Stationary ring solitons in field theory — Knots and vortons

Abstract

We review the current status of the problem of constructing classical field theory solutions describing stationary vortex rings in Minkowski space in 3 + 1 dimensions. We describe the known up to date solutions of this type, such as the static knot solitons stabilized by the topological Hopf charge, the attempts to gauge them, the anomalous solitons stabilized by the Chern-Simons number, as well as the non-Abelian monopole and sphaleron rings. Passing to the rotating solutions, we first discuss the conditions insuring that they do not radiate, and then describe the spinning Q-balls, their twisted and gauged generalizations reported here for the first time, spinning skyrmions, and rotating monopole-antimonopole pairs. We then present the first explicit construction of global vortons as solutions of the elliptic boundary value problem, which demonstrates their non-radiating character. Finally, we describe the analogs of vortons in the Bose-Einstein condensates, analogs of spinning Q-balls in the non-linear optics, and also moving vortex rings in superfluid helium and in ferromagnetics.

Dynamics of slender monopoles and anti-monopoles in non-Abelian superconductor

“…It is also interesting to study the dynamics of monopoles on a curved vortex string. Especially, one may consider a vortex ring (which is called a vorton) [50]. It is interesting to pursue the similarity between bound states of topological solitons and bound states (mesons) of elementary constituents (quarks).…”

Integrability and Prolongation Structure of a Generalized Korteweg-de Vries Equation

“…The study of solitons recently has been of interest due to their appearance in many physical applications (Radu and Volkov, 2008) and so the investigation of a new equation is of interest. The classical Korteweg-de Vries equation has many applications which extend beyond the original applications to solitary water waves (Bracken, 2004).…”

Twisted Toroidal Vortex Solitons in Inhomogeneous Media with Repulsive Nonlinearity

“…In contrast with the usual vortex rings featuring the single winding number, m , in the ( , ) x y plane, here the complex amplitude function, Counterparts of r w and i w in the FSM are two components of the triplet of real fields which is restricted to the surface of the unit sphere [10,28,29], and the twisted rings correspond to the fundamental knots (the so-called unknots) [37]. The present model also bears certain similarity to the twisted Q-balls, i.e., stationary rotating nontopological solitons in field models with polynomial nonlinearity [10,29].…”

“…Ramifications of this topic are well known in optics [1], Bose-Einstein condensates (BECs) [2,3], ferromagnetics [4], superconductors [5], semiconductors [6], nuclear matter [7], and the field theory [8][9][10]. Self-attractive nonlinearity is usually needed for the formation of localized states.…”

“…For these solitons, the phase of the wavefunction changes both along and around the vortex ring, with the corresponding topological invariant ( linking number) is the product of the number of twists ( ) s and overall vorticity ( ) m . Since this arrangement is typical for solitons of the Faddeev-Skyrme model (FSM) [10,[28][29][30], with the triplet of real scalar fields realizing the Hopf map, 3 2 :R S j  , such states are called Hopfions. So far, Hopfions were found only in systems with multicomponent wavefunctions [8,31].…”