Faddeev Model Research Articles

Revisit of the Faddeev Model in Dimension Two

Revisit of the Faddeev Model in Dimension Two

Global nonlinear stability of one kind of large solutions to evolutionary Faddeev model

It is well known that there is few results about the global classical solutions to quasilinear wave equations with large data. The famous evolutionary Faddeev model corresponding to maps from the Minkowski space $$\mathbb {R}^{1+n}$$ to the unit sphere $$\mathbb {S}^2$$ is satisfying one kind of quasilinear wave equations. In this paper, we show the global nonlinear stability of one kind of nontrivial and large classical solutions.

Cross section of neutrons from the H2(n,2n) reaction at En=15 MeV

The double-differential cross section of the deuteron breakup reaction $^{2}\mathrm{H}(n,2n$) has been studied experimentally with a neutron beam energy of 15 MeV. Special attention has been devoted to estimation of background condition and multiple scattering effect in the scattering sample. Experimental data have been compared with models based on phase-space approximation used in the ENDF/B-VIII.0 data library and in the MCNP neutron transport code, as well as with rigorous model based on Faddeev equations used for cross section evaluations in JENDL data library. It was found that experimental data are better reproduced by Faddeev model, however, the model overestimates data in the low-energy region of the neutron spectrum ($<4$ MeV).

Skyrme and Faddeev models in the low-energy limit of 4d Yang–Mills–Higgs theories

Firstly, we consider U(Nc) Yang–Mills gauge theory on R3,1 with Nf>Nc flavours of scalar fields in the fundamental representation of U(Nc). The moduli space of vacua is the Grassmannian manifold Gr(Nc,Nf). It is shown that for strong gauge coupling this 4d Yang–Mills–Higgs theory reduces to the Faddeev sigma model on R3,1 with Gr(Nc,Nf) as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we consider a Yang–Mills–Higgs model with Nf=2Nc and a Higgs potential breaking the flavour group U(Nf)=U(2Nc) to U+(Nc)×U−(Nc), realizing the simplest A2⊕A2-type quiver gauge theory. The vacuum moduli space of this model is the group manifold Uh(Nc) which is the quotient of U+(Nc)×U−(Nc) by its diagonal subgroup. When the gauge coupling constant is large, this 4d Yang–Mills–Higgs model reduces to the Skyrme sigma model on R3,1 with Uh(Nc) as target. Thus, both the Skyrme and the Faddeev model arise as effective field theories in the infrared of Yang–Mills–Higgs models.

Unphysical Energy Sheets and Resonances in the Friedrichs–Faddeev Model

We consider the Friedrichs-Faddeev model in the case where the kernel of the potential operator is holomorphic in both arguments on a certain domain of $\mathbb{C}$. For this model we, first, study the structure of the $T$- and $S$-matrices on unphysical energy sheet(s). To this end, we derive representations that explicitly express them in terms of these same operators considered exclusively on the physical sheet. Furthermore, we allow the Friedrichs-Faddeev Hamiltonian undergo a complex deformation (or even a complex scaling/rotation if the model is associated with an infinite interval). Isolated non-real eigenvalues of the deformed Hamiltonian are called the deformation resonances. For a class of perturbation potentials with analytic kernels, we prove that the deformation resonances do correspond to the scattering matrix resonances, that is, they represent the poles of the scattering matrix analytically continued to the respective unphysical sheet.

Large data global regularity for the $2+1$-dimensional equivariant Faddeev model

This article addresses the large data global regularity for the equivariant case of the $2+1$-dimensional Faddeev model and shows that it holds true for initial data in $H^s\times H^{s-1}(\mathbb R^2)$ with $s>3$.

Skyrme–Faddeev model from 5d super-Yang–Mills

We consider 5d Yang–Mills–Higgs theory with a compact ADE-type gauge group G and one adjoint scalar field on R3,1×R+, where R+=[0,∞) is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d N=(2,0) superconformal field theory on R3,1×R+×S1 along the circle S1. We show that in the low-energy limit, when momenta along R3,1 are much smaller than along R+, the 5d Yang–Mills–Higgs theory reduces to a nonlinear sigma model on R3,1 with a coset G/H as its target space. Here H is a closed subgroup of G determined by the Higgs-field asymptotics at infinity. The 4d sigma model describes an infinite tower of interacting fields, and in the infrared it is dominated by the standard two-derivative kinetic term and the four-derivative Skyrme–Faddeev term.

Knotted fields and explicit fibrations for lemniscate knots.

We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ℓ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme-Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Structure of topological solitons in nonlinear spinor model

16-spinor field is suggested to unify Skyrme and Faddeev models describing barvons and leptons as topological solitons.

Sharp Global Regularity for the 2 + 1-Dimensional Equivariant Faddeev Model

The aim of this article is to prove that for the |$2+1$|-dimensional equivariant Faddeev model, which is a quasilinear generalization of the corresponding nonlinear |$\sigma$| model, small initial data in critical Besov spaces evolve into global solutions which scatter.

Magnetic domains and waves in the Skyrme – Faddeev model

The Skyrme-Faddeev model admits exact analytical non localized solutions, which describe magnetic domain wall solutions when multivalued singularities appear or, differently, always regular periodic nonlinear waves, which may degenerate into linear spin waves or solitonic structures. Here both classes of solutions are derived and discussed and a general discusssion about the existence of integrable subsectors of the model is addressed.

Topological energy bounds for the Skyrme and Faddeev models with massive pions

A topological lower bound on the Skyrme energy which depends explicitly on the pion mass is derived. This bound coincides with the previously best known bound when the pion mass vanishes, and improves on it whenever the pion mass is non-zero. The new bound can in particular circumstances be saturated. New energy bounds are also derived for the Skyrme model on a compact manifold, for the Faddeev–Skyrme model with a potential term, and for the Aratyn–Ferreira–Zimerman and Nicole models.

Abelian Hopfions of the [formula omitted] model on [formula omitted] and a fractionally powered topological lower bound

Regarding the Skyrme–Faddeev model on R3 as a CP1 sigma model, we propose CPn sigma models on R2n+1 as generalisations which may support finite energy Hopfion solutions in these dimensions. The topological charge stabilising these field configurations is the Chern–Simons charge, namely the volume integral of the Chern–Simons density which has a local expression in terms of the composite connection and curvature of the CPn field. It turns out that subject to the sigma model constraint, this density is a total divergence. We prove the existence of a topological lower bound on the energy, which, as in the Vakulenko–Kapitansky case in R3, is a fractional power of the topological charge, depending on n. The numerical construction of the simplest ring shaped un-knot Hopfion on R5 is also discussed.

Waves in the Skyrme–Faddeev model and integrable reductions

We show that the Skyrme–Faddeev model can be reduced in different ways to completely integrable sectors; the corresponding classes of solutions can be parametrized by specific sets of arbitrary functions. Moreover, using the ansatz of a phase and pseudo-phase reduction, the corresponding ordinary nonlinear wave solutions can be integrated in terms of elliptic functions, leading to periodic solutions. The Whitham averaging method has been exploited in order to describe a slow deformation of periodic wave states, leading to a Hamiltonian system, the integrability of which has been studied.

8-Spinors and structure of solitons in generalized Mie electrodynamics

A generalization of Mie electrodynamics is considered. It includes a 8-spinor field and higher powers of the Mie invariant AμAμ. Particular topological properties of 8-spinors are indicated and are associated with the existence of the remarkable Brioschi identity of eight squares, which permits deriving a natural 8-spinor unification of the Skyrme model of baryons and the Faddeev model of leptons, these particles being treated as topological solitons. Two types of soliton configurations admitted by the model are constructed. These are charged static and neutral lightlike (luxons) ones.

Polarization of Lyman-α and Balmer-α emission in proton–hydrogen collisions: a study using first-order Born–Faddeev-type equations

A three-body Born–Faddeev model is devised to calculate the total cross sections of Balmer-α and Lyman-α emissions, for the excitation of hydrogen atoms by proton impact in the energy range of 100 keV–7 MeV. In addition, the polarization alignment factor A20 is calculated and compared against available experimental data to further test the theory. Specifically, here we use the Faddeev–Watson–Lovelace formalism to study the excitation of atomic hydrogen from its ground state to the excited states of n = 2 and 3 and magnetic sublevels l = 0, 1 and 2, wherever applicable. The first-order electronic, A(1)e, and the first-order nuclear, A(1)n, amplitudes are considered in order to calculate the excitation transition matrix (TPT), while a near-the-shell condition is assumed throughout. In addition, our results were used to calculate the first-order form factors. The present results are compared, where possible, with those of other theoretical and experimental works that are currently available in the literature.

Helical buckling of Skyrme–Faddeev solitons

Solitons in the Skyrme–Faddeev model on are shown to undergo buckling transitions as the circumference of the S 1 is varied. These results support a recent conjecture that solitons in this field theory are well-described by a much simpler model of elastic rods.

On the Wave Operators for the Friedrichs–Faddeev Model

We provide new formulae for the wave operators in the context of the Friedrichs–Faddeev model. Continuity with respect to the energy of the scattering matrix and a few results on eigenfunctions corresponding to embedded eigenvalues are also derived.

Solitonic solutions of Faddeev model

An application of the equation proposed by the present authors, which is equivalent to the static field equation of the Faddeev model, is discussed. Under some assumptions on the space and on the form of the solution, the field equation is reduced to a nonlinear ordinary differential equation of second order. By solving this equation numerically, some solitonic solutions are obtained. It is discussed that the product of two integers specifying solutions may be identified with the Hopf topological invariant.