Narrow Results

We review recent progress in the computation of leading quantum corrections to the energies of classical solitons with topological structure, including multi-soliton models in one space dimension and string configurations in three space dimensions. Taking advantage of analytic continuation techniques to efficiently organize the calculations, we show how quantum corrections affect the stability of solitons in the Shifman–Voloshin model, stabilize charged electroweak strings coupled to a heavy fermion doublet, and bind Nielsen–Olesen vortices at the classical transition between type I and type II superconductors.

We study axion strings with the electroweak gauge flux in the DFSZ axion model and show that these strings, called electroweak axions, exhibit superconductivity without fermionic zero modes. We also show that the primordial magnetic field in the early universe can induce a large electric current along the string. A pair of the strings carrying such a large current feels a net attractive force between them and can form a Y-shaped junction in the early universe, whose formation probability is roughly estimated to be 1/2.

We review our recent works on solitons in U(N_C) gauge theories with N_F(≥ N_C) Higgs fields in the fundamental representation, which possess eight supercharges. The moduli matrix is proposed as a crucial tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Since vacua are in the Higgs phase, we find domain walls (kinks) and vortices as the only elementary solitons. Stable monopoles and instantons can exist as composite solitons with vortices attached. Webs of walls are also found as another composite soliton. The moduli space of all these elementary as well as composite solitons are found in terms of the moduli matrix. The total moduli space of walls is given by the complex Grassmann manifold SU(N_F)/[SU(N_C) × SU(N_F − N_C) × U(1)] and is decomposed into various topological sectors corresponding to boundary conditions specified by particular vacua. We found charges characterizing composite solitons contribute negatively (either positively or negatively) in Abelian (non-Abelian) gauge theories. Effective Lagrangians are constructed on walls and vortices in a compact form. The power of the moduli matrix is illustrated by an interaction rule of monopoles, vortices, and walls, which is difficult to obtain in other methods. More thorough description of the moduli matrix approach can be found in our review article¹ (hep-th/0602170).

We obtain periodic and quasi-periodic wave solutions of a set of coupled non-linear Schrödinger equations (integrable Manakov model). One and two phase quasi-periodic solutions for the integrable Manakov system are given in terms of Weierstrass and Jacobian elliptic functions. The reduction of quasi-periodic solutions to elliptic functions and soliton solutions is analyzed.

Results of numerical study of lump solutions (2D solitons) of a generalised 2D Gardner equation are presented. In the moving coordinate frame the equation is:

We show that the self-trapped condensate can be dynamically stabilized in two and three dimensional free space by periodic oscillation of the scattering length. The oscillating interaction produces an effective potential that prevents the condensate from collapsing. We take into account the effect of dissipation to simulate realistic situations.

We show that solitons can be trapped in a fibre Bragg grating by a local defect provided that the soliton energy and defect strength are within a certain range of values.

We review the progress and problems in the development of soliton transmission systems. Dispersion managed soliton could be a future candidate as the information carrier. Future research prospect is presented.

The self-frequency shift effect is examined in an actively mode-locked fiber laser. We observed, with the insertion of a filter, this effect can be greatly suppressed and the range of gain for stable operation is wider.

Interactions between non-BPS non-Abelian vortices are studied in non-Abelian U(1) × SU(N) extensions of the Abelian-Higgs model in four dimensional space-time. In addition to the usual type I/II Abelian superconductors, we find two new regime which we call type I*/II*.

Starting with a quark model of nucleon structure, elaborated by one of the authors, in which the valence quarks are strongly correlated within the nucleon, the light nuclei are constructed by assuming similar correlations with the quarks of neighboring nucleons. Applying the model to larger collections of nucleons reveals the emergence of the face-centered cubic (FCC) symmetry at the nuclear level. Binding of nucleons are provided by quark loops formed by three and four nucleon correlations. The boundary of neutron stability for light and medium mass nuclei is predicted.

New soliton solutions for thick branes in 4 + 1 dimensions are considered in this article. In particular, brane models based on the sine-Gordon (SG), φ⁴ and φ⁶ scalar fields are investigated; in some cases Z₂ symmetry is broken. Besides, these soliton solutions are responsible for supporting and stabilizing the thick branes. In these models, the origin of the symmetry breaking resides in the fact that the modified scalar field potential may have non-degenerate vacuua and these non-degenerate vacuua determine the cosmological constant on both sides of the brane. At last, in order to explore the particle motion in the neighborhood of the brane, the geodesic equations along the fifth dimension are studied.

It is known that ultra-extreme Kerr-Newman (KN) solution (a >> m) produces the gravitational and EM fields of the electron and has a topological defect which may be regularized by a solitonic source, formed as a false-vacuum bubble filled by Higgs condensate in a supersymmetric superconducting state. Structure and stability of this source is determined by Bogomolnyi equations as a BPS-saturated soliton of the oscillon type. The Principal Null Congruences of the KN solution determine consistent embedding of the Dirac equation, which acquires the mass from the Higgs condensate inside the soliton, indicating that this soliton forms a bag model. Shape of this bag is unambiguously determined by BPS-bound. The bag turns out to be flexible and takes the form of a very thin disk, which is completed by a ring-string along its sharp boundary. The ring-string traveling waves generate extra deformations of the bag creating a circulating singular pole. Bag model of the KN source integrates the dressed and pointlike electron in a bag-string-quark system, which removes the conflict between the point-like electron of the Dirac theory and the required gravitational soliton model.

We review recent results concerning the interactions of solitary waves for several universal nonlinear dispersive or wave equations. Though using quite different techniques, these results are partly inspired by classical papers based on the inverse scattering theory for integrable models.

Lithium-7 exhibits a broad Feshbach resonance that we exploit to tune the interactions in a Bose-Einstein condensate (BEC). We find that the rate of photoassociation can be enhanced by several orders of magnitude by tuning close to the resonance, and use this effect to observe saturation in the rate of association of a BEC for the first time. We have also used a lithium BEC to explore the effects of disorder on the transport and coherence properties of the condensate. We also show that the scattering length goes through a shallow zero-crossing far from the resonance, where it may be made positive or negative with a magnitude of less than 0.1 a_o, and have made preliminary transport measurements in the regime of weak repulsive and attractive interactions.

Soliton solutions in a scalar field theory defined on an AdS₁₊₁ background space-time are investigated. An analytic soliton solution is obtained in a polynomial model, and the classical soliton mass is calculated. The fluctuation spectrum around the soliton solution is determined, and the one-loop quantum correction to the soliton mass is computed in the semi-classical approximation.

The dynamical model on 3+1 dimensional space-time admitting soliton solutions is discussed. The proposal soliton is localized in the vicinity of a closed contour, which could be linked and/or knotted. The topological charge is Hopf invariant. Some applications in realistic physical systems are indicated.

The soliton resolution conjecture for the focusing nonlinear Schrödinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a “statistical version” of this conjecture at mass-subcritical non-linearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis, and bare hands probability theory.

Note from Publisher: This article contains the abstract only.

The ultra extreme Kerr-Newman (KN) solution(a = J/m >> m) produces the gravitational and EM fields of the electron. It has a naked singular ring – a topological defect which may be regularized by a solitonic source forming the pseudo-vacuum bubble filled by Higgs condensate in a supersymmetric superconducting state. Structure and stability of this source is determined by Bogomolnyi equations as a BPS-saturated soliton. The Principal Null Congruences of the KN solution determine consistent embedding of the Dirac equation, which acquires the mass from the Higgs condensate inside the soliton, indicating that this soliton forms a bag model. Shape of this bag is unambiguously determined by BPS-bound. The bag turns out to be flexible and takes the form of a very thin disk, which is completed by a ring-string along its sharp boundary. The ring-string traveling waves generate extra deformations of the bag creating a circulating singular pole. Bag model of the KN source integrates the dressed and pointlike electron in a bag-string-quark system, which removes the conflict between gravity and the point-like electron of the Dirac theory.