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We propose a reformulation of SU(2) Yang-Mills theory in terms of new variables. These variables are appropriate for describing the theory in its infrared limit, and indicate that it admits knotlike configurations as stable solitons. As a consequence we arrive at a dual picture of the Yang-Mills theory where the short distance limit describes asymptotically free, massless point gluons and the large distance limit describes extended, massive knotlike solitons.In the high energy limit Yang-Mills theory is asymptotically free, and can be solved perturbatively. It describes the interactions of massless gluons which correspond to the transverse polarizations of the gauge fieldAt low energies Yang-Mills theory becomes strongly coupled. Perturbative techniques fail and nonperturbative methods must be developed. For this substantial efforts have been devoted, but numerical lattice approaches still remain the most viable tool to effectively explore the low energy theory. But in spite of our lacking theoretical understanding of low energy Yang-Mills theory, we expect that it exhibits color confinement with ensuing mass gap. The physical spectrum is supposed to describe massive composites of A µ such as glueballs. When quarks are introduced the gauge field should form string-like flux tubes which confine quarks inside hadrons.In the present Letter we propose an approach to investigate SU(2) Yang-Mills theory in the infrared limit. Our proposal is motivated by the qualitative picture developed in particular by 'tHooft and Polyakov [2], who asserted that the ultraviolet and infrared limits of a YangMills theory represent different phases, with color confinement due to a dual Meissner effect in a condensate of magnetic monopoles. This picture suggests that even though the gauge field A µ is the proper order parameter for describing the theory in its ultraviolet limit, in the infrared limit with monopole condensation some other order parameter could become more adequate. Naturally we expect, that such a change of variables may also imply certain need to reformulate the Yang-Mills action.In the high energy limit the theory is described by the standard Yang-Mills actionThis is the unique Lorentz and gauge invariant local action which is renormalizable in four dimensions and admits a Hamiltonian interpretation which identifies the transverse polarizations of A µ as the physical fields present in the ultraviolet limit.In the following we shall propose new variables for describing the infrared limit of a four dimensional SU(2) Yang-Mills theory. We shall argue that instead of A µ , in this limit the appropriate order parameter is a three component vector n a (x) (a = 1, 2, 3) with unit length n · n = 1 and classical action [3]Here m is a mass scale and e is a dimensionless coupling constant. This is the unique local and Lorentz-invariant action for the unit vector n which is at most quadratic in time derivatives so that it admits a Hamiltonian interpretation, and involves all such terms that are either relevant or marginal in the i...

We show that a charged two-condensate Ginzburg-Landau model or equivalently a Gross-Pitaevskii functional for two charged Bose condensates, can be mapped onto a version of the nonlinear O͑3͒ model. This implies in particular that such a system possesses a hidden O͑3͒ symmetry and allows for the formation of stable knotted solitons.

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We argue that protein loops can be described by topological domain-wall solitons that interpolate between ground states which are the ␣ helices and ␤ strands. We present an energy function that realizes loops as soliton solutions to its equation of motion, and apply these solitons to model a number of biologically active proteins including 1VII, 2RB8, and 3EBX ͑Protein Data Bank codes͒. In all the examples that we have considered we are able to numerically construct soliton solutions that reproduce secondary structural motifs such as ␣-helix-loop-␣-helix and ␤-sheet-loop-␤-sheet with an overall root-mean-square-distance accuracy of around 1.0 Å or less for the central ␣-carbons, i.e., close to the limits of current experimental accuracy. DOI: 10.1103/PhysRevE.82.011916 PACS number͑s͒: 87.15.Cc, 05.45.Yv Solitons are ubiquitous and widely studied objects that can be materialized in a variety of practical and theoretical scenarios ͓1,2͔. For example solitons can be deployed for data transmission in transoceanic cables, for conducting electricity in organic polymers ͓1͔, and they may also transport chemical energy in proteins ͓3͔. Solitons explain the Meissner effect in superconductivity and dislocations in liquid crystals ͓1͔. They also model hadronic particles, cosmic strings, and magnetic monopoles in high energy physics ͓2͔ and so on. The first soliton to be identified is the Wave of Translation that was observed by John Scott Russell in the Union Canal of Scotland. This wave can be accurately described by an exact soliton solution of the Korteweg-de Vries ͑KdV͒ equation ͓1͔. At least in principle it can also be constructed in an atomary level simulation where one accounts for each and every water molecule in the Canal, together with all of their mutual interactions. However, in such a Gedanken simulation it would probably become a real challenge to unravel the collective excitations that combine into the Wave of Translation without any guidance from the known soliton solution of the KdV equation: Solitons can not be constructed simply by adding up small perturbations around some ground state. Instead, a ͑topological͒ soliton emerges when non-linear interactions combine elementary constituents into a localized collective excitation that is stable against small perturbations and cannot decay, unwrap or disentangle ͓1,2͔.In this Communication we argue that topological solitons describe proteins in their native folded state ͓4,6͔. We characterize a folded protein by the Cartesian coordinates r i of its N central ␣ carbons, with i =1, ... ,N. For many biologically active proteins these coordinates can be downloaded from protein data bank ͑PDB͒ ͓7͔. Alternatively, the protein can be described in terms of its bond and torsion angles that can be computed from the PDB data. For this we introduce the tangent vector t i and the binormal vector b iTogether with the normal vector n i = b i ϫ t i we then have three vectors that are subject to the discrete Frenet equation ͓8͔.Here, T 2 and T 3 are two of the standard gen...

We introduce a novel generalization of the discrete nonlinear Schrödinger equation. It supports solitons that we utilize to model chiral polymers in the collapsed phase and, in particular, proteins in their native state. As an example we consider the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. We use its backbone as a template to explicitly construct a two-soliton configuration. Each of the two solitons describe well over 7.000 supersecondary structures of folded proteins in the Protein Data Bank with sub-angstrom accuracy suggesting that these solitons are common in nature.

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