Narrow Results

We review our proposal to generalize the standard two-dimensional flatness construction of Lax–Zakharov–Shabat to relativistic field theories in d+1 dimensions. The fundamentals from the theory of connections on loop spaces are presented and clarified. These ideas are exposed using mathematical tools familiar to physicists. We exhibit recent and new results that relate the locality of the loop space curvature to the diffeomorphism invariance of the loop space holonomy. These result are used to show that the holonomy is Abelian if the holonomy is diffeomorphism invariant.

These results justify in part and set the limitations of the local implementations of the approach which has been worked out in the last decade. We highlight very interesting applications like the construction and the solution of an integrable four-dimensional field theory with Hopf solitons, and new integrability conditions which generalize BPS equations to systems such as Skyrme theories. Applications of these ideas leading to new constructions are implemented in theories that admit volume-preserving diffeomorphisms of the target space as symmetries. Applications to physically relevant systems like Yang–Mills theories are summarized. We also discuss other possibilities that have not yet been explored.

We study magnetic flux tubes in the Higgs vacuum of the mass deformation of SU(N_c), SYM and its large N_c string dual, the Polchinski-Strassler geometry. Choosing equal masses for the three adjoint chiral multiplets, for all N_c we identify a "colour-flavour locked" symmetry, SO(3)_C+F which leaves the Higgs vacuum invariant. At weak coupling, we find explicit non-Abelian k-vortex solutions carrying a ℤ_Nc-valued magnetic flux, with topological winding 0 < k < N_c. These k-strings spontaneously break SO(3)_C+F to U(1)_C+F resulting in an S² moduli space of solutions. The world-sheet sigma model is a nonsupersymmetric ℂℙ¹ model with a theta angle θ₁₊₁ = k(N_c - k)θ₃₊₁ where θ₃₊₁ is the Yang-Mills vacuum angle. We find numerically that k-vortex tensions follow the Casimir scaling law T_k ∝ k(N_c - k) for large N_c. In the large N_c IIB string dual, the SO(3)_C+F symmetry is manifest in the geometry interpolating between AdS₅ × S⁵ and the interior metric due to a single D5-brane carrying D3-brane charge. We identify candidate k-vortices as expanded probe D3-branes formed from a collection of kD-strings. The resulting k-vortex tension exhibits precise Casimir scaling, and the effective world-sheet theta angle matches the semiclassical result. S-duality maps the Higgs to the confining phase so that confining string tensions at strong 't Hooft coupling also exhibit Casimir scaling in theory in the large N_c limit.

In this paper, we study the kink–antikink interaction in the ϕ⁶ system. This system has soliton-like solutions which fall into two categories with opposite topological charges. We consider the interaction of a kink solution with an antikink at different incident energies (velocities) and observe the outcome. In general, the collision is inelastic. There are two critical velocities v₁ and v₂ in the system, between which there are enormous number of narrow scattering windows. We study the fractal-like structure of the windows. Below v₁ the pair are absorbed into each other. Above v₁ the pair get separated after the inelastic interaction. Finally, we obtain the internal modes of the kink in this system and address its relevance to the interaction behavior.

We analyze the fluctuations of charge of the (1+1)-dimensional Dirac's fermion with charge conjugation breaking. This is done taking the separation between background soliton and antisoliton going to infinity.

In this paper, we obtain stable and metastable soliton solutions of a coupled system of two real scalar fields with five discrete points of vacua. These solutions have definite topological charges and rest energies and show classical dynamical stability. From a quantum point of view, however, the V-type solutions are expected to be unstable and decay to D-type solutions. The induced decay of a V-type soliton into two D-type ones is calculated numerically, and shown to be chiral, in the sense that the decay products do not respect left–right symmetry.

It is shown that an alternative to the standard scalar quantum electrodynamics (QED) is possible. In this new version, there is only global gauge invariance as far as the charged scalar fields are concerned, although local gauge invariance is kept for the electromagnetic field. The electromagnetic coupling has the form j_μ(A^μ +∂^μB) where B is an auxiliary field and the current j_μ is A_μ independent, so that no "sea gull terms" are introduced. As a consequence of the absence of sea gulls, it is seen that no Klein paradox appears in the presence of a strong square well potential. In a model of this kind, spontaneous breaking of symmetry does not lead to photon mass generation, instead the Goldstone boson becomes a massless source for the electromagnetic field. When spontaneous symmetry breaking takes place infrared questions concerning the theory and generalizations to global vector QED are discussed. In this framework, Q-Balls and other nontopological solitons that owe their existence to a global U(1) symmetry can be coupled to electromagnetism and could represent multiply charged particles now in search in the large hadron collider (LHC). Furthermore, we give an example where an "Emergent" Global Scalar QED can appear from an axion–photon system in an external magnetic field. Finally, formulations of Global Scalar QED that allow perturbative expansions without sea gulls are developed.

Monopoles and instantons are sheets (membranes) and strings in d = 5+1 dimension, respectively, and instanton strings can terminate on monopole sheets. We consider a pair of monopole and antimonopole sheets which is unstable to decay and results in a creation of closed instanton strings. We show that when an instanton string is stretched between the monopole sheets, there remains a new topological soliton of codimension five after the pair annihilation, i.e. a twisted closed instanton string or a knotted instanton.

We study spherically symmetric finite energy solutions of two Higgs–Chern–Simons-Yang–Mills–Higgs (HCS-YMH) models in 3+1 dimensions, one with gauge group SO(5) and the other with SU(3). The Chern–Simons (CS) densities are defined in terms of both the Yang–Mills (YM) and Higgs fields and the choice of the two gauge groups is made so that they do not vanish. The solutions of the SO(5) model carry only electric charge and zero magnetic charge, while the solutions of the SU(3) model are dyons carrying both electric and magnetic charges like the Julia–Zee (JZ) dyon. Unlike the latter, however, the electric charge in both models receives an important contribution from the CS dynamics. We pay special attention to the relation between the energies and charges of these solutions. In contrast with the electrically charged JZ dyon of the Yang–Mills–Higgs (YMH) system, whose mass is larger than that of the electrically neutral (magnetic monopole) solutions, the masses of the electrically charged solutions of our HCS-YMH models can be smaller than their electrically neutral counterparts in some parts of the parameter space. To establish this is the main task of this work, which is performed by constructing the HCS-YMH solutions numerically. In the case of the SU(3) HCS-YMH, we have considered the question of angular momentum and it turns out that it vanishes.

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

In this ppaer, we study kink soliton configurations in interacting scalar field theories containing two fields without $SO(2)$ invariance. We study a class of such theories, the well-known Montonen–Sarker–Trullinger–Bishop model is one of them. These models are interesting since the $U(1)$ current is not conserved in them due to the presence of explicit symmetry breaking terms in the action. The existence of kink soliton configurations is shown in terms of a system of first-order ordinary differential equations. Although $U(1)$ current in these models is nonconserved, our approach is general enough to study soliton configurations in a generic two interacting scalar field theory. We also discuss other benefits of this approach.

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

A three-wave resonant interaction of collective modes and related soliton excitations in a disk-shaped Bose–Einstein condensate are investigated. The phase-matching conditions for the resonant interaction are satisfied by suitably choosing the wavevectors and the frequencies of the collective modes. A set of nonlinearly coupled envelope equations describing the spatio-temporal evolution of the three-wave resonant interaction are derived by using a method of multiple-scales, and some explicit (2+1)-dimensional three-wave soliton solutions are also presented and discussed.

We study nonclassic solitonic structures on the modified oscillator — chain proposed by Peyrard and Bishop to model DNA. The two DNA's strands are linked together by hydrogen bonds that are modeled by the Morse potential. This Peyrard–Bishop model with inharmonic potential in the optical part of the Hamiltonian gives rise to several nonclassical solutions, i.e., compact-cusp and anti-peak or crowd like soliton solutions. These structures could represent not only local openings of base pairs, but also the inverse process that heals the formation of broken hydrogen bonds.

For describing some nonlinear localized excitations, the (2+1)-dimensional dispersive long wave (DLW) system is investigated with symbolic computation in this paper. Based on two different dependent variable transformations obtained through the truncated Painlevé expansion, the (2+1)-dimensional DLW system can be bilinearized or linearized. Through the Hirota bilinear method, the analytic one-, two-, three-, and N-soliton solutions are derived. On the other hand, by means of the variable separation approach, localized excitations, such as the resonant dromion, resonant solitoff, lump and compacton excitations, are obtained. Figures are plotted to illustrate the structures of those solutions.

A broad set of sufficient conditions consisting of systems of linear partial differential equations are presented which guarantee that the Wronskian determinant is the solutions of the integrable variable-coefficient Korteweg-de Vries model from Bose–Einstein condensates and fluid dynamics. The generalized Wronskian solutions provide us with a comprehensive approach to construct many exact solutions including rational solutions, solitons, negatons, positons, and complexitons.

We discuss the analogy between soliton scattering in quantum field theory and black hole/wormholes (BH/WH) production in ultrarelativistic particle collisions in gravity. It is a common wisdom of the current paradigm suggests that BH/WH formation in particles collisions will happen when a center-mass energy of colliding particles is sufficiently above the Planck scale (the transplanckian region) and the BH/WH production can be estimated by the classical geometrical cross section. We compare the background of this paradigm with the functional integral method to scattering amplitudes and, in particular, we stress the analogy of the BH production in collision of ultrarelativistic particle and appearance of breathers poles in the scattering amplitudes in the Sin–Gordon model.

A review of models describing the interactions of ultra-cold atoms and laser light is given. Both semi-classical and fully quantum models are presented with particular attention given to the introduction of local field effects. Some possible effects of self-localization and guiding, consequences of such interactions, are discussed.

We present an analytical study on intrinsic localized modes (ILMs) in the quantum $\beta$-Fermi–Pasta–Ulam lattice model with first- and second-nearest neighbor interactions by means of the semiclassical approach. We quantize the lattice model Hamiltonian by introducing vibron creation and annihilation operators, and retaining only number conserving terms. The coherent state representation is considered as the basic representation of the quantum lattice system. In order to obtain the ILM solutions, we adopt the multiple scales method combined with a quasidiscreteness approximation. It is found that, when the system parameters satisfy $K_2>4K_2^{\prime }$, at the Brillouin zone (BZ) boundary, a bright ILM occurs above the top of the harmonic wave frequency band. While for $K_2<4K_2^{\prime }$, our results indicate that at wave number $k_c$ a bright ILM occurs above the top of the harmonic wave frequency band and at the BZ boundary, the system support a dark intrinsic localized resonant mode.

We study the linear stability and collision characteristics of the solitons of the Bose–Einstein condensates (BECs) trapped in two barrier potentials. It is shown that the soliton obtained at the top of the barrier potential is stable. Especially, when the two barrier potentials are considered as the output source of the dark solitons, the collision spots and pattern (such as chase, head-on or localization and through collision) of two solitons may be manipulated by the height of the barrier potential.

The existence and properties of quantum breathers in a one-dimensional XXZ ferromagnetic Heisenberg spin chain with single-ion easy-plane anisotropy are investigated analytically in the Hartree approximation. We show that the system can support the appearance of quantum breathers, and discuss their existence conditions and properties. In addition, our results show that, for quantum breathers in this system, both the corresponding energy and magnetic moment are quantized.