddeev-Niemi model has been proven in [11], and confirmed by numerical calculations, e.g., in ([12] - [15]). …”

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confidence: 67%

Soliton stability in some knot soliton models

Adam

Sánchez-Guillén

Wereszczyński

2007

Journal of Mathematical Physics

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We study the issue of stability of static soliton-like solutions in some non-linear field theories which allow for knotted field configurations. Concretely, we investigate the AFZ model, based on a Lagrangian quartic in first derivatives with infinitely many conserved currents, for which infinitely many soliton solutions are known analytically. For this model we find that sectors with different (integer) topological charge (Hopf index) are not separated by an infinite energy barrier. Further, if variations which change the topological charge are allowed, then the static solutions are not even critical points of the energy functional. We also explain why soliton solutions can exist at all, in spite of these facts. In addition, we briefly discuss the Nicole model, which is based on a sigma-model type Lagrangian. For the Nicole model we find that different topological sectors are separated by an infinite energy barrier.

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“…In that paper, they obtain a condition for the existence of solutions for the 3D Skyrme's problem consisting in a family of strict decomposition inequalities. By modifying the proofs in [1,2] but still using the concentration-compactness approach, an existence result for minimizers of deg(φ) = ±1 can be established under the same conditions as in [4,5]. This is not surprising.…”

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“…In [1,2] an existence result for minimizers of degree ±1 was proved by using the concentration-compactness method. But as Fanghua Lin and Yisong Yang have pointed out recently [4,5], the proof of the main result contained in [1,2] is not correct. This Erratum announces that these proofs can be corrected by modifying the arguments used in [1,2].…”

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“…In a very interesting paper basically devoted to the study of the Faddeev knots ( [4])(see also [5] in 2D), F. Lin and Y. Yang have proved recently the existence of 3D Skyrmions of degree ±1 by using a different approach, which is based on a cubic decomposition of the whole space. In that paper, they obtain a condition for the existence of solutions for the 3D Skyrme's problem consisting in a family of strict decomposition inequalities.…”

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Existence of 3D Skyrmions

Esteban

2004

Commun. Math. Phys.

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The Skyrme problem consists in minimizing the energy functional E(φ) := R 3 |∇φ| 2 + i =j |∂ i φ ∧ ∂ j φ)| 2 dx in the set of functions φ : R 3 → S 3 such that deg(φ) = 1 2 π 2 R 3 det(φ, ∇φ)dx = k ∈ Z, the infimum being denoted by I k . In [1,2] an existence result for minimizers of degree ±1 was proved by using the concentration-compactness method. But as Fanghua Lin and Yisong Yang have pointed out recently [4,5], the proof of the main result contained in [1, 2] is not correct. This Erratum announces that these proofs can be corrected by modifying the arguments used in [1,2]. The method used is still the concentration-compactness principle but applied in a different, and in some sense, less usual way. The new proof, in full detail, has been electronically posted [3].In a very interesting paper basically devoted to the study of the Faddeev knots ([4])(see also [5] in 2D), F. Lin and Y. Yang have proved recently the existence of 3D Skyrmions of degree ±1 by using a different approach, which is based on a cubic decomposition of the whole space. In that paper, they obtain a condition for the existence of solutions for the 3D Skyrme's problem consisting in a family of strict decomposition inequalities. By modifying the proofs in [1, 2] but still using the concentration-compactness approach, an existence result for minimizers of deg(φ) = ±1 can be established under the same conditions as in [4,5]. This is not surprising. Indeed, the above family of strict inequalities is not only sufficient for the existence of minimizers, but it is in fact necessary and sufficient for the relative compactness of all minimizing sequences. The precise statements of the main results are:Note that in [1,2] only binary decompositions (J = 2) had to be avoided. The difference lies in the fact that we do not know anymore whether for all ∈ Z \ {0, k}, the large inequalities I k ≤ I + I k− hold or not.

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Existence of Energy Minimizers as Stable Knotted Solitons in the Faddeev Model

Cited by 68 publications

References 45 publications

Universal growth law for knot energy of Faddeev type in general dimensions

Universal growth law for knot energy of Faddeev type in general dimensions

Soliton stability in some knot soliton models

Existence of 3D Skyrmions

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