Breather-like structures in modified sine-Gordon models
BBreather-like structures in modified sine-Gordon models
L. A. Ferreira (cid:63) and Wojtek J. Zakrzewski † ( (cid:63) ) Instituto de F´ısica de S˜ao Carlos; IFSC/USP;Universidade de S˜ao PauloCaixa Postal 369, CEP 13560-970, S˜ao Carlos-SP, Brazilemail: [email protected] ( † ) Department of Mathematical Sciences,University of Durham, Durham DH1 3LE, U.K.email: [email protected]
Abstract
We report analytical and numerical results on breather-like field configurations in atheory which is a deformation of the integrable sine-Gordon model in (1 + 1) dimensions.The main motivation of our study is to test the ideas behind the recently proposed con-cept of quasi-integrability, which emerged from the observation that some field theoriespresent an infinite number of quantities which are asymptotically conserved in the scat-tering of solitons, and periodic in time in the case of breather-like configurations. Eventhough the mechanism responsible for such phenomena is not well understood yet, it isclear that special properties of the solutions under a space-time parity transformationplay a crucial role. The numerical results of the present paper give support for the ideason quasi-integrability, and it is found that extremely long-lived breather configurationssatisfy these parity properties. We also report on a mechanism, particular to the theorystudied here, that favours the existence of long lived breathers even in cases of significantdeformations of the sine-Gordon potential. a r X i v : . [ h e p - t h ] A p r Introduction
Solitons play a fundamental role in the study of non-linear phenomena because in manysituations they can be considered as the “normal modes” of the physical system in the strongcoupling regime. In fact, in special examples of gauge theories in (3 + 1) dimensions andintegrable field theories in (1 + 1) dimensions, there exist duality relations interchangingthe roles of the fundamental excitations of the fields at weak coupling, and the solitons atthe strong coupling regime [1]. In (1 + 1) dimensions, where soliton theory is much betterunderstood, the solitons are often described as those classical solutions that propagate withoutdissipation and dispersion, and when two of such solitons are scattered they do not destroyeach other. The only effect of the scattering is a shift in their positions in relation to the valuesthey would have had, had they not participated in the scattering process. The most acceptableexplanation for such behaviour is that, in practically all known soliton theories, there exists aninfinite number of conserved quantities that dramatically constrains the dynamics, and leavesno options for the solitons after the scattering but to continue being themselves. This is aremarkable fact but it certainly has an annoying drawback. It forces solitons to exist only inthe realm of the so-called exactly integrable field theories in (1 + 1) dimensions. Such theoriesare, however, very special as they possess highly non-trivial hidden symmetries and they havebeen used as convenient laboratories to study non-perturbative phenomena and so have leadto the development of new and important techniques in field theories.Recently we have observed that some non-integrable field theories in (1 + 1) dimensions,present properties similar to those of exactly integrable theories [2, 3, 4]. They have soliton-likefield configurations that behave in a scattering process in a very similar way to the true solitons, i.e. they do not destroy each other. We have also found that such theories possess an infinitenumber of quantities which are not exactly conserved in time, but are, however, asymptoticallyconserved. By that we mean that the values of these quantities do change, and change a lot,during their scattering process but they return, after the scattering, to the values they havehad before it. This is a remarkable property since from the point of view of the scattering whatmatters are the asymptotic states, and so such a theory looks a bit as an effective integrabletheory. We have also observed that some of such non-integrable theories possess breather-likesolutions that are extremely long-lived, i.e. they oscillate without loosing much energy throughradiation for very long periods of time. In addition, each from this infinite number of ’almostconserved’ quantities, when evaluated on these breather-like field configurations, does vary intime but in a steady way by oscillating between two fixed values. For these reasons we havenamed this phenomenon quasi-integrability . The mechanisms responsible for these remarkableproperties are not fully understood yet, but we believe they will play a very important rolein the study of many non-linear phenomena. Since exactly integrable theories are rare andin general do not describe realistic physical phenomena, the quasi-integrable theories may be1ery useful in their description of more realistic physical processes.In this paper we report some results of our numerical study of breather like field con-figurations in a (1 + 1)-dimensional theory of a real scalar field φ subjected to a potentialwhich is a deformation of the sine-Gordon potential. This theory has already been consideredin one of our previous papers [4], and the deformed potential depends upon two free para-meters ε and γ . The parameter ε measures the deformation of the potential away from thesine-Gordon potential. The γ -parameter, when different from zero, makes the potential notsymmetrical under the reflection φ → − φ . In [2, 4] we have argued that the phenomenonof quasi-integrability may be related to some properties of the two-soliton and breather fieldconfigurations under a very specific space-time parity transformation. When the field φ , eva-luated on such configurations, is odd under this parity transformation, we have an infinitenumber of quasi-conserved quantities, i.e. quantities which are asymptotically conserved inthe case of two-soliton solutions and oscillate in time for breather-like configurations. In orderto have this property the potential has to be even under the change φ → − φ . Thus, wewould expect the cases when γ (cid:54) = 0 to be less integrable than the cases whith γ = 0. Ournumerical simulations do confirm this expectation, but we also observe an effect which hadnot been foreseen. Due to the way we build our initial field configuration, for the numericalsimulations, from the exact sine-Gordon breather solution, the initial kinetic energy decreaseswith the increase of the ε -parameter, and so does the amplitude of oscillations of the resultingbreather-like configurations in the deformed theory. This makes this quasi-breather field tooscillate in a region where the deformed potential differs little from the sine-Gordon poten-tial. So, we find that we can have very long lived breather-like fields for theories which are(globally) large deformations of the integrable sine-Gordon model.The paper is organized as follows. In section 2, for completeness, we present our ideas aboutquasi-integrability based on a anomalous quasi-zero curvature condition (Lax equation), andgive the algebraic and dynamical arguments of why the properties of the initial field configu-rations under this space-time parity transformations are important for the quasi-integrabilityconcept. In section 3 we present the results of our numerical simulations. They have involvedusing the 4th order Runge-Kutta method to simulate the time dependence of field configura-tions which would allow us to determine and study various properties of breather-like solutionsof the full equations of motion of our models for several choices of values of the parameters ε and γ characterizing the potential. In section 4 we present our conclusions. In this paper we report some results of our numerical simulations to study breather-likesolutions of one of such quasi-integrable theories which corresponds of a particular deformation2f the sine-Gordon model introduced in [4]. It is a (1 + 1) dimensional theory of a real scalarfield φ described by the Lagrangian L = 12 (cid:2) ( ∂ t φ ) − ( ∂ x φ ) (cid:3) − V ( φ ) , (2.1)where the potential depends on two real parameters ε and γ and is given by V ( φ ) = 12 [1 + εφ ( φ − γ )] c (1 − εγφ ) sin [ ψ ( φ )] , (2.2)where ψ ( φ ) = c φ (cid:112) ε φ ( φ − γ ) . (2.3)and c = (cid:112) ε π ( π − γ ) . (2.4)Note that for ε = 0, one gets c = 1, ψ = φ , and the potential (2.2) reduces to thesine-Gordon potential V SG = 12 sin ( ψ ) . (2.5)Thus, the theory (2.1), for ε = 0, reduces to the sine-Gordon model defined by the Lagrangian L SG = 12 (cid:2) ( ∂ t ψ ) − ( ∂ x ψ ) (cid:3) − V SG . (2.6)The vacua of the sine-Gordon theory is obviously given by the constant field configurations ψ = n π , with n integer. The theory (2.1) also has infinitely many degenerate vacua but notequaly spaced like in the sine-Gordon case. However, the parameter c given in (2.4), waschosen to preserve two of these vacua. Indeed, ψ ( φ = 0) = 0 and ψ ( φ = π ) = π . Theparameter γ is important in our analysis of the quasi-integrability properties of the theories(2.1). Note that the potential (2.2) is even under the transformation φ → − φ for the case γ = 0, i.e. V γ =0 ( − φ ) = V γ =0 ( φ ) but not otherwise. As we will discuss below the quasi-integrability properties is favoured in the cases when γ = 0. In figure 1 we show the potential(2.2) for some values of ε and γ .The potential (2.2) was introduced in [4] using the techniques of [5, 6] based on ideas ofself-dual or BPS solutions. Indeed, the static one-soliton solutions of the sine-Gordon model(2.6) given by ψ = 2 ArcTan (cid:0) e ± x (cid:1) (2.7)satisfy the BPS equation ∂ x ψ = ± (cid:112) V SG (2.8)3 φ V ( φ ) V ( φ ) φ Figure 1:
Plot of the potential V ( φ ), given in (2.2), as a function of φ for ε = 0 . ε = 0 .
01 (dashed line), ε = 0 . ε = 0 . γ = 0, and those on the right to γ = 0 .
2. Note that V ( φ )is invariant under the change φ → − φ for the case γ = 0, but not otherwise. In addtion,the vacua φ = 0 and φ = π are common to all values of ε and γ . For ε (cid:54) = 0 the peaks ofthe potential grow in height, compared to those for ε = 0, for | φ | > π , irrespective ofthe value of γ . In fact, any static solution of the first order BPS equation (2.8) is a solution of the secondorder Euler-Lagrange (sine-Gordon) equation following from (2.6). If one now introduces afield transformation ψ ( φ ), it follows that the new field φ satisfies the BPS equation ∂ x φ = ±√ V (2.9)with the potential being given by V = V SG (cid:16) d ψd φ (cid:17) . (2.10)The potential (2.2) has been obtained from (2.10) by the field transformation (2.3). Itthen follows that the static solutions of (2.9) are solutions of the theory (2.1). Indeed, thestatic one-soliton solutions of (2.1) are obtained from (2.7) by applying the transformation(2.3). The transformation ψ ( φ ) maps BPS solutions of the sine-Gordon model (2.6) into BPSsolutions of the theory (2.1). Note, however, that in general, a given solution of the secondorder equation of motion of the sine-Gordon model is not necessarily mapped into a solutionof the second order Euler-Lagrange equation corresponding to (2.1).The concept of quasi-integrability does not really depend upon the fact that the quasi-integrable theories are obtained from the integrable ones by field transformations of the typedescribed above. However, many aspects of our analysis get simplified by using such a connec-tion between integrable and non-integrable theories. In particular, the initial configurations4sed in our numerical simulations for breathers, have been obtained by applying the fieldtransformation (2.3) to the exact breather solutions of the sine-Gordon model.As described in [2, 4] our concept of quasi-integrability involves a connection A µ satis-fying an anomalous zero-curvature condition. Indeed, let us consider the connection or Laxpotentials given by A + = 12 (cid:20)(cid:0) ω V − m (cid:1) b − i ω d Vd φ F (cid:21) ,A − = 12 b − − i ω ∂ − φ F , (2.11)where we have used light-cone variables x ± = 12 ( t ± x ) with ∂ ± = ∂ t ± ∂ x and ∂ + ∂ − = ∂ t − ∂ x ≡ ∂ . (2.12)The quantities b n and F n appearing in (2.11) are generators of the sl (2) loop algebra definedas b m +1 = λ m ( T + + λ T − ) , F m +1 = λ m ( T + − λ T − ) , F m = 2 λ m T , (2.13)where λ is the so-called spectral parameter of the loop algebra, and T , ± are the generators ofthe finite sl (2) algebra: [ T , T ± ] = ± T ± , [ T + , T − ] = 2 T . (2.14)It is then easy to see that the curvature of the connection (2.11) is given by F + − ≡ ∂ + A − − ∂ − A + + [ A + , A − ] = X F − i ω (cid:20) ∂ φ + ∂ V∂ φ (cid:21) F (2.15)with X = i ω ∂ − φ (cid:20) d Vd φ + ω V − m (cid:21) . (2.16)Note that the Euler-Lagrange equation following from (2.1) for a general potential V is givenby ∂ φ + ∂ V∂ φ = 0 . (2.17)Thus, the term proportional to the Lie algebra generator F vanishes when the field con-figurations satisfy the equation of motion (are ’solutions’ of the theory). For the case of the5ine-Gordon model potential V sg = mω [1 − cos ( ω φ )] = 2 mω sin (cid:16) ω φ (cid:17) ; (2.18)the remaining term in (2.15), i.e. the anomaly X given in (2.16), vanishes. In such a case thecurvature (2.15) vanishes for sine-Gordon solutions and this is what makes the sine-Gordonmodel integrable. For the potential (2.2) however, the anomaly X does not vanish irrespectiveof the choice of values of the parameters ω and m (except for the trivial case ω = 0).The infinite number of quantities conserved asymptotically can be constructed using thetechniques adapted from those of the integrable field theories. This can be done as follows.We perform the gauge transformation [4] A µ → a µ = g A µ g − − ∂ µ g g − with g = exp (cid:34) ∞ (cid:88) n =1 ζ n F n (cid:35) . (2.19)The parameters ζ n in g can then be chosen recursively starting from n = 1 onwards in such away that the component a − of the transformed connection has only terms in the direction ofthe generators b m +1 . So these terms generate an infinite dimensional abelian subalgebra ofthe sl (2) loop algebra.Note that due to the non-vanishing anomaly X the component a + has also terms in thedirection of the generators b m +1 and F m as well. For an integrable theory, like the sine-Gordonone, the anomaly X does vanish and the terms proportional to F m in a + vanish too, and thewhole connection can be made to lie in the abelian subalgebra generated by b m +1 . In thegeneral case, i.e. when the anomaly does not vanish, the transformed curvature becomes F + − → g F + − g − = ∂ + a − − ∂ − a + + [ a + , a − ] = X g F g − , (2.20)where we have used the equation of motion (2.17). Note that the commutator of any b m +1 with any given F n produces terms proportional to the F m generators only. Therefore, forevery component of the transformed curvature (2.20) in the direction of a given b m +1 we getan equation of the form ∂ + a (2 n +1) − − ∂ − a (2 n +1)+ = X γ (2 n +1) n = 0 , , , . . . (2.21)with a (2 n +1) ± and γ (2 n +1) being the coefficients of the generators b m +1 in the expansion of a ± and g F g − , respectively, in terms of the elements of the basis of the sl (2) loop algebra.The relations (2.21) constitute an infinite number of anomalous conservation laws. Indeed,6y re-expressing them in the x and t components (see (2.12)) one gets the relations d Q (2 n +1) d t = − α (2 n +1) , (2.22)where Q (2 n +1) ≡ (cid:90) ∞−∞ dx a (2 n +1) x , α (2 n +1) ≡ (cid:90) ∞−∞ dx X γ (2 n +1) . (2.23)The charges Q (2 n +1) are not conserved due to the non-vanishing anomaly X . They would, ofcourse, be conserved in an integrable theory like the sine-Gordon one for which X = 0.Note, however, that traveling solutions i.e. those which can be set at rest by a (1 + 1)dimensional Lorentz transformation the charges Q (2 n +1) are conserved. To see this we observethat in the rest frame the charges are obviously x -dependent only, and so from (2.22) one gets α (2 n +1) = 0. But from (2.21) one finds that X γ (2 n +1) is a pseudo-scalar in (1 + 1) dimensions,and so α (2 n +1) vanishes in any Lorentz frame. Therefore, for traveling solutions like the one-soliton solutions, the charges Q (2 n +1) are conserved even in non-integrable theories.Next we note a striking property that helps us to define what we mean by a quasi-integrabletheory . For some very special subsets of solutions of the theory (2.1) the charges Q (2 n +1) satisfywhat we call a mirror symmetry . For any one of the solutions in such a subset one can find aspecial point ( t ∆ , x ∆ ) in space-time, and define a parity transformation around this point: P : (cid:0) ˜ x, ˜ t (cid:1) → (cid:0) − ˜ x, − ˜ t (cid:1) with ˜ x = x − x ∆ ˜ t = t − t ∆ , (2.24)The field φ corresponding to such a solution is odd under such parity, i.e. φ → − φ + const . (2.25)To find the implications of this observation we combine our parity transformation with thefollowing order two automorphism of the sl (2) loop algebra:Σ ( b n +1 ) = − b n +1 , Σ ( F n ) = − F n , Σ ( F n +1 ) = F n +1 (2.26)to build a ZZ transformation Ω ≡ P Σ, as the composition of a space-time and internal ZZ transformations. It turns out that the A − component of the connection (2.11) is odd undersuch ZZ transformation, i.e. Ω ( A − ) = − A − . This fact can be used to show that the groupelement g used to perform the gauge transformation (2.19) is even, i.e. Ω ( g ) = g . Then, onecan use this fact to show that the factor γ (2 n +1) in the integrand of α (2 n +1) is odd under thespace-time parity, i.e. P (cid:0) γ (2 n +1) (cid:1) = − γ (2 n +1) . More details of this reasoning can be found in[4]. 7f we now assume that the potential V ( φ ) in (2.1) is even under the parity when evaluatedon the special solutions satisfying (2.25), i.e. P ( V ) = V , then it follows that the anomaly X ,given in (2.16), is also even, i.e. P ( X ) = X . Therefore we get that (cid:90) ˜ t − ˜ t dt (cid:90) ˜ x − ˜ x dx X γ (2 n +1) = 0 , (2.27)where the integration is performed on any rectangle with center in ( t ∆ , x ∆ ), i.e. ˜ t and ˜ x areany given fixed values of the shifted time ˜ t and space coordinate ˜ x , respectively, introducedin (2.24). Now, by taking ˜ x → ∞ , we conclude that the charges (2.23) satisfy the followingmirror time-symmetry around the point t ∆ : Q (2 n +1) (cid:0) t = ˜ t + t ∆ (cid:1) = Q (2 n +1) (cid:0) t = − ˜ t + t ∆ (cid:1) n = 0 , , , . . . (2.28)That is a remarkable property for the special subsets of solutions satisfying (2.25) and be-longing to a theory of type (2.1) with an even potential under the parity (2.24). Such subsetof solutions defines our quasi-integrable theory. For the case of two-soliton solutions one notethat by taking the limit ˜ t → ∞ , one gets that the charges are asymptotically conserved, i.e.have the same values before and after the scattering.For the case of sine-Gordon theory it is true that for any two-soliton solution or breathersolution it is possible to find a point in space-time ( t ∆ , x ∆ ), such that the solution is oddunder a parity transformation around such point. Let us now consider the theory (2.1) withthe potential (2.2) and expand its solutions in powers of the parameter ε around a givensolution of the sine-Gordon model φ ( − )0 which is odd under the parity φ = φ ( − )0 + ε φ + ε φ + . . . . (2.29)We now split the higher order solutions in even and odd parts as φ ( ± ) n ≡
12 (1 ± P ) φ n (2.30)It turns out that for the case where γ = 0, the equations of motion for the first order solutionare of the form ∂ φ (+)1 + ∂ V∂φ | ε =0 φ (+)1 = 0 , (2.31) ∂ φ ( − )1 + ∂ V∂φ | ε =0 φ ( − )1 = f (cid:16) φ ( − )0 (cid:17) That means that the odd part of the first order solution φ ( − )1 satisfies a non-homogeneous8quation and so it can never vanishes. On the other hand the even part φ (+)1 satisfies ahomogeneous equation and so it can vanish. In fact, if φ is a solution, so is φ − φ (+)1 = φ ( − )1 .Therefore, one can always choose a first order solution which is odd under the parity. If onemakes such choice then it turns out that the second order solution has similar properties,i.e. φ ( − )2 satisfies a non-homogeneous equation and φ (+)2 a homogeneous one. Then one canagain choose the second order solution to be odd, and the process repeats in all orders. Fora detailed account of that please see [4]. Such argument works for the potential (2.2) with γ = 0 but not otherwise. Therefore, we can say that the theory (2.1) with the potential (2.2)with γ = 0 possesses subsets of solutions which constitute a quasi-integrable theory . Thoseare the facts that we want to check with our numerical simulations which we now explain. Our numerical simulations were performed using the 4th order Runge-Kutta method of simu-lating time evolution. As in [4] we experimented with various grid sizes and numbers of pointsand most simulations were performed on lattices of 10001 lattice points with lattice spacingof 0.01 (so they covered the region of (-50.0, 50.0). Time step dt was 0.0001.The breather-like structures were placed at x ∼ ± .
00 from theirpositions hence at the edges of the grid the fields resembled the vacuum configurations whichwere modified only by waves that were emitted during the scattering.At the edges of the grid ( i.e. for 49 . < | x | < .
00) we absorbed the waves reaching thisregion (by decreasing the time change of the magnitude of the field there).In consequence, the total energy was not conserved but the only energy which was absorbedwas the energy of radiation waves. Hence the total remaining energy was effectively the energyof the field configuration which we wanted to study.To start our simulations we took a breather configuration for the sine-Gordon model (3.1)and then perfomed the change of variables (2.3) to obtain the corresponding φ field. We thenused this field and its derivative at t = 0 as the initial conditions for the simulations.The exact breather solution of the sine-Gordon model (2.6) is given by [7] ψ = 2 ArcTan (cid:20) √ − ν ν sin Γ I cosh Γ R (cid:21) , (3.1)where v is the speed of the breather, ν is its frequency ( − < ν <
1) andΓ R = √ − ν ( x − v t ) √ − v , Γ I = ν ( t − v x ) √ − v . (3.2)9n all our simulations we looked at the time dependence of breather-like field configurationsinitially at rest, i.e. with v = 0. Therefore, the initial configuration of the breather at t = 0,with v = 0 is ψ | t =0 = 0 dψdt | t =0 = 2 √ − ν cosh (cid:0) √ − ν x (cid:1) (3.3)The input for our program is the initial configuration of the φ -field defined by the transfor-mation (2.3), and so φ | t =0 = 0 dφdt | t =0 = (cid:20) dφdψ dψdt (cid:21) | t =0 = 1 (cid:112) ε π ( π − γ ) 2 √ − ν cosh (cid:0) √ − ν x (cid:1) (3.4)From (2.1) we see that the initial energy of the breather-like configuration of the model was E = 2 (cid:90) ∞−∞ dx (cid:20) (cid:2) ( ∂ t φ ) + ( ∂ x φ ) (cid:3) + V ( φ ) (cid:21) (3.5)The factor 2 in front of the integral was put to match the definition of the energy in thenumerical code. For the initial configuration (3.4) we have dφdx | t =0 = 0, and V ( φ = 0) = 0 (see(2.2)), and so the initial energy was E = 8 √ − ν [1 + ε π ( π − γ )] (3.6)Note that the energy of the initial configuration has the same ν -dependence as the sine-Gordonbreather, but it is re-scaled by the factor 1 /c , with c given by (2.4), and so it decreases withthe increase of the deformation parameter ε . This rescaling factor and its decrease with ε hasan interesting effect as we will demonstrate in the discussions of the simulations.We have performed several simulations for different values of the frequency of the breather i.e. ν in (3.1) and for various values of ε and γ . For some of these simulations we have alsocalculated the anomaly of the first non-trivial quasi-conserved charge given in (2.23), namely α (3) and Q (3) . We have chosen in the Lax potentials (2.11) the parameters as ω = 2 and m = 1. The reason for this choice was that those are the values that make the sine-Gordonpotential (2.18), for which the anomaly (2.16) vanishes, equal to (2.5) when ε is set to zero.Then from (2.21) we find that γ (3) = i ∂ − φ (3.7)Thus, using (2.23) and (2.16), we see that α (3) = − (cid:90) ∞−∞ dx ∂ − φ ∂ − φ (cid:20) d Vd φ + 4 V − (cid:21) , (3.8)10e have also computed the so-called integrated anomaly given by (see (2.22)) β (3) = − (cid:90) tt dt (cid:48) α (3) = Q (3) ( t ) − Q (3) ( t ) , (3.9)where t is the initial time of the simulation, usually taken to be zero.In the table below we summarise the main features of the simulations we had performed:Figure ε ν γ Initial Energy (eq.(3.6)) Breather behaviourFig. 2 0 .
01 0 . .
01 0 . . .
01 0 . .
01 0 . . .
01 0 . . .
01 0 .
95 0 2.27 long livedFig. 8 0 .
01 0 .
95 0 . .
01 0 .
95 0 . .
01 0 .
95 0 . . . . . . . . . . . . . . . γ = 0 tend to live longer when compared to similar breathers forthe cases where γ (cid:54) = 0. In order to see this more clearly, let us look at the simulations shownin Figs. 13, 14 and 15, all corresponding to ε = 0 . ν = 0 .
5. In Fig. 13 correspondingto the case γ = 0 we see that the energy almost goes to a constant value after the initialstabilization of the solution and the lifetime of the breather is extremely long. The simulationwas stopped after 1 . × units of time and the energy was still almost constant (in fact itdecreased all the time but extremely slowly). In Fig. 14, corresponding to the case γ = 0 . . × units of time of simulation. So,despite the fact that the fall of the energy is slow we see the γ (cid:54) = 0 effect playing an im-portant role in increasing the speed of the decrease. Now, in Fig. 15, corresponding to thecase γ = 0 .
5, we see that the energy drops much faster, and the breather starts moving after11bout 9 × units of time, and it has bounced off the edges of the grid twice during thesimulation. So, by increasing γ the phenomenon we have called quasi-integrablility , seems tohave almost disappeared. Notice also that for smaller values of ε the effect of γ (cid:54) = 0 is notso visible. However, we do notice in Figures 2 and 3, corresponding to ε = 0 .
01 and ν = 0 . γ = 0 . γ = 0. The same effect isvisible in Figures 4, 5 and 6, corresponding to ε = 0 .
01 and ν = 0 .
5. As the values of γ areincreased from 0 . . .
5, the life-times of the breathers decrease. The sameeffect is not seem however in figures 7, 8, 9 and 10, corresponding to the case ε = 0 .
01 and ν = 0 .
95. We cannot see much difference in the behaviour of their energies as the value of γ is increased. They seem to have stabilized quite well after 3 × units of time. These casesmight be feeling the influence of the second effect which we will now describe.Looking at the table above one can easily spot a correlation between the values of theenergy (3.6) of the initial configuration (3.4) used in the simulations, and the lifetime of thebreathers. The higher the energy the shorter the lifetime of the breathers. In fact, we havestarted all our simulations with φ = 0 on all sites of the grid. The energy (3.6) correspondsto the total kinetic energy given to the initial configuration. Note that the potential energyis zero because V ( φ = 0) = 0, and the elastic potential energy is also zero because dφdx = 0,on all sites of the grid at t = 0. So, the smaller is the initial kinetic energy, the smaller isthe maximal value the φ -field can reach during the oscillations of the breather. Indeed, bydeparting from φ = 0 the potential energy V increases, as seen from the plot of the potentialin Fig. 1. But if the value of φ remains small the breather oscillates only inside the part ofthe well of the potential around φ = 0 where V varies very little with ε . So, the breather canstay close to the breather solution of the sine-Gordon model which is integrable. Therefore,one would expect such a breather to live longer. Indeed, looking at Figures 7 - 16 one observesthat for energies smaller than ∼ .
3, the amplitude of oscillations of the field φ is never largerthan 0 .
6. For these values of amplitudes one sees from Fig. 1 that the field φ does not reachregions where the potential departs significantly from the sine-Gordon potential. In all thesecases the breathers live quite long, except for the case of Figure 15 where γ = 0 .
5, and asdiscussed above the lack of good parity properties of the solution makes it short-lived. Thedependence on γ only affects the initial drop of energy but is not very significant from thepoint of view of whether the breather is long-lived or not. This is clearly seen from the casesinvolving initially high ν as shown in figures 7, 8, 9 and 10. All these cases correspond to γ = 0, 0.3, 0.5 and 0.7, respectively. We note that all these breathers are long-lived. Theirinitial energies increase (but very little) with γ and, in the initial evolution decrease more forlarger γ but then the decrease slows down and the breathers appear to be very long-lived. Itwould be interesting to check whether, at some stage, their energies ’cross’ but the decreaseis so slow and the gaps are still large enough that one would have to wait extremely long toobserve this, if it ever happens, so this is not practical.12he only exception to our observation above is the case shown in Fig. 15 where γ = 0 . ν is smaller) and the effect of the lack of parity properties in this case, as discussedabove, plays an important role and makes the breather to die faster. The fact that the energy(3.6) decreases with the increase of the value of ε , makes it possible for us to find very longlived breathers for large values of ε . Note that the increase of the frequency ν of the initialconfiguration also plays a role in favouring the long lifetime of breathers since it decreases thevalue of the initial energy (3.6).We can also observe a correlation among the anomaly α (3) , given in (3.8), and the integratedanomaly β (3) given in (3.9), with the life-time of the breather solutions. In Figures 7, 8, 12and 16, where the breathers are long lived, we see that the anomaly α (3) oscillates steadilywithin a fixed interval. The integrated anomaly β (3) on the other hand presents a very slowdrift of the order of one part in 10 for 10 -10 units of time. This is quite a long range of timeintegration, and it could well be inside the numerical errors which are difficult to estimate inthese cases. In Figures 2 and 3, where the breathers are short lived we see that the anomaly α (3) does not oscillate within a fixed interval and the integrated anomaly β (3) does vary a lot.Thus, there is indeed a correlation between long lived/short lived breathers and well/badlybehaved anomalies. However, the effect of the γ parameter is not very visible in the behaviourof the anomalies. It seems that the other effect discussed above in connection with the lowinitial kinetic energy seems to be predominant in these cases.We have also observed that the energy of the breather-like solutions, after they havestabilized, seems to depend on the frequency in a way very similar to the exact sine-Gordonbreather, i.e. E ∼ √ − ν . In Figure 17 we show, for the case of ε = 0 . ν = 0 . γ = 0 . φ at x = 0, in the left figure at the beginning of the simulationand in the right figure the same time dependence of the field at a much later time. From thesetwo plots it is very clear that at first the breather oscillates with a period of ∼ T = 6 . ∼ T = 6 . ν ∼ . ν ∼ . E in = 1 . E fin = 1 . . / . . E ∼ √ − ν , so let us check what would have happened had we used this fact to estimatethe energies in this case too (as our field configuration resembles the sine-Gordon model’sbreather so well). We would have had (cid:18) E in E fin (cid:19) = 1 − ν in − ν fin . (3.10)13n our case the left and right hand sides of this formula are given by(1 . = 1 . , (3.11)1 − (0 . − (0 . = 1 . ∼ ε . In such caseswe had the initial drop of the energy (like for larger values of ε ) followed by a motion ofthe breather towards the boundaries with a reflection of it from the boundaries (producing afurther sharp drop of the energy at each reflection). This is clear from the simulation shownin figure 4 (see figures (c) and (d))Given the irregularity of the energy drop and the motion of the breathers it makes littlesense to perform a comparison of the energy loss to the increase of the frequency of theoscillation that we have performed for larger values of ε . So if we want to perform a similarcalculation we have to restrict our attention to the initial (non-moving) times of the breather,( i.e. consider it only for t up to ∼ t up to this value. Of coursethe plot of the variation of the field at x = 0 is still not very clear so in the last two figuresof Fig. 4, namely (e) and (f) we present the variation of the field at the beginning of thisinterval and at the end of it. The plots cover the range of t of 200 units and they show thatat the beginning of the simulation the system performed ∼ . t ∼ . . . . . . ) = 1 . − (0 . − (0 . = 1 . . (3.12)Hence again we have results in good agreement with our expectations.In figures 5 and 6 we present plots of the time dependence of the energy and the field at x = 0 for ε = 0 .
01 but this time for larger values of γ ; namely γ = 0 . γ = 0 . γ = 0 except that the decrease ofenergy is progressively greater. In fact, in each case, the energy start decreasing quite fastbut then the decrease slows down. Again, the breathers start moving and so, again, we could14alculate the increase of frequencies of oscillations and, like before, compare our expressionswith the decrease of energies.In the case of γ = 0 . (cid:18) . . (cid:19) ∼ . . (3.13)The frequencies are ν i = 0 . ν f = 0 . − ν i − ν f = 1 . γ = 0 . . . ν i = 0 . ν f = 0 . (cid:18) . . (cid:19) = 2 . − ν i − ν f = 1 . In this paper we have performed further studies of quasi-integrability based on the observationin [2], [3] about the behaviour of the anomaly X of the curvature (2.15) of the Lax poten-tials, which distinguishes integrable models from non-integrable ones. We have seem thatthe anomaly integrated in time (see for instance (3.9)) also vanishes in some non-integrablemodels for field configurations which possess the parity symmetries discussed in Section 2.This observation was originally made in some very specific models and here we have triedto assess its general validity. So, in [4] we constructed three classes of models (one withsymmetry, one without it and one (dependent on two parameters)) which would allow usto interpolate between the two. Our results have confirmed the validity of our assumption(and so extended the class of models in which our observation holds) and have also allowedus to study the way the anomaly varies as we move away from the models with this extrasymmetry. These results were first tested in great detail for the scattering of kinks of thesemodels. Of course, in such scatterings the kinks were interacting with each other only over15ery short periods of time (when they were close to each other). So we decided to look also atthe systems involving breather-like structures on which the kinks and anti-kinks, being boundinto breathers, interact with other all the times. In our previous work [4] we have only glancedat such configurations. As the breather-like configurations depend on many parameters in thispaper we have concentrated our attention at looking at them in detail, in particular when thesymmetry is present and when it is not. As expected, we have found that the symmetry helpsa lot in the validity of ideas of quasi-integrability. When the symmetry is present the energydecrease is much reduced and the configurations resemble, in their behaviour, the sine-Gordonbreathers. When the symmetry is broken the breathers decay quite rapidly and the range ofvalidity of quasi-integrability is much reduced.However, the symmetry is only one of many topics to investigate for the breather-likeconfigurations. We have also looked at the way the decay takes place and the dependence ofthis behaviour on various parameters of the model. One of the most interesting outcomes of oursimulations is the understanding of the way the decay takes place. The breathers increase thefrequency of their oscillations. This is quite clear from the energy of our initial configuration(3.6) but it is amazing to see as the breather-like field looses its energy this formula stillholds true, as only ν in it increases. In our simulations some breather-like configurationsstarted moving and lost their energies also by reflecting from the edges of the grid. For themthe frequency increased even more so that their total energy (consisting of the energy of theoscillation and the energy of the motion was comparable to the final energy of the field, andmuch lower than the original energy).Finally our work has lead to the discovery of the existence of many long lived breather-likefields a long way away ( i.e. for large perturbations) from the sine-Gordon model. This wasparticularly true for the cases with symmetry and so it does extend the range of validity ofquasi-integrability. Acknowledgements:
The work reported in this paper was started when the authors were“researching in pairs” at the Matematisches Forschunginstitute in Oberwolfach. They wouldlike to thank the Oberwolfach Institute for its hospitality. The work was finished when WJZwas visiting the S˜ao Carlos Institute of Physics of the University of S˜ao Paulo (IFSC/USP).His visit was supported by a grant from FAPESP. He would like to thank FAPESP for thisgrant and the IFSC/USP for its hospitality. LAF’s work was partially supported by CNPq(Brazil). 16 eferences [1] S. R. Coleman, “The Quantum Sine-Gordon Equation as the Massive Thirring Model,”Phys. Rev. D , 2088 (1975).S. Mandelstam, “Soliton Operators for the Quantized Sine-Gordon Equation,” Phys. Rev.D , 3026 (1975).C. Montonen and D. I. Olive, “Magnetic Monopoles as Gauge Particles?,” Phys. Lett. B , 117 (1977).C. Vafa and E. Witten, “A Strong coupling test of S duality,” Nucl. Phys. B , 3(1994) [hep-th/9408074].N. Seiberg and E. Witten, “Monopoles, duality and chiral symmetry breaking in N=2supersymmetric QCD,” Nucl. Phys. B , 484 (1994) [hep-th/9408099].[2] L. A. Ferreira and W. J. Zakrzewski, “The concept of quasi-integrability: a concreteexample,” Journal of High Energy Physics, JHEP , 130 (2011); [arXiv:1011.2176[hep-th]].[3] L. A. Ferreira, Gabriel Luchini and W. J. Zakrzewski, “The concept of quasi-integrabilityfor modified non-linear Schr¨odinger models” Journal of High Energy Physics,
JHEP
103 (2012); [arXiv:1206.5808 [hep-th]][4] L. A. Ferreira and W. J. Zakrzewski, “Numerical and analytical tests of quasi-integrabilityin modified Sine-Gordon models,” Journal of High Energy Physics,
JHEP , 058(2014); [arXiv:1308.4412 [hep-th]].[5] C. Adam, L. A. Ferreira, E. da Hora, A. Wereszczynski and W. J. Zakrzewski, “Someaspects of self-duality and generalised BPS theories,” Journal of High Energy Physics,
JHEP , 062 (2013); [arXiv:1305.7239 [hep-th]].[6] D. Bazeia, L. Losano, J. M. C. Malbouisson and R. Menezes, “Classical behavior ofdeformed sine-Gordon models,” Physica D , 937 (2008) [arXiv:0708.1740 [nlin.PS]].[7] L. A. Ferreira and W. J. Zakrzewski, “A Simple formula for the conserved charges of soli-ton theories,” Journal of High Energy Physics,
JHEP , 015 (2007) [arXiv:0707.1603[hep-th]]. 17
5 5.5 6 6.5 7 0 20000 40000 60000 (a) -2 -1 0 1 2 0 20000 40000 60000 (b) -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 20000 40000 60000 (c) -30 -25 -20 -15 -10 -5 0 0 20000 40000 60000 (d)
Figure 2: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).18 (a) -2 -1 0 1 2 0 10000 20000 30000 40000 50000 (b) -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 10000 20000 30000 40000 50000 (c) -35 -30 -25 -20 -15 -10 -5 0 0 10000 20000 30000 40000 50000 (d) Figure 3: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0 .
3. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).19 (a) -1.5 -1 -0.5 0 0.5 1 1.5 0 10000 20000 30000 40000 (b) (c) -1.5 -1 -0.5 0 0.5 1 1.5 0 200000 400000 600000 800000 (d) -1.5 -1 -0.5 0 0.5 1 1.5 0 50 100 150 200 (e) -1.5 -1 -0.5 0 0.5 1 1.5 47800 47850 47900 47950 48000 (f) Figure 4: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0 .
0. The plots show the time dependence of(adimensional units): (a) and (c) the energy (3.5); (b) and (d) the field φ at position x = 0in the grid. The figures (a) and (c) correspond to the moment just before when the quasi-breather has started moving. Figures (e) and (f) show a blow-up of the amplitude of the field φ at the beginning of simulation (e) and at the time just before the soliton has started moving(f). 20
4 4.5 5 5.5 6 0 200000 400000 600000 (a) -1.5 -1 -0.5 0 0.5 1 1.5 0 200000 400000 600000 (b)
Figure 5: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0 .
2. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid.21
2 3 4 5 6 0 200000 400000 600000 (a) -1.5 -1 -0.5 0 0.5 1 1.5 0 200000 400000 600000 (b)
Figure 6: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0 .
5. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid.22 .2705 2.2712.2715 2.2722.2725 2.2732.2735 2.274 0 50000 100000 150000 200000 (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50000 100000 150000 200000 (b) -0.02 -0.01 0 0.01 0.02 0.03 0 50000 100000 150000 200000 (c) -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0 50000 100000 150000 200000 (d) Figure 7: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 .
95 and γ = 0. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).23 .3105 2.3112.3115 2.3122.3125 2.3132.3135 2.314 0 50000 100000 150000 200000 (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50000 100000 150000 200000 (b) -0.02 -0.01 0 0.01 0.02 0.03 0 50000 100000 150000 200000 (c) -0.3 -0.2 -0.1 0 0.1 0 50000 100000 150000 200000 (d) Figure 8: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 .
95 and γ = 0 .
3. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).24 .3375 2.3382.3385 2.3392.3395 2.342.3405 2.341 0 20000 40000 60000 80000 100000 (a) -0.5 0 0.5 0 20000 40000 60000 80000 100000 (b) -0.02 -0.01 0 0.01 0.02 0.03 0 20000 40000 60000 80000 100000 (c) -0.15 -0.1 -0.05 0 0.05 0.1 0 20000 40000 60000 80000 100000 (d) Figure 9: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 .
95 and γ = 0 .
5. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).25 (a) -0.5 0 0.5 0 20000 40000 60000 80000 100000 (b) -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0 20000 40000 60000 80000 100000 (c) -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0 20000 40000 60000 80000 100000 (d) Figure 10: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 .
95 and γ = 0 .
7. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).26 Figure 11: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0. The plot shows the time dependence(adimensional units) of the energy (3.5). 27 (a) -0.5 0 0.5 0 50000 100000 (b)
0 0.5 0 50000 100000 (c) -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 50000 100000 (d)
Figure 12: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by of ε = 0 . ν = 0 . γ = 0. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).28 (a) -0.5 0 0.5 0 500000 1e+06 (b) Figure 13: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by of ε = 0 . ν = 0 . γ = 0. The plots show the time dependence of(adimensional units): (a) the energy (3.5); and (b) the field φ at position x = 0 in the grid.29 Figure 14: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by ε = 0 . ν = 0 . γ = 0 .
2. The plot shows the time dependence(adimensional units) of the energy (3.5). 30 (a) -0.5 0 0.5 0 100000 200000 300000 (b)
Figure 15: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by of ε = 0 . ν = 0 . γ = 0 .
5. The plots show the time dependence of(adimensional units): (a) the energy (3.5); and (b) the field φ at position x = 0 in the grid.31 (a) -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 50000 100000 150000 (b) -0.05 0 0.05 0.1 0.15 0.2 0 50000 100000 (c) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0 50000 100000 (d) Figure 16: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by of ε = 0 . ν = 0 . γ = 0. The plots show the time dependence of(adimensional units): (a) the energy (3.5); (b) the field φ at position x = 0 in the grid; (c)the anomaly (3.8) and (d) the integrated anomaly (3.9).32 -0.5 0 0.5 0 5 10 15 20 25 30 (a) -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 1.1298e+061.12985e+061.1299e+061.12995e+061.13e+06 (b) Figure 17: Breather’s simulation in the theory (2.1) with initial configuration (3.4), and withparameters given by of ε = 0 . ν = 0 . γ = 0 .
0. The plots show the time dependence of(adimensional units): (a) time dependence of the field φ at position xx