The scattering of Dirac fermions on the sine-Gordon kink is studied both analytically and numerically. To achieve invariance with respect to a discrete symmetry, the sine-Gordon model is treated as a nonlinear $σ$-model with a circular target space that interacts with fermionic isodublets through the Yukawa interaction. It is shown that the diagonal and antidiagonal parts of the fermionic wave function interact independently with the external field of the sine-Gordon kink. The wave functions of the fermionic scattering states are expressed in terms of the Heun functions. General expressions for the transmission and reflection coefficients are derived, and their dependences on the fermion momentum and mass are studied numerically. The existence condition is found for two fermionic zero modes, and their analytical expressions are obtained. It is shown that the zero modes do not lead to fragmentation of the fermionic charge, but can lead to polarization of the fermionic vacuum. The scattering of the diagonal and antidiagonal fermionic states is found to be significantly different; this difference is shown to be due to the different dependences of the energy levels of these bound states on the fermion mass, and is in accordance with Levinson's theorem.
The Casimir energy of quantum fluctuations about the classical kink configuration is computed numerically for a recently proposed lattice sine-Gordon model. This energy depends periodically on the kink position and is found to be approximately sinusoidal.
We calculate the quantum vacuum interaction energy between two kinks of the sine-Gordon equation. Using the $TGTG$--formula, the problem is reduced to the known formulas for quantum fluctuations in the background of a single kink. This interaction induces an attractive force between the kinks in parallel to the Casimir force between conducting mirrors.
In this paper we study the dynamics of explicit solutions of $2+1$-dimensional ($2$D) sine-Gordon equation. The Darboux transformation is applied to the associated linear eigenvalue problem to construct nontrivial solutions of $2$D sine-Gordon equation in terms of ratios of determinants. We obtained a generalized expression for $N$-fold transformed dynamical variable which enables us to calculate explicit expressions of nontrivial solutions. In order to explore the dynamics of kink soliton solutions explicit expressions one- and two-soliton solutions are derived for particular column solutions…
We study kink-antikink collisions in a model which interpolates smoothly between the completely integrable sine-Gordon theory, the $ϕ^4$ model, and a $ϕ^6$-like model with three degenerate vacua. We find a rich variety of behaviours, including integrability breaking, resonance windows with increasingly irregular patterns, and new types of windows near the $ϕ^6$-like regime. False vacua, extra kink modes and kink fragmentation play important roles in the explanations of these phenomena. Our numerical studies are backed up by detailed analytical considerations.
We study the scattering properties of Sine Gordon kinks on obstructions in the form of finite size potential `wells'. We model this by making the coefficient of the $\cos(ϕ)-1$ term in the Lagrangian position dependent. We show that when the kinks find themselves in the well they radiate and then interact with this radiation. As a result of this energy loss the kinks become trapped for small velocities while at higher velocities they are transmitted with a loss of energy. However, the interaction with the radiation can produce `unexpected' reflections by the well. We present two simple models …
The solitons and kinks of the generalized $sl(3, \IC)$ sine-Gordon (GSG) model are explicitly obtained through the hybrid of the Hirota and dressing methods in which the {\sl tau} functions play an important role. The various properties are investigated, such as the potential vacuum structure, the soliton and kink solutions, and the soliton masses formulae. As a reduced submodel we obtain the double sine-Gordon model. Moreover, we provide the algebraic construction of the $sl(3, \IC)$ affine Toda model coupled to matter (Dirac spinor) (ATM) and through a gauge fixing procedure we obtain the cl…
We consider $λϕ^{4}$ kink and sine-Gordon soliton in the presence of a minimal length uncertainty proportional to the Planck length. The modified Hamiltonian contains an extra term proportional to $p^4$ and the generalized Schrödinger equation is expressed as a forth-order differential equation in quasiposition space. We obtain the modified energy spectrum for the discrete states and compare our results with 1-loop resummed and Hartree approximations for the quantum fluctuations. We finally find some lower bounds for the deformations parameter so that the effects of the minimal length have the…
The CGHS model of two-dimensional dilaton gravity coupled to a sine-Gordon matter field is considered. The theory is exactly solvable classically, and the solutions of a kink and two-kink type solitons are studied in connection with black hole formation.
We study numerically the kink-fermion interactions in a 1+1 dimensional toy model, which describes sine-Gordon kinks coupled to the massless Dirac fermions with backreaction. We show that the spectrum of fermionic modes strongly depends on the choice of the coupling, in particular, there are no localized modes for a minimal Yukawa coupling. We analyze the scattering of the fermionic packet by the kink. We demonstrate that the outcome of the collision dynamically depends on the phase of the incoming fermion packet, it results in alternating regimes of positive and negative acceleration of the k…
In this paper, we numerically study the scattering of kinks in the noncanonical sine-Gordon model using Fourier spectral methods. The model depends on two free parameters, which control the localized inner structure in the energy density and the characteristics of the scattering potential. It has been conjectured that the kink solutions in the noncanonical model possess inner structures in their energy density, and the presence of these yields bound states and resonance structures for some relative velocities between the kink and the antikink. In the numerical study, we observed that the class…
In this paper we develop a procedure to compute the one-loop quantum correction to the kink masses in generic (1+1)-dimensional one-component scalar field theoretical models. The procedure uses the generalized zeta function regularization method helped by the Gilkey-de Witt asymptotic expansion of the heat function via Mellin's transform. We find a formula for the one-loop kink mass shift that depends only on the part of the energy density with no field derivatives, evaluated by means of a symbolic software algorithm that automates the computation. The improved algorithm with respect to earlie…
We study kink-antikink collisions in a particular case of the double sine-Gordon model depending on only one parameter $r$. The scattering process of large kink-antikink shows the changing of the topological sector. For some parameter intervals we observed two connected effects: the production of up to five antikink-kink pairs and up to three solitary oscillations. The scattering process for small kink-antikink has several possibilities: the changing of the topological sector, one-bounce collision, two-bounce collision, or formation of a bion state. In particular, we observed for small values …
We study various solutions of the Sine Gordon model in (1+1) dimensions. We use the Hirota method to construct some of them and then show that the wobble, discussed in detail by Kälberman, is one of such solutions. We concentrate our attention on a kink and its bound states with one or two breathers. We study their stability and some aspects of their scattering properties on potential wells and on fixed boundary conditions.
We study collisions of two, three, and four kinks of the double sine-Gordon model. The initial conditions are taken in a special form in order to provide collision of all kinks in one point. We obtain dependences of the maximal energy densities on the model parameter. We also analyze the final states observed in these collisions.
Calculations of quantum corrections to soliton masses generally require both the vacuum sector and the soliton sector to be regularized. The finite part of the quantum correction depends on the assumed relation between these regulators when both are taken to infinity. Recently, in the case of quantum kinks, a manifestly finite prescription for the calculation of the quantum corrections has been proposed, which uses the kink creation operator to relate the two sectors. In this note, we test this new prescription by calculating the one-loop correction to the Sine-Gordon soliton mass, reproducing…
We studied the kink-antikink collision process for the "double sine-Gordon" (DSG) equation in 1+1 dimensions at different values of the potential parameter $R>0$. For small values of $R$ we discuss the problem of resonance frequencies. We give qualitative explanation of the frequency shift in comparison with the frequency of the discrete level in the potential well of isolated kink. We show that in this region of the parameter $R$ the effective long-range interaction between kink and antikink takes place.
The solitons and kinks of the SU(3) generalized sine-Gordon model (GSG) are shown to describe the baryonic spectrum of two-dimensional quantum chromodynamics (QCD$_{2}$). The GSG model arises in the low-energy effective action of bosonized QCD$_{2}$ with unequal quark mass parameters. The GSG potential for $N_{f}=3$ flavors resembles the potential of the effective chiral lagrangian proposed by Witten to describe low-energy behavior of four dimensional QCD. Among the attractive features of the GSG model are the variety of soliton and kink type solutions for QCD$_{2}$ unequal quark mass paramete…
We consider the sl(3,C) affine Toda model coupled to matter (Dirac spinor) (ATM) and through a gauge fixing procedure we obtain the classical version of the generalized sl(3,C) sine-Gordon model (cGSG) which completely decouples from the Dirac spinors. The GSG models are multifield extensions of the ordinary sine-Gordon model. In the spinor sector we are left with Dirac fields coupled to cGSG fields. Based on the equivalence between the U(1) vector and topological currents, which holds in the theory, it is shown the confinement of the spinors inside the solitons and kinks of the cGSG model pro…
We study bosonisation in the massive Thirring and sine-Gordon models at finite temperature and nonzero fermion chemical potential. Both canonical operator and path integral approaches are used to prove the equality of the partition functions of the two models at finite $T$ and zero chemical potential, as it has been recently shown. This enables the relationship between thermal normal ordering and path-integral renormalisation to be specified. Furthermore, we prove that thermal averages of zero-charge operators can also be identified. At nonzero chemical potential and temperature we show, in pe…
In this paper, we first use semi-classical methods to study quantum field theoretical aspects of the integrable noncommutative sine-Gordon model proposed in [hep-th/0406065]. In particular, we examine the fluctuations at quadratic order around the static kink solution using the background field method. We derive equations of motion for the fluctuations and argue that at O(theta^2) the spectrum of fluctuations remains essentially the same as that of the corresponding commutative theory. We compute the one-loop two-point functions of the sine-Gordon field and the additional scalar field present …
A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(ϕ)=p(1+\cos\frac{2ϕ}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity \textbf{22}, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin half fermions on these branes in the presence of two types of scalar-fermion couplings: $η\barΨϕΨ$ and $η\barΨ\sinϕΨ$. By presenting the mass-independent potentials in the corresponding Schrödinger equations, we obtain the lowest Kaluza--Klein (KK) modes and a continuous gapless spectr…
We study the collision of a kink and an antikink in the double sine-Gordon model with and without the excited vibrational mode. In the latter case, we find that there is a limited range of the parameters where the resonance windows exist, despite the existence of a vibrational mode. Still, when the vibrational mode is initially excited, its energy can turn into translational energy after the collision. This creates one-bounce as well as a rich structure of higher-bounce resonance windows that depend on the wobbling phase being in or out of phase at the collision and the wobbling amplitude bein…
We study the kink-antikink scattering within the double sine-Gordon model. In the numerical simulations we found a critical value $v_{cr}$ of the initial velocity $v_{in}$, which separates two different scenarios: at $v_{in}<v_{cr}$ the kinks capture each other and form a bound state, while at $v_{in}>v_{cr}$ the kinks pass through each other and escape to infinities. We obtain non-monotonous dependence of $v_{cr}$ on the model parameter $R$. Besides that, at some initial velocities below $v_{cr}$ we observe formation and interaction of the so-called oscillons (new phenomenon), as well as esca…
We study the scattering of kink and antikink of the double sine-Gordon model. There is a critical value of the initial velocity $v_{cr}$ of the colliding kinks, which separates different regimes of the collision. At $v_{in}>v_{cr}$ we observe kinks reflection, while at $v_{in}<v_{cr}$ their interaction is complicated with capture and escape windows. We obtain the dependence of $v_{cr}$ on the parameter of the model. This dependence possesses a series of local maxima, which has not been reported by other authors. At some initial velocities below the critical value we observe a new phenomenon --…
Sine-Gordon kinks are a much studied integrable system that possesses multi-soliton solutions. Recent studies on sine-Gordon kinks with space-dependent square-well-type potentials have revealed interesting dynamics of a single kink interacting with wells and barriers. In this paper, we study a class of smooth space-dependent potentials and discuss the dynamics of one kink in the presence of different wells. We also present values for the critical velocity for different types of barriers. Furthermore, we study two kinks interacting with various wells and describe interesting trajectories such a…
At one loop, we provide an explicit formula for the ground state of the one-soliton sector in the Sine-Gordon theory. The state is given in the basis of eigenstates of the field operator, or equivalently as a Schrodinger wave functional. The formula readily generalizes to other solitons in other models and as an example we also provide the ground state of the kink in the (1+1)-dimensional $ϕ^4$ double well.
Some properties of the non-commutative (NC) versions of the generalized sine-Gordon model (NCGSG) and its dual massive Thirring theory are studied. Our method relies on the NC extension of integrable models and the master lagrangian approach to deal with dual theories. The master lagrangian turns out to be the NC version of the so-called affine Toda model coupled to matter related to the group GL(n), in which the Toda field $g \subset GL(n), (n=2, 3)$. Moreover, as a reduction of GL(3) NCGSG one gets a NC version of the remarkable double sine-Gordon model.
We use a recently constructed linearized soliton sector perturbation theory to calculate the form factors relevant to the elastic scattering of ultrarelativistic mesons off of nonrelativistic kinks. Both localized kink wave packets and also delocalized momentum eigenstate kinks are considered. In the delocalized case, the leading term is just the classical kink solution, as was found by Goldstone and Jackiw. The leading delocalized quantum correction agrees with that found by Gervais, Jevicki and Sakita in the $ϕ^4$ model and Weisz in the Sine-Gordon model. In the case of localized kink wave p…
We construct the soliton solutions in the symmetric space sine-Gordon theories. The latter are a series of integrable field theories in 1+1-dimensions which are associated to a symmetric space F/G, and are related via the Pohlmeyer reduction to theories of strings moving on symmetric spaces. We show that the solitons are kinks that carry an internal moduli space that can be identified with a particular co-adjoint orbit of the unbroken subgroup H of G. Classically the solitons come in a continuous spectrum which encompasses the perturbative fluctuations of the theory as the kink charge becomes …
We consider non-topological, "bell-shaped" localized and regular solutions available in some 1+1 dimensional scalar field theories. Several properties of such solutions are studied, namely their stability and the occurence of fermion bound states in the background of a kink and a kink-anti-kink solutions of the sine-Gordon model.
We study kink-antikink scattering in the sine-Gordon model in the presence of interactions with an additional scalar field, $ψ$, that is in its quantum vacuum. In contrast to the classical scattering, now there is quantum radiation of $ψ$ quanta and the kink-antikink may form bound states that resemble breathers of the sine-Gordon model. We quantify the rate of radiation and map the parameters for which bound states are formed. Even these bound states radiate and decay, and eventually there is a transition into long-lived oscillons.
We construct the restricted sine-Gordon theory by truncating the sine-Gordon multi-soliton Hilbert space for the repulsive coupling constant due to the quantum group symmetry $SL_q(2)$ which we identify from the Korepin's $S$-matrices. We connect this restricted sine-Gordon theory with the minimal ($c<1$) conformal field theory ${\cal M}_{p/p+2}$ ($p$ odd) perturbed by the least relevent primary field $Φ_{1,3}$. The exact $S$-matrices are derived for the particle spectrum of a kink and neutral particles. As a consistency check, we compute the central charge of the restricted theory in the UV l…
We have studied the space-reflection symmetries of some soliton solutions of deformed sine-Gordon models in the context of the quasi-integrability concept. Considering a dual pair of anomalous Lax representations of the deformed model we compute analytically and numerically an infinite number of alternating conserved and asymptotically conserved charges through a modification of the usual techniques of integrable field theories. The charges associated to two-solitons with a definite parity under space-reflection symmetry, i.e. kink-kink (odd parity) and kink-antikink (even parity) scatterings …
In this paper we construct a one-parametric family of (1+1)-dimensional one-component scalar field theory models supporting kinks. Inspired by the sine-Gordon and $ϕ^4$ models, we look at all possible extensions such that the kink second-order fluctuation operators are Schrödinger differential operators with Pöschl-Teller potential wells. In this situation, the associated spectral problem is solvable and therefore we shall succeed in analyzing the kink stability completely and in computing the one-loop quantum correction to the kink mass exactly. When the parameter is a natural number, the fam…
We study a discrete Darboux transformation and construct the multi-soliton solutions in terms of ratio of determinants for integrable discrete sine-Gordon equation. We also calculate explicit expressions of single, double, triple, quad soliton solutions as well as single and double breather solutions of discrete sine-Gordon equation. Dynamical features of discrete kinks and breathers have also been illustrated.
The interaction of a kink and a monochromatic plane wave in one dimensional scalar field theories is studied. It is shown that in a large class of models the radiation pressure exerted on the kink is negative, i.e. the kink is {\sl pulled} towards the source of the radiation. This effect has been observed by numerical simulations in the $ϕ^4$ model, and it is explained by a perturbative calculation assuming that the amplitude of the incoming wave is small. Quite importantly the effect is shown to be robust against small perturbations of the $ϕ^4$ model. In the sine-Gordon (sG) model the time a…
We derive analytic expressions of the semiclassical energy levels of Sine-Gordon model in a strip geometry with Dirichlet boundary condition at both edges. They are obtained by initially selecting the classical backgrounds relative to the vacuum or to the kink sectors, and then solving the Schodinger equations (of Lame' type) associated to the stability condition. Explicit formulas are presented for the classical solutions of both the vacuum and kink states and for the energy levels at arbitrary values of the size of the system. Their ultraviolet and infrared limits are also discussed.
Sine-Gordon deformed defects that exhibit unusual phenomenological features on the topological charge are investigated. The possibility of a smooth and continuous transition between topological (non null charge) and non-topological (null charge) scenarios of deformed defects supported by sine-Gordon structures is evinced by the analytical calculation of topological charges and localized energy distributions. By describing cyclic deformation chains, we show that a triggering sine-Gordon model simultaneously supports kink and lump-like defects, whose topological mass values are closed by trigono…
We consider SUSY sine-Gordon theory in the framework of perturbed conformal field theory. Using an argument from Zamolodchikov, we obtain the vacuum structure and the kink adjacency diagram of the theory, which is cross-checked against the exact S matrix prediction, first-order perturbed conformal field theory (PCFT), the NLIE method and truncated conformal space approach. We provide evidence for consistency between the usual Lagrangian description and PCFT on the one hand, and between PCFT, NLIE and a massgap formula conjectured by Baseilhac and Fateev, on the other. In addition, we extend th…
In this work we deform the phi^4 model with distinct deformation functions, to propose a diversity of sine-Gordon-like models. We investigate the proposed models and we obtain all the topological solutions they engender. In particular, we introduce non-polynomial potentials which support some exotic two-kink solutions.
We study scattering properties of topological solitons in two classes of models, which are generalizations of the Sine-Gordon model and which have recently been proposed by Bazeia et al. These two classes of models depend on an integer parameter n which, when n=2(for the first class) and n=1 (for the second class), reduce to the Sine-Gordon model. We take the soliton solutions of these models (generalizations of the 'kink' solution of the Sine-Gordon model) and consider their scattering on potential holes and barriers. We present our results for n=1,...6. We find that, like in the Sine Gordon …
We study a family of classical string solutions with large spins on R x S^3 subspace of AdS_5 x S^5 background, which are related to Complex sine-Gordon solitons via Pohlmeyer's reduction. The equations of motion for the classical strings are cast into Lame equations and Complex sine-Gordon equations. We solve them under periodic boundary conditions, and obtain analytic profiles for the closed strings. They interpolate two kinds of known rigid configurations with two spins: on one hand, they reduce to folded or circular spinning/rotating strings in the limit where a soliton velocity goes to ze…
Moduli spaces - finite-dimensional, collective coordinate manifolds - for kinks and antikinks in $ϕ^4$ theory and sine-Gordon theory are reconsidered. The field theory Lagrangian restricted to moduli space defines a reduced Lagrangian, combining a potential with a kinetic term that can be interpreted as a Riemannian metric on moduli space. Moduli spaces should be metrically complete, or have an infinite potential on their boundary. Examples are constructed for both kink-antikink and kink-antikink-kink configurations. The naive position coordinates of the kinks and antikinks sometimes need to b…
Nonperturbative, oscillatory, winding number one solutions of the Sine-Gordon equation are presented and studied numerically. We call these nonperturbative shape modes {\sl wobble} solitons. Perturbed Sine-Gordon kinks are found to decay to {\sl wobble} solitons.
