The concept of quasi-integrability: a concrete example
L. A. Ferreira , and Wojtek J. Zakrzewski
Instituto de Física de São Carlos; IFSC/USP;
Universidade de São Paulo
Caixa Postal 369, CEP 13560-970, São Carlos-SP, Brazil
Department of Mathematical Sciences,
University of Durham, Durham DH1 3LE, U.K.
We use the deformed sine-Gordon models recently presented by Bazeia et al  to discuss possible definitions of quasi-integrability. We present one such definition and use it to calculate an infinite number of quasi-conserved quantities through a modification of the usual techniques of integrable field theories. Performing an expansion around the sine-Gordon theory we are able to evaluate the charges and the anomalies of their conservation laws in a perturbative power series in a small parameter which describes the “closeness” to the integrable sine-Gordon model. Our results indicate that in the case of the two-soliton scattering the charges are conserved asymptotically, i.e. their values 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 existence of long lived breather-like and wobble-like states in these models.
Solitons and integrable field theories play a central role in the study of many non-linear phenomena. Indeed, it is perhaps correct to say that many non-perturbative and exact methods known in field theories are in one way or the other related to solitons. The reason for that is twofold. On one hand, the appearance of solitons in a given theory is often related to a high degree of symmetries and so to the existence of a large number of conservation laws. On the other hand, in a large class of theories the solitons possess a striking property. They become weakly 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 relevant normal modes in the strong coupling (non-perturbative) regime of the theory. Such relation between the strong and weak coupling regimes have been observed in some (1+1) dimensional field theories, as, for example, in the equivalence of the sine-Gordon and Thirring models , as well as in four dimensional supersymmetric gauge theories where monopoles (solitons) and fundamentals gauge particles exchange roles in the so-called duality transformations .
The exact methods to study solitons in -dimensional field theories involve many algebraic and geometrical concepts, but the most important ingredient is the so-called zero curvature condition or the Lax-Zakharov-Shabat equation . All theories known to possess exact soliton solutions admit a representation of their equations of motion as a zero curvature condition for a connection living in an infinite dimensional Lie (Kac-Moody) algebra [5, 6]. In fact, in dimensions such zero curvature condition is a conservation law, and the conserved quantities are given by the eigenvalues of the holonomy of the flat connection calculated on a spatial (fixed time) curve. On the other hand, many techniques, like the dressing transformation method, for the construction of exact solutions are based on the zero curvature condition. In dimensions higher than two the soliton theory is not so well developed, even though many exact results are known for some -dimensional theories such as the models , as well as in four dimensional gauge theories where instanton and self-dual monopoles are the best examples . Some approaches have been proposed for the study of integrable field theories in higher dimensions based on generalizations of the two dimensional methods like the tetrahedron equations  and of the concept of zero curvature involving connections in loop spaces .
Another important aspect of integrable field theories is that they serve as good approximations to many physical phenomena. In fact, there is a vast literature exploring many aspects and applications of perturbations around integrable models. In this paper we want to put forward a technique that, so far as we know, has not been explored yet and which suggests that some non-integrable theories often possess many important properties of fully integrable ones. We put forward and develop the concept of quasi-integrability for theories 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 connection in an infinite dimensional Lie algebra. In other words, we have an anomalous zero curvature condition that leads to an infinite number of quasi-conservation (almost conservation) laws.
Moreover, in practice, in physical situations, like the scattering of solitons, these charges are effectively conserved. The striking property we have discovered is that as the scattering process takes place the charges do vary in time. However, after the solitons have separated from each other the charges return to the values they had prior to the scattering. Effectively what 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 we believe that if our results are indeed robust then such asymptotic charges could play a role in many important properties of the theory like the factorization of the S-matrix.
We introduce our concept of quasi-integrability through a concrete example involving a real scalar field theory in dimensions which is a special deformation of the sine-Gordon model. The scalar field of our theory is subjected to the potential
where is a real parameter which in the case reduces the potential to that of the sine-Gordon model, i.e. .
The potential (1.1) has an infinite number of degenerate vacua that allow the existence of solutions with non-trivial topological charges. It is worth noticing that the positions of the vacua are independent of and so they are the same as in the sine-Gordon model, i.e. , with being any integer.
The model with the potential (1.1) is fully topological (i.e. it satisfies its Bogomolnyi bound for any ) and so its one soliton field configurations are known in an explicit form. They are given by:
where the velocity is given in units of the speed of light, and the signs correspond to the kink (), and anti-kink (), with topological charges , and respectively. In Figure 1 we plot the fields of a one soliton configuration for various values on . We see from this plot that the case does not appear to be very special; all soliton fields look very similar and the solitons for different values of differ only in their slopes.
We are going to use this model to back up our discussion of quasi-integrability and so next we look at , with small. In Figure 2 we plot the potential (1.1) for . Of course, in this case the kinks solutions are given by (1.2) with replaced by . In Figure 3 we plot the one kink solutions (1.2) for the potentials shown in Figure 2, i.e. for . Note that they connect the vacua to , as goes from to , with the slope of the kink increasing as the value of decreases.
In this paper we study the concept of quasi-integrability in the context of the
theory (1.1) from the analytical and numerical points
We first consider a real scalar field theory with a very general potential , and construct a connection based on the loop algebra which, as a consequence of its equations of motion, satisfies an anomalous zero curvature condition. Using a modification of the methods employed in integrable field theories we construct and infinite number of quasi-conserved charges for such a theory, i.e.
where the anomalies are non-zero due to the non-flatness of the connection . The charges are in fact conserved, i.e. , and linear combinations of them correspond to the energy and momentum.
We then restrict ourselves to the case of the potential (1.1) and set up a perturbative expansion around the sine-Gordon theory. We expand all the quantities, equations of motion, field , charges and anomalies, in powers of the parameter , related to appearing in (1.1) by . For instance, we have
The anomalies vanish in the lowest order (order zero) in because they correspond to the sine-Gordon theory which is integrable and so all the charges are conserved during the dynamics of all field configurations.
In this paper we concentrate our attention on the evaluation of the first non-trivial charge and its anomaly, namely and , but our calculations can easily be extended to the other charges. We considered the case of the scattering of two kinks and also of a kink/anti-kink in the theory (1.1), where the solitons are far apart in the distant past and future, and collide when . We found that the first order anomaly , vanishes when integrated over the whole time axis. Therefore, from (1.3) we see that
Consequently, the scattering of the solitons happen in a way that, to first order in at least, the charge is asymptotically conserved. That is a very important result, and if one can extend it to higher orders and higher charges one would prove that effectively the scattering of solitons in 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 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. In other words, the first order anomaly vanishes when integrated over a period , and so from (1.3) we find that
where is the angular frequency of the breather. That means that the period of the charge is half of that of the breather.
We have performed many numerical simulations of the full theory (1.1) using a fourth order Runge-Kuta method, and using various lattice grids to make sure that the results are not contaminated by numerical artifacts. We have found reliable results with lattice grids of at least 3001 points, where the kinks were of size points. Most of the simulations in this paper were performed with lattices of 10001 points (i.e. well within this reliability). The main results we have found are the following: We have found that if does not get very close to unity the kinks and the kink/anti-kink scatter without destroying themselves and preserve their original shapes, given in (1.2). For small values of the anomaly integrates to zero for large values of the time interval, and so the charge is asymptotically conserved within our numerical errors. This is an important confirmation of our analytical result described above, 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) also possesses very long lived breather solutions for , which correspond to non-integrable models. These long-lived breathers were obtained by starting the simulations with a field configuration corresponding to a kink and an anti-kink. As they get close to each other they interact and readjust their profiles and some radiation is emitted in this process. We absorbed this radiation at the boundaries of the grid and the system stabilized to a breather-like configuration. For the resultant field configuration was the exact (and analytically known) breather while for small those breather-like fields lived for millions of units of time. As one changed and made it come close to unity the quasi-breathers radiated more and for even larger values they eventually died. We also looked at the anomalies for such breather-like configurations and have found a good agreement with our analytical results described above. The anomaly, integrated in time, does oscillate and for small values of the charge is periodic in time. That is, again, in agreement with the analytical result (1.6). Notice however, that the numerical result is stronger in the sense that it corresponds to the full charge and not only to its first order approximation as in (1.6).
We have also performed similar numerical simulations of wobbles  which correspond to configurations of a breather and a kink. Again, such configurations were obtained by starting the simulation with two kinks and an anti-kink. As the three solitons interact and adjust their profiles they radiate energy and this radiation had been absorbed at the boundaries of the grid. Eventually the system has evolved to a breather and a kink and for small values of the resultant configuration was quite stable, and for it agreed with the analytically known configuration of a wobble. Again, we believe that this is a very interesting result which shows that non-integrable theories can support such kinds of solutions.
Our results open up the way to investigate large classes of models which are not really exactly integrable but which possess properties which are very similar to those of integrable field theories. We believe that they will have applications in many non-linear phenomena of physical interest.
The paper is organized as follow: in section 2 we introduce the quasi-zero curvature condition, based on the loop algebra, for a real scalar field theory subject to a generic potential, and construct an infinite number of quasi-conserved quantities. In section 3 we perform the expansion of the theory (1.1) around the sine-Gordon model, and evaluate the first non-trivial charge and its corresponding anomaly. The numerical simulations, involving the two solitons scattering, breathers and a wobble, are presented in section 4. In section 5 we present our conclusions; the details of the loop algebra, charge calculations and -expansion are presented in the appendices.
2 The quasi zero curvature condition
We shall consider Lorentz invariant field theories in -dimensions with a real scalar field and equation of motion given by
where is the scalar potential. Thus we want to study the integrability properties of such theory using the techniques of integrable field theories [4, 5, 6]. We then start by trying to set up a zero curvature representation of the equations of motion (2.1), and so we introduce the Lax potentials as
Our Lax potentials live on the so-called loop algebra with generators and , with integer; their commutation relations are given in Appendix A. The parameters and are constants, and they play a special role in our analysis. Note, that the dynamics governed by (2.1) does not depend upon them, but since they appear in (2.2) they will play a role in the quasi-conserved quantities that we will construct through the Lax equations. In the expression above we have used light cone coordinates , where , and .
The curvature of the connection (2.2) is given by
As in the case of the sine-Gordon model where the potential is given by
we find that , given by (2.4), vanishes when we take and . Then the curvature (2.3) vanishes when the equations of motion (2.1) hold. The vanishing of the curvature allows us to use several powerful techniques to construct conserved charges and exact solutions. We want to analyze what can be said about the conservation laws for potentials when does not vanish but can be considered small.
In general, the conserved charges can be constructed using the fact that the path ordered integral of the connection along a curve , namely , is path independent when the connection is flat [5, 6, 13]. Here, we will use a more refined version of this technique and try to gauge transform the connection into the abelian subalgebra generated by the . 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 loop algebra is graded, with being the grades determined by the grading operator (see appendix A for details)
We perform a gauge transformation
with the group element being an exponentiation of generators lying in the positive grade subspace generated by the ’s, i.e.,
where we have used the equations of motion (2.1) to drop the term proportional to in (2.3). The component of the connection (2.2) has terms with grade and . Therefore, under (2.7) it is transformed into which has terms with grades ranging from to . Decomposing into grades we get from (2.7) and (2.8) that
Next we note that one can choose the parameters recursively by requiring that the component in the direction on cancels out in . Thus we can put that , and so on. In the appendix B we give the first few ’s obtained that way. In consequence, the component is rotated into the abelian subalgebra generated by the . Note that this procedure has not used the equations of motion (2.1). We then have
and the first three components are
With the ’s determined this way we perform the transformation of the component of the connection (2.2). Since the ’s are polynomials of -derivatives of (see appendix B) and since there will be terms involving -derivatives of ’s, we use the equations of motion to eliminate terms involving . Due to the nonvanishing of the anomaly term in (2.3) we are not able to transform into the abelian subalgebra generated by the . We find that is of the form
The next step is to decompose the curvature (2.9) into the component lying in the abelian subalgebra generated by and one lying in the subspace generated by . Since the equation of motion (2.1) has been imposed, it turns out that the terms proportional to the ’s in the combination , exactly cancel out. We are then left with terms in the direction of the only. Therefore, the transformed curvature (2.9) leads to equations of the form
with being linear in the anomaly given in (2.4), and the first three of them being given by
Working with the and variables we have that (2.15) takes the form and so we find that
As we are interested in finite energy solutions of the theory (2.1) we are concerned with field configurations satisfying the boundary conditions
Therefore from (2.2) we see that
where is the value of the potential at the global minimum which, in general, is taken to be zero. As we have seen the parameters of the gauge (2.7) and (2.8) are polynomials in -derivatives of the field (see appendix B). Therefore, for finite energy solutions we see that as , and so
Of course, the theory (2.1) is invariant under space-time translations and so its energy momentum tensor is conserved. The conserved charge is in fact a combination of the energy and momentum of the field configuration. In section 3 we will analyze the anomalies for a concrete perturbation of the sine-Gordon model, and we will show that even though the charges are not exactly conserved they lead to very important consequences for the dynamics of the soliton solutions.
A result that we can draw for general potentials, thus, is the following. For static finite energy solutions the charges are obviously time independent, and as a consequence of (2.22) one sees that the anomalies vanish, i.e. . Under a -dimensional Lorentz transformation where one finds that the connection (2.2) does not really transform as a vector. However, consider the internal transformation
where is the grading operator introduced in (A.3). Then, one notices that , given in (2.2), transforms as a vector under the combination of the external Lorentz transformation and the internal transformation (2.23). For the same reasons the transformed connection , defined in (2.7), is also a vector under the combined transformations. Consequently, the anomalies , introduced in (2.15), are pseudo-scalars under the same combined transformation. Therefore, in any Lorentz reference frame the integrated anomalies , defined in (2.18), satisfy
where by a travelling solution we mean any solution that can be put at rest by a Lorentz boost. Even though this result may look trivial, it can perhaps shed some light on the nature of the anomalies . In fact, as we will see in our concrete example of section 3, the anomalies vanish in multi-soliton solutions when the solitons they describe are far apart and so when they are not in interaction with each other. The anomalies seem to be turned on only when the interaction takes place among the solitons.
2.1 A second set of quasi conserved charges
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 Lax potentials are obtained from (2.2) by interchanging with , and by reverting the grades of the generators. Then we introduce the Lax potentials
In this case using the commutation relations of appendix A we observe that the curvature of such a connection is
The construction of the corresponding charges follows the same procedure as in section 2. We perform the gauge transformation
with the group element being
and analogously to the case of section 2, we choose the ’s to cancel the ’s components of . We then have
The transformed curvature (2.30) leads to equations of the form
with being linear in the anomaly , given in (2.27), and the first three are given by
Following the same reasoning as in section 2, we find that for finite energy solutions we have the quasi conservation laws
3 The expansion around the sine-Gordon model
The construction of quasi conserved charges of section 2 was performed for a very general potential, and no estimates were done on how small the anomaly of the zero curvature condition really is. We now turn to the problem of evaluating the anomalies , introduced in (2.18), and to discuss the usefulness of the quasi conservation laws (2.22). In order to do that we choose a specific potential which is a perturbation of the sine-Gordon potential and that preserves its main features like infinite degenerate vacua and the existence of soliton-like solutions. So we consider the potential given in (1.1) and we put i.e. we take
Using the results of appendix D, where we give the detailed calculations of such expansion, we have that the order zero field must satisfies the sine-Gordon equation, i.e.
On the other hand the first order field has to satisfy the equation
We shall consider here only the anomalies for the charges constructed in section 2 (the analysis for the charges constructed in section 2.1 is very similar). We expand the anomaly introduced in (2.4) as
and we also expand the parameters
Then we find that
Using the results of appendix D we find that vanishes by an appropriate choice of parameters, i.e.
With such a choice the first order contribution to reduces to (again using the results of appendix D)
and so we see that does not depend upon .
Since the anomalies , introduced in (2.18), are linear in and since , it follows that their zero order contribution vanishes, as it should since sine-Gordon is integrable. Thus we write our anomalies as
and the first order contribution to the first two of them are (remember that )
with given in (3.9). Thus, the first order anomalies do not depend on the first order field . The first order charges, however, do depend upon . To see this we expand the charges as
Then we find that are conserved and correspond to the charges of the sine-Gordon model, and involve only. As an example we present the first charge at first order
which, indeed, does depend on .
We can now evaluate the anomaly, to first order, for some physical relevant solutions of the theory (2.1) with the potential given by (3.1). As we have stressed this earlier the first order anomaly depends only upon the zero order field which is an exact solution of the sine-Gordon equation (3.3).
3.1 Anomaly for the kink
First we look at the case of one kink. The kink solution is given by (1.2) with and it is an exact solution of (2.1) for given by (3.1). The first order anomaly depend upon the kink solution of the sine-Gordon equation (3.3) which is given by
This expression can be integrated explicitly using the fact that
Therefore, the first order anomalies vanish, i.e. , agreeing with the general result shown in (2.24).
3.2 Anomalies for the 2-soliton solutions
The soliton/anti-soliton scattering
Let us consider a 2-soliton solution corresponding, for , to a soliton moving to the right with speed and located at at , and an anti-soliton moving to the left with speed and located at at . For the roles of soliton and anti-soliton are interchanged. The solution at order zero in the -expansion is given by a solution of the sine-Gordon equation (3.3) given by
where we have introduced
Note that given in (LABEL:eq:firstanomaly2sol), is an odd function of due to the term in front of the integral. All other terms involving in (LABEL:eq:firstanomaly2sol) appear as or , and so are even in . Consequently we see that
Thus, in the scattering of the soliton and anti-soliton the charge at first order is conserved asymptotically. From the physical point of view that is as effective as in the case of the integrable sine-Gordon theory. The solitons have to scatter preserving higher charges (at least in first order approximation).
The soliton/soliton scattering
Next we consider a 2-soliton solution corresponding, for , to a soliton moving to the right with speed and located at at , and another soliton moving to the left with speed and located at at . For the roles of soliton and anti-soliton are interchanged. The solution, at order zero in the -expansion, is again given by a solution of the sine-Gordon equation (3.3), namely