Narrow Results

A novel approach to the analysis of a noncommutative Chern–Simons gauge theory with matter coupled in the adjoint representation has been discussed. The analysis is based on a recently proposed closed form Seiberg–Witten map which is exact in the noncommutative parameter.

We review the recent experimental and theoretical advances in the generation of matter wave solitons in Bose–Einstein condensates. In particular, the controlled generation and dynamics of stable bright solitons by mean of Feshbach resonance techniques is discussed in details. Several aspects are taking into account, including the variation of the scattering length due to Feshbach resonance, the safe parameters against the collapse and the experimental implications of our scenario.

The Maxwell–Chern–Simons model with scalar matter in the adjoint representation is analyzed from an alternative approach which is regular in the θ→0 limit. This method is complementary to the usual operator formalism applied to explore the nonperturbative solutions which give singular results in the θ→0 limit. The absence of any regular non-trivial lumpy solutions satisfying B–P–S bound has been conclusively demonstrated.

One possible solution of the cosmological constant problem involves a so-called q-field, which self-adjusts so as to give a vanishing gravitating vacuum energy density (cosmological constant) in equilibrium. We show that this q-field can manifest itself in other ways. Specifically, we establish a propagating mode (q-wave) in the nontrivial vacuum and find a particular soliton-type solution in flat spacetime, which we call a q-ball by analogy with the well-known Q-ball solution. Both q-waves and q-balls are expected to play a role for the equilibration of the q-field in the very early universe.

Non-Abelian strings are considered in non-supersymmetric theories with fermions in various appropriate representations of the gauge group U$(N)$. We derive the electric charge quantization conditions and the index theorems counting fermion zero modes in the string background both for the left-handed and right-handed fermions. In both cases we observe a non-trivial $N$ dependence.

The multi-component noncommutative coupled dispersionless (NC-CD) system is presented. It has been shown that multi-component NC-CD system is integrable in the sense of exhibiting its Lax pair, zero-curvature representation, Darboux transformation and multisoliton solutions. Explicit expressions of multisoliton solutions of this noncommutative system have been computed and results have been compared with their commutative counterparts.

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

We develop an alternative derivation of the renormalized expression for the one-loop soliton quantum mass corrections in (1 + 1)-dimensional scalar field theories. We regularize implicitly such quantity by subtracting and adding its corresponding tadpole graph contribution and use the renormalization prescription that such added term vanishes with adequate counterterms. As a result, we get a finite unambiguous formula for the soliton quantum mass corrections up to one-loop order, which turns to be independent of the chosen regularization scheme.

We consider a ϕ⁴ model in 1+1 dimensions modified by the addition of an extra kinetic term similar to the Skyrme term in higher dimensions and a potential term. We generalize the procedures of the leading order method to include elliptic functions and obtain cnoidal wave type solutions of the familiar ϕ⁴ model. As a limit case, we re-obtain the well-known solitary and travelling wave solutions of this model from the constructed cnoidal wave solutions. A further kink solution of the pure ϕ⁴ model is also obtained. We obtain soliton solutions of the modified ϕ⁴ model and show that it possesses two ϕ⁴ kink solutions, with the same velocity and total energy, as well as a double-kink solution. We study some properties of the modified model. In particular, we see that the two ϕ⁴ kinks have a definite velocity and this velocity is a critical velocity for the double-kink structures. Finally, we conclude the paper with some features and comments.

Recently we have introduced a matrix model depending on two coupling constants g² and λ, which contains the fuzzy sphere as a background; to obtain the classical limit g² must depend on N in a precise way. In this paper we show how to obtain the classical solitons of the N → ∞ limit imposing the development ; as a consequence at finite N one obtains a noncommutative version of the solitons for the fuzzy sphere.

We have examined the deformation of a generic non-Abelian gauge theory obtained by replacing its Lie group by the corresponding quantum group. This deformed gauge theory has more degrees of freedom than the theory from which it is derived. By going over from point particles in the standard theory to solitonic particles in the deformed theory, it is proposed that we interpret the new degrees of freedom as being descriptive of the non-locality of the deformed theory. It also turns out that the original Lie algebra gets replaced by two dual algebras, one of which lies close to and approaches the original Lie algebra in a correspondence limit, while the second algebra is new and disappears in this same correspondence limit. The exotic field particles associated with the second algebra can be interpreted as quark-like constituents of the solitons, which are themselves described as point particles in the first algebra. These ideas are explored for q-deformed SU(2) and GL_q(3).

The behavior of solitons in integrable theories is strongly constrained by the integrability of the theory, that is by the existence of an infinite number of conserved quantities that these theories are known to possess. As a result, the soliton scattering of such theories is expected to be trivial (with no change of direction, velocity or shape). In this paper we present an extended review on soliton scattering of two spatial dimensional integrable systems which have been derived as dimensional reductions of the self-dual Yang–Mills equations and whose scattering properties are highly nontrivial.

We review some generalizations of 't Hooft and Mandelstam ideas on confinement for theories with non-Abelian unbroken gauge groups.

The Seiberg–Witten limit of fermionic N = 2 string theory with nonvanishing B-field is governed by noncommutative self-dual Yang–Mills theory (ncSDYM) in 2+2 dimensions. Conversely, the self-duality equations are contained in the equation of motion of N = 2 string field theory in a B-field background. Therefore finding solutions to noncommutative self-dual Yang–Mills theory on ℝ^2,2 might help to improve our understanding of nonperturbative properties of string (field) theory. In this paper, we construct nonlinear soliton-like and multi-plane wave solutions of the ncSDYM equations corresponding to certain D-brane configurations by employing a solution generating technique, an extension of the so-called dressing approach. The underlying Lax pair is discussed in two different gauges, the unitary and the Hermitian gauge. Several examples and applications for both situations are considered, including Abelian solutions constructed from GMS-like projectors, noncommutative U(2) soliton-like configurations and interacting plane waves. We display a correspondence to earlier work on string field theory and argue that the solutions found here can serve as a guideline in the search for nonperturbative solutions of nonpolynomial string field theory.

We show that the least energy conditions in the gauged nonlinear sigma model with Chern–Simons term lead to exact soliton-like solutions which look like domain walls. In fact, they are string-like solutions on the 2D plane. We will derive and discuss the corresponding solutions, and compute their energy and charge per unit length. The spin of the solutions is shown to vanish.

The equations of motion for a theory described by a Chern–Simons type of action in two dimensions are obtained and investigated. The equation for the classical, continuous Heisenberg model is used as a form of gauge constraint to obtain a result which provides a completely integrable dynamics and which partially fixes the gauge degrees of freedom. Under a particular form of the spin connection, an integrable equation which can be analytically extended to a form of the nonlinear Schrödinger equation is obtained. Some explicit solutions are presented, and in particular a soliton solution is shown to lead to an integrable two-dimensional model of gravity.

Hydrodynamics is the appropriate "effective theory" for describing any fluid medium at sufficiently long length scales. This paper treats the vacuum as such a medium and derives the corresponding hydrodynamic equations. Unlike a normal medium the vacuum has no linear sound-wave regime; disturbances always "propagate" nonlinearly. For an "empty vacuum" the hydrodynamic equations are familiar ones (shallow water-wave equations) and they describe an experimentally observed phenomenon — the spreading of a clump of zero-temperature atoms into empty space. The "Higgs vacuum" case is much stranger; pressure and energy density, and hence time and space, exchange roles. The speed of sound is formally infinite, rather than zero as in the empty vacuum. Higher-derivative corrections to the vacuum hydrodynamic equations are also considered. In the empty-vacuum case the corrections are of quantum origin and the post-hydrodynamic description corresponds to the Gross–Pitaevskii equation. We conjecture the form of the post-hydrodynamic corrections in the Higgs case. In the (1+1)-dimensional case the equations possess remarkable "soliton" solutions and appear to constitute a new exactly integrable system.

We formulate the nonlinear isovector model in a curved background and calculate the spherically symmetric solutions for weak and strong coupling regimes. The question whether gravity has appreciable effects on the structure of solitons will be examined, in the framework of the calculated solutions, by comparing the flat-space and curved-space solutions. It turns out that in the strong coupling regime, gravity has essential effects on the solutions. It is also shown that the asymptotic form of the metric conforms with that of the charged Reissner–Nordstrom metric. The dimensionless coupling constant of the model has a limit, beyond which a horizon appears in the solutions, indicating the presence of black hole solutions.

Networks or webs of domain walls are admitted in Abelian or non-Abelian gauge theory coupled to fundamental Higgs fields with complex masses. We examine the dynamics of the domain wall loops by using the moduli space approximation. This talk is based on works in collaboration with M.Eto, T.Fujimori, M.Nitta, K.Ohashi, and N.Sakai^1,2.