The concept of quasi-integrability: a concrete example
aa r X i v : . [ h e p - t h ] N ov The concept of quasi-integrability: a concrete example
L. A. Ferreira ⋆ , and Wojtek J. Zakrzewski † ( ⋆ ) Instituto de F´ısica de S˜ao Carlos; IFSC/USP;Universidade de S˜ao PauloCaixa Postal 369, CEP 13560-970, S˜ao Carlos-SP, Brazil ( † ) Department of Mathematical Sciences,University of Durham, Durham DH1 3LE, U.K.
Abstract
We use the deformed sine-Gordon models recently presented by Bazeia et al [1] todiscuss possible definitions of quasi-integrability. We present one such definition and useit to calculate an infinite number of quasi-conserved quantities through a modificationof the usual techniques of integrable field theories. Performing an expansion aroundthe sine-Gordon theory we are able to evaluate the charges and the anomalies of theirconservation laws in a perturbative power series in a small parameter which describesthe “closeness” to the integrable sine-Gordon model. Our results indicate that in thecase of the two-soliton scattering the charges are conserved asymptotically, i.e. theirvalues are the same in the distant past and future, when the solitons are well separated.We back up our results with numerical simulations which also demonstrate the exis-tence of long lived breather-like and wobble-like states in these models.
Introduction
Solitons and integrable field theories play a central role in the study of many non-linear phe-nomena. Indeed, it is perhaps correct to say that many non-perturbative and exact methodsknown in field theories are in one way or the other related to solitons. The reason for that istwofold. On one hand, the appearance of solitons in a given theory is often related to a highdegree of symmetries and so to the existence of a large number of conservation laws. On theother hand, in a large class of theories the solitons possess a striking property. They becomeweakly coupled when the interaction among the fundamental particles of the theory is strong,and vice-versa. Therefore, the solitons are the natural candidates to describe the relevantnormal modes in the strong coupling (non-perturbative) regime of the theory. Such relationbetween the strong and weak coupling regimes have been observed in some (1+1) dimensionalfield theories, as, for example, in the equivalence of the sine-Gordon and Thirring models [2],as well as in four dimensional supersymmetric gauge theories where monopoles (solitons) andfundamentals gauge particles exchange roles in the so-called duality transformations [3].The exact methods to study solitons in (1 + 1)-dimensional field theories involve manyalgebraic and geometrical concepts, but the most important ingredient is the so-called zerocurvature condition or the Lax-Zakharov-Shabat equation [4]. All theories known to possessexact soliton solutions admit a representation of their equations of motion as a zero curvaturecondition for a connection living in an infinite dimensional Lie (Kac-Moody) algebra [5, 6].In fact, in (1 + 1) dimensions such zero curvature condition is a conservation law, and theconserved quantities are given by the eigenvalues of the holonomy of the flat connectioncalculated on a spatial (fixed time) curve. On the other hand, many techniques, like thedressing transformation method, for the construction of exact solutions are based on thezero curvature condition. In dimensions higher than two the soliton theory is not so welldeveloped, even though many exact results are known for some 2-dimensional theories suchas the CP N models [7], as well as in four dimensional gauge theories where instanton andself-dual monopoles are the best examples [8]. Some approaches have been proposed for thestudy of integrable field theories in higher dimensions based on generalizations of the twodimensional methods like the tetrahedron equations [9] and of the concept of zero curvatureinvolving connections in loop spaces [10].Another important aspect of integrable field theories is that they serve as good appro-ximations to many physical phenomena. In fact, there is a vast literature exploring manyaspects and applications of perturbations around integrable models. In this paper we wantto put forward a technique that, so far as we know, has not been explored yet and whichsuggests that some non-integrable theories often possess many important properties of fullyintegrable ones. We put forward and develop the concept of quasi-integrability for theories1 f i e l d x ’Kink0.1’’Kink1.0’’Kink1.5’’Kink2.0’’Kink2.5’’Kink7.0’’Kink12.0’ Figure 1:
Plots of solitons for various values on n that do not admit a representation of their equations of motion in terms of the Lax-Zakharov-Shabat equation, but which can, nevertherless, be associated with an almost flat connectionin an infinite dimensional Lie algebra. In other words, we have an anomalous zero curvaturecondition that leads to an infinite number of quasi-conservation (almost conservation) laws.Moreover, in practice, in physical situations, like the scattering of solitons, these chargesare effectively conserved. The striking property we have discovered is that as the scatteringprocess takes place the charges do vary in time. However, after the solitons have separatedfrom each other the charges return to the values they had prior to the scattering. Effectivelywhat we have is the asymptotic conservation of an infinite number of non-trivial charges.There are still several aspects of this observation that have to be better understood but webelieve that if our results are indeed robust then such asymptotic charges could play a role inmany important properties of the theory like the factorization of the S-matrix.We introduce our concept of quasi-integrability through a concrete example involving areal scalar field theory in (1 + 1) dimensions which is a special deformation of the sine-Gordonmodel. The scalar field ϕ of our theory is subjected to the potential V ( ϕ, n ) = 2 n tan ϕ [1 − | sin ϕ | n ] (1.1)where n is a real parameter which in the case n = 2 reduces the potential to that of thesine-Gordon model, i.e. V ( ϕ,
2) = [1 − cos (4 ϕ )].This potential (1.1) is a slight modification of that introduced by D. Bazeia et al, [1], inthe sense that we take the absolute value of sin ϕ to allow n to take real and not only integervalues. 2 - - Figure 2:
Plots of the potential (1.1) against ϕ , for three values of the parameter ε ,where n = 2 + ε , namely ε = − . ε = 0 . ε = 0 . The potential (1.1) has an infinite number of degenerate vacua that allow the existenceof solutions with non-trivial topological charges. It is worth noticing that the positions ofthe vacua are independent of n and so they are the same as in the sine-Gordon model, i.e. ϕ vac . = m π , with m being any integer.The model with the potential (1.1) is fully topological ( i.e. it satisfies its Bogomolnyibound for any n ) and so its one soliton field configurations are known in an explicit form.They are given by: ϕ = arcsin " e e /n , Γ = ± ( x − v t − x ) √ − v , (1.2)where the velocity v is given in units of the speed of light, and the signs correspond to thekink (+), and anti-kink ( − ), with topological charges +1, and − n . We see from this plotthat the n = 2 case does not appear to be very special; all soliton fields look very similar andthe solitons for different values of n differ only in their slopes.We are going to use this model to back up our discussion of quasi-integrability and so nextwe look at n = 2+ ε , with ε small. In Figure 2 we plot the potential (1.1) for ε = − .
5; 0 .
0; 0 . n replaced by 2 + ε . InFigure 3 we plot the one kink solutions (1.2) for the potentials shown in Figure 2, i.e. for ε = − .
5; 0 .
0; 0 .
5. Note that they connect the vacua ϕ vac = 0 to ϕ vac . = π , as x goes from −∞ to + ∞ , with the slope of the kink increasing as the value of ε decreases.In this paper we study the concept of quasi-integrability in the context of the theory (1.1)3 - Figure 3:
Plots of the kink solutions (1.2) against x , with t = 0 and x = 0, for threevalues of the parameter ε , where n = 2 + ε , namely ε = − . ε = 0 . ε = 0 . from the analytical and numerical points of view . Our approach and the main results of thispaper can be summarised as follows (more details are given in the following sections):We first consider a real scalar field theory with a very general potential V ( ϕ ), and constructa connection A µ based on the sl (2) loop algebra which, as a consequence of its equations ofmotion, satisfies an anomalous zero curvature condition. Using a modification of the methodsemployed in integrable field theories we construct and infinite number of quasi-conservedcharges for such a theory, i.e. d Q (2 n +1) d t = − α (2 n +1) ( t ) n = 0 , ± , ± , . . . , (1.3)where the anomalies α (2 n +1) ( t ) are non-zero due to the non-flatness of the connection A µ .The charges Q ( ± are in fact conserved, i.e. α ( ± = 0, and linear combinations of themcorrespond to the energy and momentum.We then restrict ourselves to the case of the potential (1.1) and set up a perturbativeexpansion around the sine-Gordon theory. We expand all the quantities, equations of motion,field ϕ , charges and anomalies, in powers of the parameter ε , related to n appearing in (1.1)by n = 2 + ε . For instance, we have Q (2 n +1) = Q (2 n +1)0 + ε Q (2 n +1)1 + O (cid:16) ε (cid:17) , α (2 n +1) = ε α (2 n +1)1 + O (cid:16) ε (cid:17) . (1.4)The anomalies vanish in the lowest order (order zero) in ε because they correspond to the Preliminary results of our approach have already been given in [11] Q (2 n +1)0 are conserved during thedynamics of all field configurations.In this paper we concentrate our attention on the evaluation of the first non-trivial chargeand its anomaly, namely Q (3) and α (3) , but our calculations can easily be extended to the othercharges. We considered the case of the scattering of two kinks and also of a kink/anti-kink inthe theory (1.1), where the solitons are far apart in the distant past and future, and collidewhen t ∼
0. We found that the first order anomaly α (3)1 , vanishes when integrated over thewhole time axis. Therefore, from (1.3) we see that Q (3)1 ( t = + ∞ ) = Q (3)1 ( t = −∞ ) . (1.5)Consequently, the scattering of the solitons happen in a way that, to first order in ε at least,the charge is asymptotically conserved. That is a very important result, and if one can extendit to higher orders and higher charges one would prove that effectively the scattering of solitonsin the theory (1.1) takes place in the sdame way as if the theory were a truly integrable theory.We have also analyzed the first order charge Q (3)1 for the breather solution of the theory (1.1),and found that even though the charge is not conserved, it oscillates around a fixed value. Inother words, the first order anomaly vanishes when integrated over a period πν , and so from(1.3) we find that Q (3)1 ( t ) = Q (3)1 (cid:18) t + πν (cid:19) , (1.6)where ν is the angular frequency of the breather. That means that the period of the chargeis half of that of the breather.We have performed many numerical simulations of the full theory (1.1) using a fourthorder Runge-Kuta method, and using various lattice grids to make sure that the results arenot contaminated by numerical artifacts. We have found reliable results with lattice grids ofat least 3001 points, where the kinks were of size ∼ i.e. well within this reliability). The mainresults we have found are the following: We have found that if | ε | does not get very closeto unity the kinks and the kink/anti-kink scatter without destroying themselves and preservetheir original shapes, given in (1.2). For small values of | ε | the anomaly α (3) integrates to zerofor large values of the time interval, and so the charge Q (3) is asymptotically conserved withinour numerical errors. This is an important confirmation of our analytical result describedabove, and is valid for the full charge and not only for its first order approximation as in (1.5).One of the important discoveries of our numerical simulations is that the theory (1.1) alsopossesses very long lived breather solutions for ε = 0, which correspond to non-integrablemodels. These long-lived breathers were obtained by starting the simulations with a fieldconfiguration corresponding to a kink and an anti-kink. As they get close to each other5hey interact and readjust their profiles and some radiation is emitted in this process. Weabsorbed this radiation at the boundaries of the grid and the system stabilized to a breather-like configuration. For n = 2 the resultant field configuration was the exact (and analyticallyknown) breather while for ε small those breather-like fields lived for millions of units of time.As one changed ε and made it come close to unity the quasi-breathers radiated more and foreven larger values they eventually died. We also looked at the anomalies for such breather-likeconfigurations and have found a good agreement with our analytical results described above.The anomaly, integrated in time, does oscillate and for small values of ε the charge is periodicin time. That is, again, in agreement with the analytical result (1.6). Notice however, thatthe numerical result is stronger in the sense that it corresponds to the full charge and not onlyto its first order approximation as in (1.6).We have also performed similar numerical simulations of wobbles [12] which correspond toconfigurations of a breather and a kink. Again, such configurations were obtained by startingthe simulation with two kinks and an anti-kink. As the three solitons interact and adjust theirprofiles they radiate energy and this radiation had been absorbed at the boundaries of thegrid. Eventually the system has evolved to a breather and a kink and for small values of ε theresultant configuration was quite stable, and for n = 2 it agreed with the analytically knownconfiguration of a wobble. Again, we believe that this is a very interesting result which showsthat non-integrable theories can support such kinds of solutions.Our results open up the way to investigate large classes of models which are not reallyexactly integrable but which possess properties which are very similar to those of integrablefield theories. We believe that they will have applications in many non-linear phenomena ofphysical interest.The paper is organized as follow: in section 2 we introduce the quasi-zero curvature con-dition, based on the sl (2) loop algebra, for a real scalar field theory subject to a genericpotential, and construct an infinite number of quasi-conserved quantities. In section 3 weperform the expansion of the theory (1.1) around the sine-Gordon model, and evaluate thefirst non-trivial charge and its corresponding anomaly. The numerical simulations, involvingthe two solitons scattering, breathers and a wobble, are presented in section 4. In section5 we present our conclusions; the details of the sl (2) loop algebra, charge calculations and ε -expansion are presented in the appendices. 6 The quasi zero curvature condition
We shall consider Lorentz invariant field theories in (1 + 1)-dimensions with a real scalar field ϕ and equation of motion given by ∂ ϕ + ∂ V ( ϕ ) ∂ ϕ = 0 , (2.1)where V ( ϕ ) is the scalar potential. Thus we want to study the integrability properties of suchtheory using the techniques of integrable field theories [4, 5, 6]. We then start by trying toset up a zero curvature representation of the equations of motion (2.1), and so we introducethe Lax potentials as A + = 12 "(cid:16) ω V − m (cid:17) b − i ω d Vd ϕ F ,A − = 12 b − − i ω ∂ − ϕ F . (2.2)Our Lax potentials live on the so-called sl (2) loop algebra with generators b n +1 and F n , with n integer; their commutation relations are given in Appendix A. The parameters ω and m are constants, and they play a special role in our analysis. Note, that the dynamics governedby (2.1) does not depend upon them, but since they appear in (2.2) they will play a rolein the quasi-conserved quantities that we will construct through the Lax equations. In theexpression above we have used light cone coordinates x ± = ( t ± x ), where ∂ ± = ∂ t ± ∂ x , and ∂ + ∂ − = ∂ t − ∂ x ≡ ∂ .The curvature of the connection (2.2) is given by F + − ≡ ∂ + A − − ∂ − A + + [ A + , A − ] = X F − i ω " ∂ ϕ + ∂ V∂ ϕ F (2.3)with X = i ω ∂ − ϕ " d Vd ϕ + ω V − m . (2.4)As in the case of the sine-Gordon model where the potential is given by V SG = 116 [1 − cos (4 ϕ )] (2.5)we find that X , given by (2.4), vanishes when we take ω = 4 and m = 1. Then the curvature(2.3) vanishes when the equations of motion (2.1) hold. The vanishing of the curvature allowsus to use several powerful techniques to construct conserved charges and exact solutions. Wewant to analyze what can be said about the conservation laws for potentials when X does notvanish but can be considered small. 7n general, the conserved charges can be constructed using the fact that the path orderedintegral of the connection along a curve Γ, namely P exp hR Γ dσ A µ d x µ d σ i , is path independentwhen the connection is flat [5, 6, 13]. Here, we will use a more refined version of this techniqueand try to gauge transform the connection into the abelian subalgebra generated by the b n +1 .We follow the usual procedures of integrable field theories discussed for instance in [14, 15, 16].An important ingredient of the method is that our sl (2) loop algebra G is graded, with n beingthe grades determined by the grading operator d = T + 2 λ ddλ (see appendix A for details) G = X n G n ; [ G m , G n ] ⊂ G m + n ; [ d , G n ] = n G n . (2.6)We perform a gauge transformation A µ → a µ = g A µ g − − ∂ µ g g − (2.7)with the group element g being an exponentiation of generators lying in the positive gradesubspace generated by the F n ’s, i.e. , g = exp " ∞ X n =1 ζ n F n (2.8)with ζ n being parameters to be determined as we will explain below. Under (2.7) the curvature(2.3) is transformed as F + − → g F + − g − = ∂ + a − − ∂ − a + + [ a + , a − ] = X g F g − , (2.9)where we have used the equations of motion (2.1) to drop the term proportional to F in(2.3). The component A − of the connection (2.2) has terms with grade 0 and −
1. Therefore,under (2.7) it is transformed into a − which has terms with grades ranging from − ∞ .Decomposing a − into grades we get from (2.7) and (2.8) that a − = 12 b − (2.10) − ζ [ b − , F ] − i ω ∂ − ϕ F − ζ [ b − , F ] + 14 ζ [ [ b − , F ] , F ] − i ω ∂ − ϕ ζ [ F , F ] − ∂ − ζ F ... − ζ n [ b − , F n ] + . . . . Next we note that one can choose the parameters ζ n recursively by requiring that thecomponent in the direction on F n − cancels out in a − . Thus we can put that ζ = i ω ∂ − ϕ ,8nd so on. In the appendix B we give the first few ζ n ’s obtained that way. In consequence,the component a − is rotated into the abelian subalgebra generated by the b n +1 . Note thatthis procedure has not used the equations of motion (2.1). We then have a − = 12 b − + ∞ X n =0 a (2 n +1) − b n +1 (2.11)and the first three components are a (1) − = − ω ( ∂ − ϕ ) , (2.12) a (3) − = − ω ( ∂ − ϕ ) − ω ∂ − ϕ∂ − ϕ,a (5) − = − ω ( ∂ − ϕ ) − ω ∂ − ϕ ( ∂ − ϕ ) − ω ( ∂ − ϕ ) ( ∂ − ϕ ) − ω ∂ − ϕ∂ − ϕ. With the ζ n ’s determined this way we perform the transformation of the A + component ofthe connection (2.2). Since the ζ n ’s are polynomials of x − -derivatives of ϕ (see appendix B)and since there will be terms involving x + -derivatives of ζ n ’s, we use the equations of motionto eliminate terms involving ∂ + ∂ − ϕ . Due to the nonvanishing of the anomaly term X in (2.3)we are not able to transform a + into the abelian subalgebra generated by the b n +1 . We findthat a + is of the form a + = ∞ X n =0 a (2 n +1)+ b n +1 + ∞ X n =2 c ( n )+ F n (2.13)where a (1)+ = 12 h ω V − m i , (2.14) a (3)+ = 14 ω ∂ − ϕ d Vd ϕ − iω∂ − ϕX,a (5)+ = − iω ( ∂ − ϕ ) X + 516 ω ∂ − ϕ ( ∂ − ϕ ) d Vd ϕ − iω∂ − ϕ∂ − X + 12 iω∂ − ϕ∂ − X − iω∂ − ϕX + 14 ω ∂ − ϕ d Vd ϕ with X given in (2.4), and V being the potential (see (2.1)). See appendix B for more details,including the terms involving c ( n )+ .The next step is to decompose the curvature (2.9) into the component lying in the abeliansubalgebra generated by b n +1 and one lying in the subspace generated by F n . Since theequation of motion (2.1) has been imposed, it turns out that the terms proportional to the F n ’s in the combination − ∂ − a + + [ a + , a − ] − X g F g − , exactly cancel out. We are then left9ith terms in the direction of the b n +1 only. Therefore, the transformed curvature (2.9) leadsto equations of the form ∂ + a (2 n +1) − − ∂ − a (2 n +1)+ = β (2 n +1) n = 0 , , , . . . (2.15)with β (2 n +1) being linear in the anomaly X given in (2.4), and the first three of them beinggiven by β (1) = 0 , (2.16) β (3) = iω ∂ − ϕ X,β (5) = iω (cid:20) ω ( ∂ − ϕ ) ∂ − ϕ + ∂ − ϕ (cid:21) X. Working with the x and t variables we have that (2.15) takes the form ∂ t a (2 n +1) x − ∂ x a (2 n +1) t = − β (2 n +1) and so we find that d Q (2 n +1) d t = − α (2 n +1) + a (2 n +1) t | x = ∞ x = −∞ (2.17)with Q (2 n +1) ≡ Z ∞−∞ dx a (2 n +1) x , α (2 n +1) ≡ Z ∞−∞ dx β (2 n +1) . (2.18)As we are interested in finite energy solutions of the theory (2.1) we are concerned withfield configurations satisfying the boundary conditions ∂ µ ϕ → V ( ϕ ) → global minimum as x → ±∞ . (2.19)Therefore from (2.2) we see that A + → (cid:16) ω V vac . − m (cid:17) b , A − → b − as x → ±∞ , (2.20)where V vac . is the value of the potential at the global minimum which, in general, is taken tobe zero. As we have seen the parameters ζ n of the gauge (2.7) and (2.8) are polynomials in x − -derivatives of the field ϕ (see appendix B). Therefore, for finite energy solutions we seethat g → x → ±∞ , and so a ( − t → ,a (1) t → (cid:16) ω V vac . − m (cid:17) as x → ±∞ , (2.21) a (2 n +1) t → n = 1 , , . . .
10e can also investigate this behaviour more explicitly by analyzing (2.12), (2.14), (2.4) and(2.19). Consequently, for finite energy solutions satisfying (2.19), we have that d Q (1) d t = 0 , d Q (2 n +1) d t = − α (2 n +1) n = 1 , , . . . (2.22)Of course, the theory (2.1) is invariant under space-time translations and so its energy mo-mentum tensor is conserved. The conserved charge Q (1) is in fact a combination of the energyand momentum of the field configuration. In section 3 we will analyze the anomalies α (2 n +1) for a concrete perturbation of the sine-Gordon model, and we will show that even though thecharges are not exactly conserved they lead to very important consequences for the dynamicsof the soliton solutions.A result that we can draw for general potentials, thus, is the following. For static finiteenergy solutions the charges Q (2 n +1) are obviously time independent, and as a consequenceof (2.22) one sees that the anomalies vanish, i.e. α (2 n +1) = 0. Under a (1 + 1)-dimensionalLorentz transformation where x ± → γ ± x ± one finds that the connection (2.2) does not reallytransform as a vector. However, consider the internal transformation A µ → γ d A µ γ − d (2.23)where d is the grading operator introduced in (A.3). Then, one notices that A µ , given in (2.2),transforms as a vector under the combination of the external Lorentz transformation and theinternal transformation (2.23). For the same reasons the transformed connection a µ , definedin (2.7), is also a vector under the combined transformations. Consequently, the anomalies β (2 n +1) , introduced in (2.15), are pseudo-scalars under the same combined transformation.Therefore, in any Lorentz reference frame the integrated anomalies α (2 n +1) , defined in (2.18),satisfy α (2 n +1) = 0 for any static or a travelling finite energy solution (2.24)where by a travelling solution we mean any solution that can be put at rest by a Lorentzboost. Even though this result may look trivial, it can perhaps shed some light on the natureof the anomalies α (2 n +1) . In fact, as we will see in our concrete example of section 3, theanomalies vanish in multi-soliton solutions when the solitons they describe are far apart andso when they are not in interaction with each other. The anomalies seem to be turned ononly when the interaction takes place among the solitons. Note that we can also construct a second set of quasi conserved charges for the theories(2.1) using another zero curvature representation of their equations of motion. The new Lax11otentials are obtained from (2.2) by interchanging x + with x − , and by reverting the gradesof the generators. Then we introduce the Lax potentials˜ A − = 12 "(cid:16) ω V − m (cid:17) b − − i ω d Vd ϕ F − , ˜ A + = 12 b − i ω ∂ + ϕ F . (2.25)In this case using the commutation relations of appendix A we observe that the curvature ofsuch a connection is˜ F + − ≡ ∂ + ˜ A − − ∂ − ˜ A + + h ˜ A + , ˜ A − i = ˜ X F − + i ω " ∂ ϕ + ∂V∂ϕ F (2.26)with ˜ X = − i ω ∂ + ϕ " d Vd ϕ + ω V − m . (2.27)The construction of the corresponding charges follows the same procedure as in section 2. Weperform the gauge transformation˜ A µ → ˜ a µ = ˜ g ˜ A µ ˜ g − − ∂ µ ˜ g ˜ g − (2.28)with the group element being ˜ g = exp " ∞ X n =1 ζ − n F − n (2.29)and analogously to the case of section 2, we choose the ζ − n ’s to cancel the F − n ’s componentsof ˜ a + . We then have ∂ + ˜ a − − ∂ − ˜ a + + [ ˜ a + , ˜ a − ] = ˜ X ˜ g F − ˜ g − (2.30)where we have used the equation of motion (2.1) to cancel the component of ˜ F + − in thedirection of F . The details of the calculations are given in the appendix C. The transformedconnection takes the form˜ a + = 12 b + ∞ X n =0 ˜ a ( − n − b − n − , ˜ a − = ∞ X n =0 ˜ a ( − n − − b − n − + ∞ X n =2 ˜ c ( − n )+ F − n . The transformed curvature (2.30) leads to equations of the form ∂ + ˜ a ( − n − − − ∂ − ˜ a ( − n − = ˜ β ( − n − n = 0 , , , . . . (2.31)12ith ˜ β (2 n +1) being linear in the anomaly ˜ X , given in (2.27), and the first three are given by˜ β ( − = 0 , ˜ β ( − = iω ∂ ϕ ˜ X, ˜ β ( − = iω (cid:20) ω ( ∂ + ϕ ) ∂ ϕ + ∂ ϕ (cid:21) ˜ X. Following the same reasoning as in section 2, we find that for finite energy solutions we havethe quasi conservation laws d ˜ Q ( − d t = 0 , d ˜ Q ( − n − d t = −
12 ˜ α ( − n − n = 1 , , . . . (2.32)with ˜ Q ( − n − ≡ Z ∞−∞ dx ˜ a ( − n − x , ˜ α ( − n − ≡ Z ∞−∞ dx ˜ β ( − n − . (2.33) The construction of quasi conserved charges of section 2 was performed for a very generalpotential, and no estimates were done on how small the anomaly of the zero curvature con-dition really is. We now turn to the problem of evaluating the anomalies α (2 n +1) , introducedin (2.18), and to discuss the usefulness of the quasi conservation laws (2.22). In order to dothat we choose a specific potential which is a perturbation of the sine-Gordon potential andthat preserves its main features like infinite degenerate vacua and the existence of soliton-likesolutions. So we consider the potential given in (1.1) and we put n = 2 + ε i.e. we take V ( ϕ, ε ) = 2(2 + ε ) tan ϕ h − | sin ϕ | ε i . (3.1)In order to analyze the role of zero curvature anomalies we shall expand the equation ofmotion (2.1) for the potential (3.1), as well as the solutions, in powers of ε . We then write ϕ = ϕ + ϕ ε + ϕ ε + . . . (3.2)and ∂ V∂ ϕ = ∂ V∂ ϕ | ε =0 + " dd ε ∂ V∂ ϕ ! ε =0 ε + . . . = ∂ V∂ ϕ | ε =0 + " ∂ V∂ε∂ϕ + ∂ V∂ϕ ∂ϕ∂ε ε =0 ε + . . . ϕ must satisfies the sine-Gordon equation, i.e. ∂ ϕ + 14 sin (4 ϕ ) = 0 . (3.3)On the other hand the first order field ϕ has to satisfy the equation ∂ ϕ + cos (4 ϕ ) ϕ = sin( ϕ ) cos( ϕ ) h ϕ ln (cid:16) sin ( ϕ ) (cid:17) + cos ( ϕ ) i . (3.4)We shall consider here only the anomalies for the charges constructed in section 2 (the anal-ysis for the charges constructed in section 2.1 is very similar). We expand the anomaly X introduced in (2.4) as X = X + X ε + X ε + . . . (3.5)and we also expand the parameters ω = ω + ω ε + ω ε + . . .m = m + m ε + m ε + . . . . (3.6)Then we find that X = i ω ∂ − ϕ " d Vd ϕ | ε =0 + ω V | ε =0 − m . (3.7)Using the results of appendix D we find that X vanishes by an appropriate choice of param-eters, i.e. X = 0 when ω = 4 and m = 1 . (3.8)With such a choice the first order contribution to X reduces to (again using the results ofappendix D) X = i ∂ − ϕ h − ϕ ln (cid:16) sin ϕ (cid:17) − cos ϕ − m + 8 (cid:18) ω − (cid:19) sin ϕ cos ϕ (cid:21) (3.9)and so we see that X does not depend upon ϕ .Since the anomalies α (2 n +1) , introduced in (2.18), are linear in X and since X = 0, itfollows that their zero order contribution vanishes, as it should since sine-Gordon is integrable.Thus we write our anomalies as α (2 n +1) = α (2 n +1)1 ε + α (2 n +1)2 ε + . . . (3.10)and the first order contribution to the first two of them are (remember that α (1) = 0) α (3)1 = i ω Z ∞−∞ dx X ∂ − ϕ , (3.11) α (5)1 = i ω Z ∞−∞ dx X (cid:20) ω ( ∂ − ϕ ) ∂ − ϕ + ∂ − ϕ (cid:21) X given in (3.9). Thus, the first order anomalies do not depend on the first order field ϕ . The first order charges, however, do depend upon ϕ . To see this we expand the chargesas Q (2 n +1) = Q (2 n +1)0 + Q (2 n +1)1 ε + Q (2 n +1)2 ε + . . . (3.12)Then we find that Q (2 n +1)0 are conserved and correspond to the charges of the sine-Gordonmodel, and involve ϕ only. As an example we present the first charge at first order Q (3)1 = Z ∞−∞ dx h ∂ − ϕ ) ( ω ∂ − ϕ + 4 ∂ − ϕ ) + ∂ − ϕ ( ω ∂ − ϕ + 2 ∂ − ϕ )+ 2 ∂ − ϕ ∂ − ϕ + 14 sin (4 ϕ ) (cid:16) ω ∂ − ϕ + 2 ∂ − ϕ (cid:17) − ∂ − ϕ ∂ + ∂ − ϕ − i X ∂ − ϕ (cid:21) which, indeed, does depend on ϕ .We can now evaluate the anomaly, to first order, for some physical relevant solutions ofthe theory (2.1) with the potential given by (3.1). As we have stressed this earlier the firstorder anomaly depends only upon the zero order field ϕ which is an exact solution of thesine-Gordon equation (3.3). First we look at the case of one kink. The kink solution is given by (1.2) with n = 2 + ε andit is an exact solution of (2.1) for V given by (3.1). The first order anomaly depend upon thekink solution of the sine-Gordon equation (3.3) which is given by ϕ = arctan ( e x ) . (3.13)Inserting this expression into (3.11) and (3.9) we find that α (3)1 = α (5)1 = Z ∞−∞ dx sinh x cosh x (cid:20) e x ln (cid:18) e x cosh x (cid:19) + e − x (cid:21) . (3.14)This expression can be integrated explicitly using the fact that dd x x ) + e x ( e x −
3) ln (cid:16) e x cosh x (cid:17) ( x ) = sinh x cosh x (cid:20) e x ln (cid:18) e x cosh x (cid:19) + e − x (cid:21) and so Z ∞−∞ dx sinh x cosh x (cid:20) e x ln (cid:18) e x cosh x (cid:19) + e − x (cid:21) = 0 . (3.15)Therefore, the first order anomalies vanish, i.e. α (3)1 = α (5)1 = 0, agreeing with the generalresult shown in (2.24). 15 .2 Anomalies for the 2-soliton solutions Let us consider a 2-soliton solution corresponding, for η = 1, to a soliton moving to the rightwith speed v and located at x = − L at t = 0, and an anti-soliton moving to the left withspeed v and located at x = L at t = 0. For η = − ε -expansion is given by a solution of thesine-Gordon equation (3.3) given by ϕ = ArcTan " η v cosh y sinh τ (3.16)with y = x √ − v , τ = v t − L √ − v + η ln v. (3.17)Putting this expression into (3.11) and (3.9) we find that the first anomaly at first order is α (3)1 = 8 v (1 − v ) / sinh τ cosh τ Z ∞−∞ dx h v (cid:16)(cid:16) v (cid:17) Ω + 4 v (cid:17) cosh y − v Ω i ×× " − v cosh y Λ ln v cosh y Λ ! − sinh τ Λ − m + 8 (cid:18) ω − (cid:19) v cosh y Λ sinh τ Λ , (3.18)where we have introducedΛ = sinh τ + v cosh y , Ω = sinh τ − v cosh y . (3.19)Note that α (3)1 given in (3.18), is an odd function of τ due to the term sinh τ in front of theintegral. All other terms involving τ in (3.18) appear as cosh τ or sinh τ , and so are evenin τ . Consequently we see that Z ∞−∞ dt α (3)1 = 0 . (3.20)We point out that this result is independent of the values of ω and m which appear in theexpression for α (3)1 . Note that, from (2.22), (3.10) and (3.12), we have that d Q (3)1 d t = − α (3)1 (3.21)and so Q (3)1 ( t = ∞ ) = Q (3)1 ( t = −∞ ) . (3.22)Thus, in the scattering of the soliton and anti-soliton the charge at first order is conservedasymptotically. From the physical point of view that is as effective as in the case of theintegrable sine-Gordon theory. The solitons have to scatter preserving higher charges (at leastin first order approximation). 16 .2.2 The soliton/soliton scattering Next we consider a 2-soliton solution corresponding, for η = 1, to a soliton moving to theright with speed v and located at x = − L at t = 0, and another soliton moving to the leftwith speed v and located at x = L at t = 0. For η = − ε -expansion, is again given by a solutionof the sine-Gordon equation (3.3), namely ϕ = ArcTan " − η cosh τ v sinh y , (3.23)where y = x √ − v + η ln v, τ = v t − L √ − v . (3.24)Following the same procedure as in the case of soliton/anti-soliton solution, by putting (3.23)into (3.11) and (3.9) we find that the first anomaly, at first order, is α (3)1 = 8 v (1 − v ) / sinh τ cosh τ Z ∞−∞ dx h v (cid:16)(cid:16) v + 3 (cid:17) Ω − v (cid:17) sinh y + 2 v Ω i × " − τ Λ ln cosh τ Λ ! − v sinh y Λ − m + 8 (cid:18) ω − (cid:19) cosh τ Λ v sinh y Λ (3.25)with Λ = cosh τ + v sinh y , Ω = cosh τ − v sinh y . (3.26)Again, one notices that α (3)1 given in (3.25) is odd in τ . Indeed, except for the factor sinh τ infront of the integral, all other terms are even in τ since they involve only cosh τ . Consequently,we again have Z ∞−∞ dt α (3)1 = 0 (3.27)and such result is independent of the values of ω and m . Again, using (3.21) we see that Q (3)1 ( t = ∞ ) = Q (3)1 ( t = −∞ ). So, the solitons scatter preserving higher charges asymptoti-cally, like in the case of soliton/anti-soliton scattering discussed above. As we show in the next section the theory (2.1) with potential (3.1) has long lived breather-likesolutions. Hence, next we evaluate the anomaly, to first order, for such a solution. For that we17eed the solution for the zero order field ϕ which is a breather solution for the sine-Gordonequation (3.3), i.e. ϕ = arctan √ − ν ν sin ( ν t )cosh (cid:16) √ − ν x (cid:17) , (3.28)where ν is the frequency of the breather (0 < ν < α (3)1 = − ν (cid:16) − ν (cid:17) sin (2 ν t ) (cid:20) I ( ν, t ) − m I ( ν, t ) + 8 (cid:18) ω − (cid:19) I ( ν, t ) (cid:21) (3.29)with I ( ν, t ) = Z ∞−∞ dx h (cid:16) − ν (cid:17) sinh (cid:16) √ − ν x (cid:17) Ω (3.30)+ cosh (cid:16) √ − ν x (cid:17) (cid:16)(cid:16) − ν (cid:17) Ω − ν (cid:16) − ν (cid:17)(cid:17)i ×× − − ν ) sin ( ν t )Λ ln (1 − ν ) sin ( ν t )Λ ! − ν cosh (cid:16) √ − ν x (cid:17) Λ and I ( ν, t ) = Z ∞−∞ dx h (cid:16) − ν (cid:17) sinh (cid:16) √ − ν x (cid:17) Ω+ cosh (cid:16) √ − ν x (cid:17) (cid:16)(cid:16) − ν (cid:17) Ω − ν (cid:16) − ν (cid:17)(cid:17)i (3.31)and I ( ν, t ) = Z ∞−∞ dx h (cid:16) − ν (cid:17) sinh (cid:16) √ − ν x (cid:17) Ω+ cosh (cid:16) √ − ν x (cid:17) (cid:16)(cid:16) − ν (cid:17) Ω − ν (cid:16) − ν (cid:17)(cid:17)i ×× ν (1 − ν ) sin ( ν t ) cosh (cid:16) √ − ν x (cid:17) Λ , (3.32)where we have denotedΛ = ν cosh (cid:16) √ − ν x (cid:17) + (cid:16) − ν (cid:17) sin ( ν t ) , Ω = ν cosh (cid:16) √ − ν x (cid:17) − (cid:16) − ν (cid:17) sin ( ν t ) . (3.33)Note that the time dependence of the integrals I j ( ν, t ), j = 1 , ,
3, comes only through thefactor sin ( ν t ) = [1 − cos (2 ν t )]. Since these integrals are multiplied by the factor sin (2 ν t )in (3.29), we conclude that α (3)1 is periodic in time with period T ≡ πν . In addition, we observe18hat I j ( ν, t ) = I j ( ν, − t ), and so α (3)1 ( t ) = − α (3)1 ( − t ), due to the overall factor sin (2 ν t ) in(3.29). Consequently, we have that Z t + Tt dt ′ α (3)1 ( t ′ ) = Z T/ − T/ dt ′ α (3)1 ( t ′ ) = 0 , (3.34)where we have used the fact that R t + Tt = R − T/ t + R T/ − T/ + R t + TT/ , and so the first and thirdintegrals cancel due to the fact that α (3)1 ( t ) = α (3)1 ( t + T ). Consequently, we find from (3.21)that the charge (to first order) is periodic in time Q (3)1 ( t ) = Q (3)1 (cid:18) t + πν (cid:19) . (3.35) To check our results on the anomaly we have decided to perform various simulations of theBazeia at al model - studying two kinks, a kink-antikink, and a system involving two kinksand an antikink.In all our numerical work the time evolution was simulated by the fourth order Runge -Kuta method. We used various lattice grids (to make sure that our results were not contam-inated by any numerical artefacts, the issue here was the size of the lattice and the latticestep). We found that to have reliable results the lattice grid (given that the kinks were ofsize ∼ ±
5) had to stretch to, at least, ±
50. Hence most of our work was performed usingeven larger grids and the results given in this paper were obtained in simulations in which thelattice contained 10001 equally spaced points and stretched from -75 to 75. At the edge ofthe grid (in practice from -71 to -75 and from +71 to +75) we absorbed the kinetic energy ofthe fields. During the scattering process there was some radiation sent out towards the edgesof the grid and it is this radiation that our procedure absorbed (so that we would not haveany reflection of the radiation from the boundaries). Thus our procedure had the effect ofsimulating an infinite grid in which we looked only at the fields in a finite region. Thus, dueto this absorption, the total energy seen in our simulations would decrease but this decreasecould be associated with the system radiating some energy towards the boundaries and theenergy seen by us corresponded to the energy of the system that we have tried to describe.
First we looked at the interaction between 2 kinks. To study this we placed two kinks at somedistance from each other and then performed a simulation to see what happens.19igure 4: Trajectories: a) n = 2, b) n = 1 . n = 2 and n = 1 .
9. The kinkswere initially placed at d = ± . n and each time the situation was the same. Looking at the plots of thetrajectories we do not see much difference between n = 2 and n = 1 . v = 0 . n = 2 and n = 1 .
9. Initially the kinks were placed at d = ± . d , n and their velocties.Incidentally, the sine-Gordon model ( i.e. the model with n = 2) possesses a solutiondescribing two moving kinks and our results (for n = 2) reproduce them very well and,surprise, surprise, the model does not have any static solutions involving more than one kink.On the other hand, the sine-Gordon moving kinks solutions are known in an explicit form,and because these kinks are described by explicit functions it is often said that the ”kinkspass through each other”. This is clearly wrong when one looks at the energy density of the20igure 5: Trajectories: a) n = 2, b) n = 1 . i.e. the two peaks of the energy density never form a doublepeak); in practice, the functions which describe each kink switch after the kinks’ interaction.We have also looked at the scattering of two kinks from the point of view of the integrabilitydiscussed in the previous section - i.e. from the point of view of the anomalies.To do this we considered the scattering of two kinks for values of n close to 2. We lookedat various positions of kinks and various velocities. All results were qualitatively similar sohere we present our results for v = .
5. The kinks were initially placed at ± .
5. We performedmany simulations of the dynamics of such systems. In each case, as mentioned above, thekinks came close to each other, reflected and then moved to the boundaries with essentiallythe original velocity. Thus the scattering was very elastic. Looking at the scattering in moredetail it was easy to see that, strictly speaking, there was also some radiation emitted duringthe scattering and that the amount of this emitted radiation increased with the increase of | ε | = | n − | ; however, even for n = 1 this radiation constituted less than 2% of the totalenergy. Hence the scattering was very elastic.We have also looked at the values of the first anomaly and its time integrated value forthese scatterings.In fig 6. we present a representative selection of our results. Fig. 6a and 6b presentthe time dependence of the anomaly and its time integrated form for n = 2, as seen in oursimulations. Of course we know that for n = 2 the anomaly vanishes so our results provide thetest of our numerics. We note that our values of the anomaly are very small - i.e. consistentwith zero. Next we looked at the values of n = 2 for which the anomaly does not vanish. Infig 6c and 6d present our results for the anomaly and its integrated form for n = 1 .
99 and fig6e and 6f present similar results for n = 2 .
01. Fig 6g and 6h refer to the case of n = 1 . n = 2 . n = 3 . From these results we see very clearly that for all values of n (with the exception of n = 3)the integrated anomaly is approximately zero. This supports our analytical results and itshows that (for small ε ) the unintegrated total anomaly is approximately proportional to ε = n −
2. This is supported further by the observation that the anomaly changes sign as ε → − ε . The second order (in ε ) effects are comparable to those of the first order and theexpansion in ε clearly does not converge for ε ∼
1. This last point is very clear from thecase of n = 3 in which case ε = 1. Of course it would be nice to understand why all theterms in the ε power series expansion are comparable in magnitude; at this stage we have nounderstanding of this fact. Next we have looked at the kink - antikink configurations and breathers. In the sine-Gordonmodel we do have breathers and their analytical form is well known. They are in fact boundstates of a kink and an antikink. This is all well known; what is perhaps less known, is thatone can generate breathers by taking a kink and an antikink and placing them not too close toeach other and then let the configuration evolve in time. As the kink and the antikink attractthey move towards each other, alter their shape and, at the same time, emit some radiationand become a breather. Interestingly, they do not annihilate but do form a breather. If wethen absorb the energy at the boundaries the system stabilises and essentially stops emittingfurther energy as the fields have taken the shape of a breather which is a time dependentsolution of the model.It is sometimes thought that the existence of breathers and of other similar configurations(wobbles etc) is, at least in part, associated with the integrability of the sine-Gordon model.Actually, as we have stressed this before, the models of Bazeia et al [1] do not appear tobe integrable for any n other than 2; so we have decided to apply our procedure to look atconfigurations of a kink and an antikink for other values of n . However, before we discusssome of our results obtained in such cases let us first present them for the sine-Gordon model.In fig. 7 we present the time dependence of the energy of the field configuration whichinvolved a kink and an antikink initially placed at d = ± d = ± x < Figure 6: Anomalies and the corresponding integrated anomalies (from left to rightand then down) for n = 2 . n = 1 . n = 2 . n = 1 . n = 2 .
10 and n = 3 . e n e r gy e n e r gy -5 -4 -3 -2 -1 0 0 2000 4000 6000 8000time p o s i t i o n Figure 7: (ab) Time dependence of total energies; (c). Position of the kink.by E = 2 E √ − ω and its frequency ω is related to the intial extend given by d .Next, we repeated the same procedure of generating breathers for field configurationscorresponding to other values of n . In fig 8. we present our results for two values of n , namely n = 1 and n = 3 .
1, in which the kink and the antikink were initially placed at d = ± .
0. Ourplots give the time dependence of the total energy of the configuration (after the absorptionat the boundaries has eliminated the radiation reaching the boundaries).We note a fundamental difference; for n = 1 the energy seems to ‘stabilise’ around somefinite nonzero value while for n = 3 . n and for different distancesbetween kinks and antikinks) running them for very long times. We have found that for somevalues of n the fields annihilate very quickly; while for the others the fields evolved towardsbreather-like configurations. This was not much dependent on the distance between the initialkinks but depended much more on n . In fact, as the distance d increased the whole process,like for the sine-Gordon model, was slower, the initial radiation was smaller and the generatedbreather was larger (and so its oscillations were slower). Looking at the dependence on n itwas clear that the closer n was to 2 the more stable the breather was (this was true fromabout ∼ . ∼ . n (very close to 2, like 2 .
01 or so) theresultant configuration was almost indistinguishable from a breather. In fact, in all such cases,the energy kept decreasing but this decrease was infinitesimal. Thus we could say that we hada quasi-breather ( i.e. a long-lived breather). As the lifetime of such a quasi-breather couldbe counted in millions of units of time, such fields, for practical (but not purely mathematical)reasons were not very different from a breather.24
0 1 2 3 4 5 6 7 8 9 10 11 12 0 20000 40000 60000time e n e r gy
0 1 2 3 4 5 6 0 10000 20000 30000 40000time e n e r gy Figure 8: Time dependence of the total energy (a) n = 1, (b) n = 3 . n = 2 .
01) which demonstrate theexistence of our quasi-breathers. In fig 9ab we present the plots of the field configuration fortwo values of t , namely t = 355500 and t = 356300. We see that the fields look very muchlike those of the n = 2 breather.In fig. 10 ab and c we present the time dependence of the energy of the configurationon t , a detail of this dependence at large t and the time dependence of the value of field at x = 0. Note the extremely large values of t in the plots of the energy density. Note also theirregularity of the energy decrease. The energy gradually appears to decrease less and lessand then suddenly drops and changes its slope of decrease. It then continues in the same wayuntil the slope changes again etc. We do not understand these changes but, in any case, thetotal decrease of the energy is still very modest and it is clear that the quasi-breather is notgoing to “die” soon.The plot in fig. 10c presents the time variation of the field at x = 0. As the field issymmetric around x = 0 this plot demonstrates the frequency of the oscillation.We have performed similar simulations starting with the initial kinks and antikinks atother distances from each other and for other values of n . The results were qualitatively thesame; the further the initial structures were - the slower was the decrease in energy ( i.e thelonger the life-time of the breather). The same was true when we considered n further awayfrom n = 2. This once again suggests that the models for n = 2 (but close to 2) are quasi-integrable as discussed in the previous sections. Hence we have also looked at the behaviourof the anomaly for our quasi-breathers.In fig. 11 ab we present the plots of the anomaly and the time integrated anomaly (at25 -1.5 -1 -0.5 0-60 -40 -20 0 20 40 0 0.5 1 1.5-60 -40 -20 0 20 40 Figure 9: Field configurations (of n = 2 .
01) for (a) t = 355500 and (b) t = 356300 Figure 10: The time dependence of the total energy (a), of the details of thisdependence at the larger values of t (b) and of the values of the field at x = 0 seenin a similation of a kink and an antikink in the n = 2 . n = 2 .
01, c) andd) the same for n = 2 . n = 2 .
01 quasi-breather and in fig. 11 cd the similar plots for n = 2 . i.e. close to zero). For n = 2 .
01 this value is so small that it isdifficult to see that it changes at all, for n = 2 . Finally we looked at wobbles i.e. , fields involving a kink and a breather. In the sine-Gordonmodel they are again well known and, in fact, one has their analytical form. Of course, asbefore, we can generate them, numerically, from field configurations involving an antikink andtwo kinks (or vice-versa). However, as these configurations have an excess of energy, which isemitted when an antikink a kink form a breather, this energy can be, in part, converted into themotion of the remaining kink (or of the breather). Hence it is much harder, by comparisonwith pure breathers (where one can exploit the symmetry of the initial configuration), togenerate non-moving wobbles. We have performed many simulations and the resultant fieldssometimes were static but most of the time were moving. Clearly, the result of the simulationdepends on the excess of energy - so further the initial structures were from each other themore likely there were to remain static. But this, in turn, slowed down the process of thegeneration of the breather. In addition, the futher n was from n = 2 the more radiation wassent out by the system and more likely it was that this radiation would set in motion the kinkor the breather. However, for n close to 2 we did manage to obtain wobbles and in the plotsgiven below we show some of our results. 27igure 12: a) Total energy, b) Potential energyFigure 13: a) t = 0, b) t = 6400, c) t = 12800First we present our results for n = 2, i.e. for the sine-Gordon model.In fig. 12 we plot the time dependence of the total and of the potential energy seen inthe simulation involving the kink, the antikink and the kink originally located at -15.5, 0 and+15. In the following figure we exhibit the field configurations for three values of t , namelyfor t = 0, t = 6400 and t = 12800. We note a fast decrease of the total energy over the initialperiod and then stability. The potential energy is virtually always close to 12 and then itdecreases to just over 4 when the breather is ‘breathing’, i.e. when almost all its energy iskinetic. Note that, for the breather, the flow of energy between the kinetic and the potentialenergies is very uneven; most of the time the breather’s energy is mainly potential and theperiods over which the kinetic energy dominates are relatively short.Next we present our results for the case of n = 2 i.e. n = 2 .
01. In this case the energycontinues to decrease but this decrease is very slow. In fig 14c we present the details ofthe plot of the total energy for larger values of t . We clearly see the decrease - hence the28 Figure 14: a) Total energy, b) Potential energy, c) Total energy for large values of t ( n = 2 . n = 2 . n = 2 and n = 2 . n = 2 .
01 wobble and slowly decreasing when compared to thatof n = 2 . n = 1 . t ; a) t = 0, b) t = 6400, c) t = 1280029 -2 -1 0 1 2 3 -50 0 50 0 0.5 1 1.5 2 2.5 3 -50 0 50 0 0.5 1 1.5 2 2.5 3 -50 0 50 Figure 16: Fields at 3 values of t (for n = 1 . t = 500, b) t = 2000, c) t = 5000. In this paper we have made the first steps to introduce the concept of quasi-integrabilityand discussed it on the example of the models of Bazeia et al [1]. We showed that whenthe models are close to being intergrable and so can be compared with them then one canintroduce many quantities, which in the integrable case are conserved and which, in the non-integrable case, are not conserved. One can then calculate their anomalies that are responsiblefor this nonconservation, in a power series of the difference of the quasi-integrable models fromtheir integrable neighbours. In the models of Bazeia et al [1] this difference is provided by ε = n − i.e. the field configurationslike those of breathers or wobbles. And, conveniently, the models of Bazeia et al do possesssuch configurations.In fact, the models of Bazeia et al, which depend on a parameter n (which when n = 2reduce to the integrable sine-Gordon models) have many very similar properties and can beused to discuss the concept of quasi-integrability. All models ( i.e. for any n ) have one kinksolutions and their scattering properties are very similar. Moreover, no other analytic solutionsof these models (when n = 2) are known.However, the models can be studied numerically. When we studied these models for n = 2but close to 2 we have found that the models do, indeed, possess long-lived breather-like field30onfigurations; i.e. when we have constructed breather-like field configurations and let themevolve they gradually emitted some energy but this process was extremely slow; and so wecan claim that these models (for n close to 2) possess ’very long-lived’ breather-like solutions.Their life-time is closely related to how close n is to 2 and when n < . n > . i.e. states involving a breather and a kink) andthe situation was found to be similar although the range of n for which such states appearedto be long lived was smaller. This is partly related to our construction of such states; wegenerated them all by taking initial configurations consisting of kinks and antikinks and thenevolving them and absorbing, at the boundaries, any energy emitted by the configuration.For the wobble-like states, as the configuration was less symmetric the energy was emittednon-symmetrically and this often lead to more perturbation of the resultant (wobble-like) fieldconfiguration.Thus, in addition to supporting our studies of quasi-integrability, our numerical resultsdemonstrated also the existence of long-lived breather-like and wobble-like states.31 cknowledgements: LAF and WJZ thank the Royal Society for a grant which set up theircollaboration on the topics of this paper. LAF is also partially supported by a CNPq grantwhile WJZ acknowledges a FAPESP grant which supported his visit to IFSC/USP.
A The algebra
We consider the sl (2) algebra[ T , T ± ] = ± T ± , [ T + , T − ] = 2 T . (A.1)We take the following basis for the corresponding loop algebra b m +1 = λ m ( T + + λ T − ) , F m +1 = λ m ( T + − λ T − ) , F m = 2 λ m T . (A.2)The algebra is [ b m +1 , b n +1 ] = 0 , [ F m +1 , F n +1 ] = 0 , [ F m , F n ] = 0 , [ b m +1 , F n +1 ] = − F m + n +1) , [ b m +1 , F n ] = − F m + n )+1 , [ F m +1 , F n ] = − b m + n )+1 . We have a grading operator d = T + 2 λ ddλ (A.3)such that [ d , b m +1 ] = (2 m + 1) b m +1 , [ d , F m ] = m F m . (A.4) B The gauge transformation (2.7)
The first six parameters ζ n of the gauge transformation (2.7), determined through (2.10) aregiven by ζ = 12 iωϕ (0 , ,ζ = 12 iωϕ (0 , , = 16 i (cid:16) ω ( ϕ (0 , ) + 3 ωϕ (0 , (cid:17) ,ζ = 16 i (cid:16) ϕ (0 , ) ϕ (0 , ω + 3 ϕ (0 , ω (cid:17) ,ζ = 130 i (cid:16) ω ( ϕ (0 , ) + 30 ω ϕ (0 , ( ϕ (0 , ) + 40 ω ( ϕ (0 , ) ϕ (0 , + 15 ωϕ (0 , (cid:17) ,ζ = 130 i (cid:16) ϕ (0 , ) ϕ (0 , ω + 40( ϕ (0 , ) ω + 145 ϕ (0 , ϕ (0 , ϕ (0 , ω + 35( ϕ (0 , ) ϕ (0 , ω + 15 ϕ (0 , ω (cid:17) , where ϕ (0 ,n ) ≡ ∂ n − ϕ .The first few components of transformed gauge potentials introduced in (2.7) are given by a − = 12 b − + b (cid:20) − ω ( ∂ − ϕ ) (cid:21) + b (cid:20) − ω ( ∂ − ϕ ) − ω ∂ − ϕ∂ − ϕ (cid:21) + b (cid:20) − ω ( ∂ − ϕ ) − ω ∂ − ϕ ( ∂ − ϕ ) − ω ( ∂ − ϕ ) ( ∂ − ϕ ) − ω ∂ − ϕ∂ − ϕ (cid:21) + .... and a + = b (cid:20) (cid:16) ω V − m (cid:17)(cid:21) + b " ω ∂ − ϕ d Vd ϕ − iω∂ − ϕX + b " − iω ( ∂ − ϕ ) X + 516 ω ∂ − ϕ ( ∂ − ϕ ) d Vd ϕ − iω∂ − ϕ∂ − X + 12 iω∂ − ϕ∂ − X − iω∂ − ϕX + 14 ω ∂ − ϕ d Vd ϕ + F X + F ∂ − X + F (cid:20) ω ( ∂ − ϕ ) X + ∂ − X (cid:21) + F (cid:20) ω ( ∂ − ϕ ) ∂ − X + 12 ω ∂ − ϕ∂ − ϕX + ∂ − X (cid:21) + F (cid:20) ω ( ∂ − ϕ ) X + 32 ω ( ∂ − ϕ ) ∂ − X + 52 ω ∂ − ϕ∂ − ϕ∂ − X + ω ∂ − ϕ∂ − ϕX + ∂ − X (cid:21) + ... i.e. , those that do no vanish dueto the anomaly X introduced in (2.4), are given by X g F g − − [ a + , a − ] = b h iω∂ − ϕX i + b (cid:20) iω ( ∂ − ϕ ) ∂ − ϕX + iω∂ − ϕX (cid:21) + F [ − ∂ − X ]+ F h − ∂ − X i + F (cid:20) − ω ( ∂ − ϕ ) ∂ − X − ω ∂ − ϕ∂ − ϕX − ∂ − X (cid:21) + F (cid:20) − ω ∂ − ϕ∂ − ϕ∂ − X − ω ( ∂ − ϕ ) ∂ − X − ω ( ∂ − ϕ ) X − ω ∂ − ϕ∂ − ϕX − ∂ − X (cid:21) + ... C The gauge transformation (2.28)
The first six parameters ζ − n of the gauge transformation (2.28) and (2.29) are given by ζ − = 12 iωϕ (1 , ,ζ − = 12 iωϕ (2 , ,ζ − = 16 i (cid:16) ω ( ϕ (1 , ) + 3 ωϕ (3 , (cid:17) ,ζ − = 16 i (cid:16) ϕ (1 , ) ϕ (2 , ω + 3 ϕ (4 , ω (cid:17) ,ζ − = 130 i (cid:16) ω ( ϕ (1 , ) + 30 ω ϕ (3 , ( ϕ (1 , ) + 40 ω ( ϕ (2 , ) ϕ (1 , + 15 ωϕ (5 , (cid:17) ,ζ − = 130 i (cid:16) ϕ (1 , ) ϕ (2 , ω + 40( ϕ (2 , ) ω + 145 ϕ (1 , ϕ (2 , ϕ (3 , ω + 35( ϕ (1 , ) ϕ (4 , ω + 15 ϕ (6 , ω (cid:17) , where ϕ ( n, ≡ ∂ n + ϕ .The first few components of transformed gauge potentials introduced in (2.28) are givenby˜ a + = 12 b b − (cid:20) − ω ( ∂ + ϕ ) (cid:21) + b − (cid:20) − ω ( ∂ + ϕ ) − ω ∂ ϕ∂ + ϕ (cid:21) + b − (cid:20) − ω ( ∂ + ϕ ) − ω ∂ ϕ ( ∂ + ϕ ) − ω ( ∂ ϕ ) ( ∂ + ϕ ) − ω ∂ ϕ∂ + ϕ (cid:21) + ... and˜ a − = b − (cid:20) (cid:16) ω V − m (cid:17)(cid:21) + b − " ω ∂ ϕ d Vd ϕ + 12 iω∂ + ϕ ˜ X + b − " ω ∂ ϕ ( ∂ + ϕ ) d Vd ϕ + 14 ω ∂ ϕ d Vd ϕ + 38 iω ( ∂ + ϕ ) ˜ X + 12 iω∂ + ϕ∂ ˜ X − iω∂ ϕ∂ + ˜ X + 12 iω∂ ϕ ˜ X (cid:21) + F − h − ˜ X i + F − h − ∂ + ˜ X i + F − (cid:20) − ω ( ∂ + ϕ ) ˜ X − ∂ ˜ X (cid:21) + F − (cid:20) ω (cid:16) − ( ∂ + ϕ ) (cid:17) ∂ + ˜ X − ω ∂ + ϕ∂ ϕ ˜ X − ∂ ˜ X (cid:21) + F − (cid:20) − ω ( ∂ + ϕ ) ˜ X − ω ( ∂ + ϕ ) ∂ ˜ X − ω ∂ ϕ∂ + ϕ∂ + ˜ X − ω ∂ ϕ∂ + ϕ ˜ X − ∂ ˜ X (cid:21) + ... The anomalous terms of the gauge transformation (2.28), i.e. , those that do no vanish dueto the anomaly ˜ X introduced in (2.27), are given by˜ X ˜ g F − ˜ g − − [ ˜ a + , ˜ a − ] = b − h iω∂ ϕ ˜ X i + b − (cid:20) iω ( ∂ + ϕ ) ∂ ϕ ˜ X + iω∂ ϕ ˜ X (cid:21) + F − h − ∂ + ˜ X i + F − h − ∂ ˜ X i + F − (cid:20) − ω ( ∂ + ϕ ) ∂ + ˜ X − ω ∂ + ϕ∂ ϕ ˜ X − ∂ ˜ X (cid:21) F − (cid:20) − ω ∂ + ϕ∂ ϕ∂ + ˜ X − ω ( ∂ + ϕ ) ∂ ˜ X − ω ( ∂ ϕ ) ˜ X − ω ∂ + ϕ∂ ϕ ˜ X − ∂ ˜ X (cid:21) + ... D The ε -expansion V = V | ε =0 + d Vd ε | ε =0 ε + . . . = V | ε =0 + " ∂V∂ε + ∂V∂ϕ ∂ϕ∂ε ε =0 ε + ∂ V∂ε + 2 ∂ V∂ε∂ϕ ∂ϕ∂ε + ∂V∂ϕ ∂ ϕ∂ε + ∂ V∂ϕ ∂ϕ∂ε ! ε =0 ε + . . . (D.1)Analogously, we have ∂ V∂ ϕ = ∂ V∂ ϕ | ε =0 + " dd ε ∂ V∂ ϕ ! ε =0 ε + . . . = ∂ V∂ ϕ | ε =0 + " ∂ V∂ε∂ϕ + ∂ V∂ϕ ∂ϕ∂ε ε =0 ε + ∂ V∂ε ∂ϕ + 2 ∂ V∂ε∂ϕ ∂ϕ∂ε + ∂ V∂ϕ ∂ ϕ∂ε + ∂ V∂ϕ ∂ϕ∂ε ! ε =0 ε + . . . (D.2)and ∂ V∂ ϕ = ∂ V∂ ϕ | ε =0 + " dd ε ∂ V∂ ϕ ! ε =0 ε + . . . = ∂ V∂ ϕ | ε =0 + " ∂ V∂ε∂ϕ + ∂ V∂ϕ ∂ϕ∂ε ε =0 ε + ∂ V∂ε ∂ϕ + 2 ∂ V∂ε∂ϕ ∂ϕ∂ε + ∂ V∂ϕ ∂ ϕ∂ε + ∂ V∂ϕ ∂ϕ∂ε ! ε =0 ε + . . . (D.3)Calculating we have V | ε =0 = 18 sin (2 ϕ ) = 116 [1 − cos (4 ϕ )] ∂ V∂ ϕ | ε =0 = 14 sin (4 ϕ ) ∂ V∂ ϕ | ε =0 = cos (4 ϕ )36 V∂ϕ | ε =0 = − ϕ ) ∂ V∂ϕ | ε =0 = −
16 cos (4 ϕ ) (D.4)and ∂ V∂ε | ε =0 = −
12 sin ( ϕ ) h ( ϕ ) log | sin( ϕ ) | + cos ( ϕ ) i (D.5) ∂ V∂ ϕ∂ε | ε =0 = −
14 sin(2 ϕ ) h ϕ log | sin( ϕ ) | + cos(2 ϕ ) + 1 i ∂ V∂ ϕ ∂ε | ε =0 = −
12 [4 (cos(2 ϕ ) − cos(4 ϕ )) log | sin( ϕ ) | + cos(2 ϕ ) + 1] ∂ V∂ ϕ ∂ε | ε =0 = sin(2 ϕ ) [ − ( − | sin( ϕ ) | +4 cos(2 ϕ )(4 log | sin( ϕ ) | +1) + 1)]and ∂ V∂ε | ε =0 = 14 tan ( ϕ ) h sin ( ϕ ) (cid:16) | sin( ϕ ) | − | sin( ϕ ) | +3 (cid:17) + sin ( ϕ ) (cid:16) − | sin( ϕ ) | +8 log | sin ϕ | − (cid:17) + 3 i ∂ V∂ϕ∂ε | ε =0 = 132 tan( ϕ ) sec ( ϕ ) h
24 log | sin( ϕ ) | +16 log | sin( ϕ ) | + 8 cos(6 ϕ ) log | sin( ϕ ) | (D.6)+ cos(2 ϕ ) (cid:16) −
40 log | sin( ϕ ) | +4 log | sin( ϕ ) | +23 (cid:17) + 8 cos(4 ϕ )(log | sin( ϕ ) | − − ϕ ) log | sin( ϕ ) | + cos(6 ϕ ) + 16 i ∂ V∂ϕ ∂ε | ε =0 = 116 sec ( ϕ ) h
44 log | sin( ϕ ) | +16 log | sin( ϕ ) | + 10 cos(6 ϕ ) log | sin( ϕ ) | +4 cos(8 ϕ ) log | sin( ϕ ) | + 2 cos(2 ϕ ) (cid:16) −
29 log | sin( ϕ ) | + log | sin( ϕ ) | +6 (cid:17) − ϕ ) log | sin( ϕ ) | + cos(4 ϕ )(3 −
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