some comments on quasi-integrability

Vol. 67 (2011) REPORTS ON MATHEMATICAL PHYSICS No. 2

SOME COMMENTS ON QUASI-INTEGRABILITY1

L. A. FERREIRAInstituto de Fısica de Sao Carlos; IFSC/USP;

Universidade de Sao PauloCaixa Postal 369, CEP 13560-970, Sao Carlos-SP, Brazil

(e-mail: [email protected])

and

WOJTEK J. ZAKRZEWSKIDepartment of Mathematical Sciences,

University of Durham, Durham DH1 3LE, U.K.(e-mail: [email protected])

(Received September 3, 2010)

In this paper we present our preliminary results which suggest that some field theory modelsare ‘almost’ integrable; i.e. they possess a large number of ‘almost’ conserved quantities. Firstwe demonstrate this, in some detail, on a class of models which generalise sine-Gordon modelin (1+1) dimensions. Then, we point out that many field configurations of these models looklike those of the integrable systems and others are very close to being integrable. Finally weattempt to quantify these claims looking in particular, both analytically and numerically, at somelong lived field configurations which resemble breathers.

Keywords: solitons, integrability, quasi-integrability, field theory, stability.

1. IntroductionThe concept of integrability is very hard to define properly for field theories.

In finite-dimensional systems it involves the existence of a number of conservedquantities; so large, that one then completely constrains the dynamics. As field theoriesdescribe systems with infinite number of degrees of freedom, their integrability mustinvolve an infinite number of conserved quantities. However, one can easily missa few of them and still have an infinite number of them. Hence, for field theories,the attempts to describe integrability have involved defining an appropriate Lax pairand generating these conserved quantities through it.The problem is, however, even more complicated if a given theory is not integrable

but, in some sense, being close to being integrable. Would we then have a smallernumber of conserved quantities, or will some (or many) of them be nonconserved,

1Presented by W. J. Zakrzewski at the 42nd Symposium on Mathematical Physics, Torun, 19–22 June, 2010.

[197]

198 L. A. FERREIRA and W. J. ZAKRZEWSKI

i.e. their changes being small (or even very small) but strictly speaking non zero.And will this be true for all fields, or just some of them? And what about fieldconfigurations which are ‘interesting’, whatever this may mean?In this paper we present some of our thoughts on this problem. We will recall

a few observations and use them to try to define the concept of ‘quasi-integrability’ or‘almost integrability’. Our thoughts are based on the observations that if the models in-volve some extended objects like solitons and in their scattering processes such objectsemerge from their scattering regions almost unaffected and the processes generate verylittle radiation, then we can think of the model as quasi-integrable. In fact, the sine-Gordon model, which is integrable possesses solutions which describe the scattering ofsolitons. The solitons scatter but this scattering does not lead to the emission of any ra-diation. This non-emission of radiation is thought of being forbidden by the conservedquantities and so may be considered as being an effective test of the integrability ofthe model. For had we not known that sine-Gordon model is integrable, we could haveconsidered this non-emission of radiation as a practical test of ‘integrability’. But whatfield configurations should we have looked at? Solitonic ones? Or more general ones?When we look at scattering of solitons in higher dimensions (say (2+1)) we

often see very little (or virtually no) radiation. This is true in Ward’s model,some Baby–Skyrme models and their superpositions. Of these the Ward model isintegrable and the others are not but, as Sutcliffe has shown, a superposition of theWard model and the Baby–Skyrme model, from the type of the scattering and theemission of radiation point of view, behaves like the pure Ward model (as long theadmixture of the Baby–Skyrme model is not too large).This suggests that we are on a correct path and in this paper we describe our

attempts to formalise these observations. To do this we need a class of models which istopological, and which possesses solitons. We want this class to include an integrablemodel so that we can compare our results to the results of this model. And, to startoff, we want to look at the problem in (1+1) dimensions as then all the calculationscan be performed more easily. Luckily, a class of such models was presented by D.Bazeia et al. [1] and we discuss them in Section 2. Section 3 recalls some results onthe scattering of solitons (kinks) in these models. Then we present our preliminaryresults on quasi-integrability and test them experimentally on the scattering ofbreather-like structures in the Bazeia’s models. We end with some further comments.

2. Deformed sine-Gordon modelsRecently, D. Bazeia et al. [1] have presented two classes of models which

are generalisations of the sine-Gordon model. These models depend on an integerparameter n, which when n = 2 (for the first class) or n = 1 (the second one),reduce to the sine-Gordon model.The two classes of models differ by their potentials. The potentials for the first

class of models are given by

V (ϕ) = 2λ2

n2tan2 (ϕ)

(1− sinn (ϕ)

)2. (1)

SOME COMMENTS ON QUASI-INTEGRABILITY 199

Note that when n = 2 this expression reduces to the potential of the sine-Gordonmodel, i.e.

V → λ2

8sin2 (2ϕ) .

The potentials of the second class are given by

V (ϕ) = λ2

2n2ϕ2−2n sin2

(ϕn

)(2)

And again, this model reduces to the sine-Gordon one when n = 1, i.e.V → λ2

2 sin2 (ϕ).

Incidentally, although Bazeia et al. consider these models for integer n, it is easyto check that one can consider also noninteger n. In this case all sin(ϕ) have to bereplaced by | sin(ϕ)| and all is OK. This observation allows us, for the first classof models, to take n close to 2 and so consider perturbations of the sine-Gordonmodel.The models possess topological soliton (kink) solutions for any n. They are

given by:• For the first class

ϕ (x, t) = sin−1[exp (2λγ (x − x0 − ut))

1+ exp (2λγ (x − x0 − ut))

] 1n

. (3)

Such solutions satisfy the boundary conditions:

ϕ (x →−∞) −→ 0, ϕ (x →∞) −→ π

2. (4)

It is easy to check that when n = 2 we have the well-known kink solutionsof the sine-Gordon model. Below we show the plots of these kinks forn = 1, . . . , 6.

• For the second class we haveϕ (x, t) = [

2 tan−1 (exp (λγ (x − x0 − ut)))] 1

n . (5)They satisfy the boundary conditions

ϕ (x →−∞) −→ 0, ϕ (x →∞) −→ π1n . (6)

And, again, when n = 1 this expression reduces to the kink solution of thesine-Gordon model.

In this paper we will concentrate on looking only at the type I models. We havestudied various scattering properties of these kinks for n integer or not. Looking atone kink and its properties we have found that:• A kink is stable. If we perturb it, it emits the extra energy and returns to itsproper shape.

• When we scattered a kink on a potential well or a barrier it behaved the sameway independently of whether n = 2 or not. Thus the kinks could scatterthrough the well, get trapped in it or get reflected.


Fig. 1. Solitons for a few values of n.

• From these properties we do not see any difference of whether the kinks areof the integrable model (n = 2) or not.

So what are the differences between n = 2 and n �= 2? They are:• Sine-Gordon model (n = 2) has multi-kink solutions and breathers.• Sine-Gordon model is integrable.• In fact, these two properties may be related. The integrability of the modelallows us to construct these ’extra solutions’.

Had we not known about the integrability of the sine-Gordon what would wehave done?Not clear. Probably we would have looked at two kinks and checked whether

they attract or repel. Then we would have done the same for a kink and ananti-kink. This would have allowed us to determine forces between them. However,would this tell us anything about the integrability?

3. Kink-kink and kink-antikink interactions

The easiest to study are the interactions between 2 kinks for n even; say forn = 2 and n = 4. We can place two kinks at some distance from each other andsee what happens. When we do this we find that kinks repel.Below we present the trajectories of two kinks, initially at rest, seen in simulations

for n = 2 and n = 4. The kinks were initially placed at d = ±6.5.


(a) (b)

Fig. 2. Trajectories of kinks for 2 models: (a) n = 2, (b) n = 4.

It is clear from these plots that the kinks repel. Looking at the plots we do notsee much difference between n = 2 and n = 4.Incidentally, in these and all other our simulations, we have used finite grids and

placed our kinks long way away from the boundaries. Moreover, when the scatteringproduced any radiation, this radiation was absorbed when it reached the boundaries.This was done to reproduce as much as possible the behaviour on infinite gridsand so to eliminate all effects due to the reflection from the boundaries. Hence,when we speak of the decrease of energy we mean by this the decrease due tothis absorption—i.e. the decrease of energy in the region of the kinks, with thedifference being sent out (and absorbed) at the boundary.Of course, similar results were obtained also when we took noninteger n.Next we sent the kinks towards each other with some velocities. In the figure

below we present the trajectories of kinks sent to each other with velocity v = 0.7for n = 2 and n = 2.01. Initially the kinks were placed at d = ±15.5.Again we see a very similar behaviour for both values of n. The kinks clearly

come to each other and then they get repelled from each other. At larger velocitiesthey get closer before their repulsion sends them back. Similar results were obtainedfor other values of d, n and their velocities.Incidentally, the sine-Gordon model (i.e. the model with n = 2) possesses moving

two kink solutions and our results (for n = 2) reproduce them. On the other hand,these moving kinks solutions are known in an explicit form, and because these kinksare described by explicit functions it is often said that the ,,kinks pass through eachother”. This is clearly wrong. When one looks at the energy density of the movingkinks one sees very clearly that they never come on top of each other (the twopeaks of energy never form a double peak); in fact, the functions which describea given kink switch after their interaction.


(a) (b)

Fig. 3. Trajectories of kinks initially sent towards each other in two models: (a) n = 2, (b) n = 2.01.

Next we look at kink-antikink configurations and breathers. In the sine-Gordonmodel we do have breathers and their analytical form is well known. They are infact bound states of a kink and an antikink. This is all well known; what is knownperhaps less is that we can generate breathers by taking a kink and an antikink andplacing them not too close to each other. Then they attract, alter their shape and,as they do that, emit some radiation and become a breather. Interestingly, they donot annihilate but do form a breather. Hence we can use the same procedure forn �= 2 models for which we do not know any breather like states.Below we present our results for the sine-Gordon model. In the Fig. 4(a) we

present the time dependence of the energy of the field configuration which involvedthe kink and antikink initially placed at d = ±3 (Fig. 4(a)) and d = ±5 (Fig. 4(b)).In Fig. 4(c) we present the plot of the position of the kink (or antikink) for x < 0.We note that in all cases we have a very small initial drop of the total energy

of the configuration to the energy of the resultant breather. The final energy of thebreather is given by E = 2E0

√1− ω2 and its frequency ω is related to the initial

extend given by d.Hence, we have repeated the same procedure of generating breathers for other

values of n. In Fig. 5 we present our results for two values of n, namely n = 1and n = 2.9 in which the kink and the antikink were initially placed at d = ±4.0.Our plots present the time dependence of the total energy of the configuration.We note a fundamental difference—for n = 1 the energy seems to stabilise

around some finite nonzero value while for n = 2.9 it goes to zero.We have performed many simulations (for different values of n and different

distances between kinks and antikinks) running them for very long times and have


(a) (b)

(c)

Fig. 4. (a) Total energy for kinks at d = ±3; (b) the same for d = ±5; (c) position of the kink (or antikink)for x < 0.

found similar results; namely, for a range of n, from about ∼ 0.8 to around ∼ 2.8the configuration evolved towards a long-living breather-like state. This state hada fixed energy only for n = 2, for other values the energy decreased but very veryslowly indeed. For larger values of n, or for n very small the system annihilatedand the energy quickly dropped towards zero.This suggests to us that the models for n �= 2 (but close to 2) are quasi-

integrable, i.e. that for them we have long lived breathers and that such timedependent configurations will be a good testing ground of any quantitative claimsabout this quasi-integrability.


(a) (b)

Fig. 5. (a) Total energy for n = 1; (b) total energy for n = 2.9.

4. Quasi-integrabilityHere we exploit the ideas used to demonstrate integrability of models such as the

sine-Gordon one, based on the zero curvature condition or the Lax–Zakharov–Shabatequation [2–4]. Such ideas were used in many formulations of this problem, andwe shall introduce a quasi zero curvature condition for models closely related tothe sine-Gordon model, and construct quasi conserved quantities using techniquessimilar to those described in [5–8].Thus we take a general equation for a scalar field in (1 + 1) dimensions with

a potential V (ϕ),∂2ϕ + ∂V (ϕ)

∂ϕ= 0 (7)

and introduce the following 2× 2 matrices:

T+ =⎛⎝0 1

0 0

⎞⎠ , T− =

⎛⎝0 0

1 0

⎞⎠ , T0 =

12

⎛⎝1 0

0 −1

⎞⎠ . (8)

These matrices satisfy the sl(2) algebra[T3, T±] = ±T±, [T+, T−] = 2T3. (9)

We also introduce the following basis for the corresponding loop algebra, basedon the so-called spectral parameter λ,

b2m+1 = λm(T+ + λT−), F2m+1 = λm(T+ − λT−), (10)and

F2m = 2λmT3. (11)


Then we define the Lax pair potentials as

A+ =12

[Hb1 −

i

ω

dH

dϕF1

], (12)

andA− =

12b−1 −

i

2ω∂−ϕF0, (13)

whereH = ω2V −m2. (14)

Here m2 and ω are constants and ∂± = ∂t ± ∂x are derivatives w.r.t. the lightcone variables.By using the equations of motion (7) we find that the curvature of the potential

is given by∂+A− − ∂−A+ + [A+, A−] = XF1, (15)

whereX = i

21ω

∂−ϕ

[d2H

dϕ2+ ω2H

]. (16)

Note that for the sine-Gordon model, where

V = 116[1− cos (4ϕ)] ,

we see that X = 0, if we choose ω = 4 and m2 = 1. For other models of the class(1), i.e. when n �= 2, there are no choices of the parameters ω and m2 such thatX vanishes. However, we now show that even for X �= 0 and n not too far from2 one can construct an infinite number of quasi-conserved charges.We use the methods of [5–8] to gauge transform the potentials A± into an

abelian subalgebra generated by b2m+1, plus terms proportional to the anomaly Xand lying in the subspace generated by Fn. So we consider the following gaugetransformation

Aμ → aμ = gAμg−1 − ∂μgg−1, (17)where the group element g is of the form

g = exp[ ∞∑

n=1ζnFn

]. (18)

First we transform the A− component of the potential by choosing the parametersζn to cancel the terms in a− proportional to the Fn’s. Then we use the equationsof motion to calculate a+ using the determined ζn’s. We find that

a− =12b−1 +

∞∑n=0

a(2n+1)− b2n+1,

a+ =12Hb1 +

∞∑n=1

a(2n+1)+ b2n+1 +

∞∑n=2

c(n)+ Fn, (19)


with the coefficients c(n)+ involving terms proportional to the anomaly X and its

x−-derivatives. The curvature of the new potential is then given by

∂+a− − ∂−a+ + [a+, a−] = XgF1g−1. (20)

The non-vanishing of the commutator term [a+, a−], due to X, and the anomalyterm itself, i.e. XgF1g

−1, is what prevents the charges from being conserved. Indeed,from (20) one has that

d

dt

∫ ∞

−∞dxax = −

12

∫ ∞

−∞dx

[XgF1g

−1 − [a+, a−]], (21)

where we have used the fact that at |x=±∞= 0, with a± = at ± ax . Calculating theintegrand on the right-hand side of this expression we find

XgF1g−1 − [a+, a−] = b3[iω∂2−ϕX] + b5[

32iω3(∂−ϕ)2∂2−ϕX + iω∂4−ϕX]

+∞∑

n=3d(2n+1)b2n+1 +

∞∑n=2

h(n)Fn. (22)

Since we do not have terms in the direction of b1 it follows that the componentof ax in the direction of b1 leads to a conserved charge, which is in fact a linearcombination of the energy and momentum. The other components of ax lead toquasi-conserved charges.Hence, the first anomaly (i.e. the first correction to the conserved quantity) is

given by the integral of the first term i.e. by

iω

∫ ∞

−∞dx∂2−ϕX. (23)

In the sine-Gordon case (n = 2) we have X = 0, and this expression leads toa conserved quantity. For other values of n the anomaly is nonzero and so we havea correction to this quantity, which is now not conserved. This correction can becalculated explicitly by choosing appropriately ω and m2

Note that we can also calculate other anomalies (given by the terms proportionalto higher powers in λ) Note also that we can make other choices of our gaugepotentials and then perform appropriate gauge transformations.

4.1. Some comments and some numerical studies

Let us summarise our results.Our procedure has provided us with a method of calculating an infinite set of

conserved quantities for n = 2. They are given by what we called the anomaly andare proportional to the increasing powers of λ. As the anomaly (for n = 2) vanishesthey give us the corresponding conserved quantities. In the n �= 2, the anomaly doesnot vanish so the corresponding quantities are not conserved. However, we expectthe anomaly to be small (for n close to 2) and so the corresponding quantities will


be approximately conserved. This, we believe, corresponds to a possible definitionof quasi-integrability.To test our ideas we have also looked at the expressions of the first anomaly

for various field configurations. Our results can be summarised as follows:• Static field configurations do not contribute to the anomaly.• The same holds for ‘separated’ field configurations.• Thus the most interesting to study are ‘breather-likes’ states.For the breather-like states we have found that as their fields are oscilatory (in

time)—so is the anomaly. The expression for the anomaly oscillates a lot, so wehave to look at its ’average’. To get this average we can integrate the anomalyover time. Then the oscillations cancel and we see the drift of its average value.Moreover, the range of the oscillations sets the scale for the value of the average.Below we present our first plots of the anomaly. We note, as expected, that

although the oscillations are quite sizeable the average value changes little. Inthe plots in the next figure we present the time dependence of our quasi-breatherobtained for n = 2.01. The three curves show, respectively, the time dependence ofthe energy, the first anomaly and the integrated anomaly.We have also studied this for other values of n and in the next figure we

show the time dependence of the anomaly and of the time integrated anomaly ofa corresponding quasi-breather for n = 2.7. We note that the results are qualitativelythe same although they exibit some quantitative differences. We plan to report onthis further in the near future.

5. SummaryIn this paper we have presented our preliminary results of our study and our

attempts to define the concept of quasi-integrability. Our work has been based onthe models of Bazeia et al. which are a good laboratory for studying such problems.The reason for this is that all these models have one kink solutions and theirscattering, in all models, is very similar. Moreover, the models do not appear topossess ‘breather-like’ solutions (with the exception of the integrable sine-Gordonone)—although the n ∼ 2 models possess long lived breather-like states.Hence, having suggested how to define the concept of quasi-integrability, we

have looked the ‘almost conserved’ quantities in these models. For static and othersimple field configurations these quantities were conserved; the interesting resultswere obtained for ‘breather-like’ states that these models possess as well. For themwe have found interesting results and we are now working further on trying toanalyse them further. One of the ideas we are working on at the moment involvesexpanding all our expressions around n = 2; i.e. we have put n = 2 − ε and weare calculating all our expressions in powers of ε. Unfortunately, this process isvery complicated and, so far, we have not got very far in this calculations (whichare very tedious and involve rather complicated expressions). We hope to havesome results on this soon and, if we are successful, we will then present them ina separate publication.


(a) (b)

(c)

Fig. 6. (a) Total energy; (b) the first anomaly; (c) the integrated first anomaly for n = 2.01.

However, we feel that the results we have presented in this paper are veryinteresting even though they are still somewhat preliminary. Clearly, a lot stillremains to be done, both on these and on other models.

AcknowledgementsLAF is partially supported by a CNPq grant. WJZ wishes to thank the organisers

of the Symposium for giving him the opportunity to present this talk. He also


(a) (b)

Fig. 7. The anomaly (a) for n = 2.7 and the integrated anomaly (b).

acknowledges a FAPESP grant supporting his visit to IFSC/USP during which thispaper has been written up.

REFERENCES

[1] D. Bazeia et al.: Physica D 237 (2008), 937; arXiv:0708.1740.[2] P. D. Lax: Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math.

21 (1968), 467–490.V. E. Zakharov and A. B. Shabat, Zh. Exp. Teor. Fiz. 61 (1971), 118–134; English transl. Soviet Phys.JETP 34 (1972), 62–69.

[3] V. E. Zakharov and A. B. Shabat: Zh. Exp. Teor. Fiz. 61 (1971), 118–134; English transl. Soviet Phys.JETP 34 (1972), 62–69.

[4] L. D. Faddeev: Integrable models in (1+1)-dimensional quantum field theory, Proc. of Summer School ofTheoretical Physics, Les Houches, France, Aug 2–Sep 10, 1982. Published in Les Houches Sum. School561 (1982) (QC175:G7:1982).

[5] D. I. Olive and N. Turok: Local conserved densities and zero curvature conditions for toda lattice fieldtheories, Nucl. Phys. B 257 (1985), 277.

[6] D. I. Olive and N. Turok: The Toda lattice field theory hierarchies and zero curvature conditions inKac–Moody algebras, Nucl. Phys. B 265 (1986), 469.

[7] H. Aratyn, L. A. Ferreira, J. F. Gomes and A. H. Zimerman: The Conserved charges and integrability ofthe conformal affine Toda models, Mod. Phys. Lett. A 9 (1994), 2783; [arXiv:hep-th/9308086].

[8] L. A. Ferreira and W. J. Zakrzewski: A simple formula for the conserved charges of soliton theories, JHEP0709 (2007), 015; [arXiv:0707.1603 [hep-th]].

some comments on quasi-integrability

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