Soliton Stability in a Generalized Sine-Gordon Potential
aa r X i v : . [ h e p - t h ] S e p Soliton stability in a generalized sine-Gordon potential
Rub´en Cordero ∗ and Roberto D. Mota † Abstract
We study stability of a generalized sine-Gordon model with two coupled scalarfields in two dimensions. Topological soliton solutions are found from the first-orderequations that solve the equations of motion. The perturbation equations can be cast interms of a Schr¨odinger-like operators for fluctuations and their spectra are calculated.
PACS numbers: 0.350.-z, 11.10.EfKey words: topological defects, solitons, sine-Gordon, stability.
It is well known that in field theories when a discrete symmetry is broken domain walls arise.Domain walls have been observed in condensed matter, for example, in liquid crystals. Inthe cosmological context domain walls could appear in phase transitions in the early universeand have some important consequences (Vilenkin and Shellard, 1994).In the domain walls context, there exist classical static configurations with finite mini-mum localized energy, see for instance (Bogomol’nyi, 1976; Prasad, 1975). Several authorshave been interested in coupled scalar fields systems due to their important physical prop-erties. For example, Peter showed in (Peter, 1996) that surface current-carrying domainwall arises when a bosonic charge carrier is coupled to the Higgs field forming the wall. In(Bazeia, et al., 1997; Riazi, et al., 2001) was studied linear stability of soliton solutions for ∗ Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Ed. 9, Unidad ProfesionalAdolfo L´opez Mateos, 07738 M´exico D F, Mexico. [email protected]. † Unidad Profesional Interdisciplinaria de Ingenier´ıa y Tecnolog´ıas Avanzadas, IPN. Av. InstitutoPolit´ecnico Nacional 2580, Col. La Laguna Ticom´an, Delegaci´on Gustavo A. Madero, 07340 M´exico D.F. Mexico. [email protected] × In this paper we consider the generalization of the sine-Gordon model for two scalar inter-acting fields given by the following Lagrangian L = 12 ( ∂ µ Φ) + 12 ( ∂ µ X ) −
12 cos Φ(1 + α sin X ) −
12 cos X (1 + α sin Φ) , (1)where α is a dimensional parameter, and all other dimensional parameters are set equal tounity. For 0 ≤ Φ , X ≤ π , the potential in (1) has three minima at Φ = X = π/
2; Φ = π/ X = 3 π/ X = 3 π/
2, one maximum at Φ = 0, X = π/
2, and three saddle pointsat Φ = π/ X = π ; Φ = π/ X = 0 and Φ = 3 π/ X = π/ (cid:3) Φ + ∂∂ Φ V = 0 , (cid:3) X + ∂∂X V = 0 (2)which become for a static configurationsΦ ′′ = − cos Φ sin Φ(1 + α sin X ) + α cos X (1 + α sin X ) cos Φ (3) X ′′ = − cos X sin X (1 + α sin Φ) + α cos Φ(1 + α sin Φ) cos X, (4)where primes means derivatives with respect to space variable.The form of the energy of the system can be written as E s = Z ∞−∞ "(cid:18) d Φ dz − W Φ (cid:19) + (cid:18) dXdz − W X (cid:19) dz + (cid:12)(cid:12)(cid:12) Z ∞−∞ ∂∂z W [Φ( z ) , X ( z )] dz (cid:12)(cid:12)(cid:12) (5)where W [Φ( z ) , X ( z )] is the corresponding superpotential of (1), which turns out to be W = − sin Φ − sin X − α (sin Φ)(sin X ) . (6)In (Shifman and Voloshin, 1998) this superpotential was referred as a generalization of thesine-Gordon model. It is periodic in both Φ and X ; for α = 0 it describes two decoupled3elds, representing each of them a supergeneralization of the sine-Gordon model. If α = 0the fields Φ and X start interacting with each other. Inside the periodicity domain 0 ≤ Φ , X ≤ π , − W has one maximum at Φ = X = π/
2, one minimum at Φ = X = 3 π/ π/ X = 3 π/ π/ π/ α .The lower bound for the energy is achieved if Φ and X satisfyΦ ′ = − cos Φ(1 + α sin X ) X ′ = − cos X (1 + α sin Φ) . (7)For the case X = π/
2, we have d Φ dz = − cos Φ(1 + α ) , (8)whose solution is Φ = − tan − (cid:18) c e z (1+ α ) − e − z (1+ α ) (cid:19) . (9)Other possible solution of equations (7) is obtained for X = 3 π/ α by − α . Interchanging the fields X and Φ in thelast equations we get the solution for Φ = π/ π/ We are interested in determining the classical stability of this system under small fluctuationsaround a static configuration. In order to investigate the linear stability of the interactingfields we proceed in the usual way by considering small perturbations around the staticscalar fields Φ( z, t ) = Φ( z ) + η ( z, t ) (10) X ( z, t ) = X ( z ) + ξ ( z, t ) . (11)The stability equations can be written in a Schr¨odinger-like equation S l Ψ n = ω n Ψ n (12)4here n = 0 , , .. . The differential operator S l is given by S l = − d dz + ∂ ∂ Φ V ∂ ∂ Φ ∂X V ∂ ∂ Φ ∂X V − d dz + ∂ ∂X V | Φ=Φ( z ) ,X = X ( z ) ≡ − d dz I × + V P H (13)and the two components wave functions areΨ n = Φ n ( z ) X n ( z ) , (14)where we have expanded the fluctuations α ( z, t ) and β ( z, t ) in terms of normal modes η ( z, t ) = X n a n η n ( z ) e iω n t (15) ξ ( z, t ) = X n b n ξ n ( z ) e iω n t . (16)Notice that in the case when the differential operator S l is diagonal the perturbation fieldscould be expanded in terms of different frequencies.The SUSY QM approach to linear stability consists in realizing a 2 × V P H W + W ′ = V P H . (17)The existence of W that satisfies this equation ensures the existence of the first order self-adjoint differential operators D ± = ± I ddz + W ( z ) (18)that factorize the operator S l = D + D − . This fact implies the stability for equal fluctuationfrequencies, since 0 ≤ |D − Ψ n | = ( D − Ψ n ) † ( D − Ψ n ) = hD + D − i = h S l i = ω n .For our case the matrix elements of V P H are given by( V P H ) = − (cos Φ − sin Φ)(1 + α sin X ) + α cos X cos Φ (19) − α cos X (1 + α sin Φ) sin Φ( V P H ) = − (cos X − sin X )(1 + α sin Φ) + α cos Φ cos X − α cos Φ(1 + α sin X ) sin X ( V P H ) = ( V P H ) = − α cos Φ sin Φ(1 + α sin X ) cos X (20) − α cos X sin X (1 + α sin Φ) cos Φ , W min = (1 + α sin X ) sin Φ − α cos Φ cos X − α cos Φ cos X (1 + α sin Φ) sin X . (21)For the sector X = π/
2, the fluctuation potential term becomes V min = − (1 + α ) (cos Φ − sin Φ) 00 (1 + α sin Φ) − α cos Φ(1 + α ) , (22)so, the corresponding superpotential is W min = (1 + α ) sin Φ 00 (1 + α sin Φ) . (23)We point out the existence of another self-adjoint and non-negative second-order differ-ential operator S ′ l = D − D + which plays the role of the supersymmetric partner operator of S l in SUSY QM. The operators S l and S ′ l have the same energy spectrum except for theground state. The study of stability for the general case is very difficult. However, in order to haveanalytical results in the case of X = π/ π/ c = 1 in equation (9) i. e. tan Φ = sinh z (1 + α ). Since the differential operator S l is diagonal we could have differentfluctuation frequencies that can be determined from the perturbation equations − d η n dz − (1 + α ) (cid:0) z (1 + α ) − (cid:1) η n = ω n η n (24)and − d ξ n dz + (cid:0) α − α tanh z (1 + α ) − α (1 + 2 α )sech z (1 + α ) (cid:1) ξ n = ω n ξ n . (25)Performing the variable change y = z (1 + α ), equation (24) transforms to the Rosen-Morseequation (Morse and Feshbach, 1953). We find that the fluctuation frequencies are ω n = (1 + α ) (cid:0) − (1 − n ) (cid:1) . (26)6owever the bound states exist only for n < η = (1 + α )sech z (1 + α ) with eigenvalue ω = 0 is stable.By means of the same variable change the equation (25) can be cast as a Rosen-Morseequation whose eigenvalues are ω n = 1 + α − (1 + α ) "(cid:18) α + 12(1 + α ) − ( n + 1 / (cid:19) (27) − α (1 + α ) (cid:16) α +12(1+ α ) − ( n + 1 / (cid:17) which are the frequencies for possible bound states. However, for this case we have no boundstates because n must be less than zero for both α > α < X = π/
2, tan Φ = − sinh z (1 + α ) ( the same is true for Φ = π/
2, tan X = − sinh z (1 + α )). Conclusions
We have applied the SUSY QM formalism to study the linear stability of the Shifman gener-alization of the sine-Gordon model. We have shown that stability for soliton configurationsis ensured by solving the Riccati equation for the 2 × X = π/ π/ Acknowledgments
R D Mota would like to thank the Departamento de Matem´aticas del Centro de Investigaci´ony Estudios Avanzados del IPN where he was a visitor during the preparation of this work.7his work was partially supported by SNI-M´exico, CONACYT grant CO1-41639, COFAA-IPN, EDI-IPN, and CGPI project number 20030642.
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