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We discuss the impact of the breaking of the Lorentz symmetry on the usual oscillons, the so-called flat-top oscillons, and the breathers. Our analysis is performed by using a Lorentz violation scenario rigorously derived in the literature. We show that the Lorentz violation is responsible for the origin of a kind of deformation of the configuration, where the field configuration becomes oscillatory in a localized region near its maximum value. Furthermore, we show that the Lorentz breaking symmetry produces a displacement of the oscillon along the spatial direction; the same feature is present in the case of breathers. We also show that the effect of a Lorentz violation in the flat-top oscillon solution is responsible by the shrinking of the flat-top. Furthermore, we find analytically the outgoing radiation; this result indicates that the amplitude of the outgoing radiation is controlled by the Lorentz breaking parameter, in such a way that this oscillon becomes more unstable than its symmetric counterpart; however, it still has a long living nature.

The study of nonlinear systems is becoming an area of increasing interest along the last few decades [

In the context of the field theory it is quite common the appearance of solitons [

An important feature of a large number of interesting nonlinear models is the presence of topologically stable configurations, which prevents them from decaying due to small perturbations. Among other types of nonlinear field configurations, there is a specially important class of time-dependent stable solutions, the breathers appearing in the Sine-Gordon like models. Another time-dependent field configuration whose stability is granted for by charge conservation are the

Oscillons are quite general configurations and are found in inflationary cosmological models [

The usual oscillon aspect is typically that of a bell shape which oscillates sinusoidally in time. Recently, Amin and Shirokoff [

At this point, it is interesting to remark that Segur and Kruskal [

On the other hand, some years ago, Kostelecký and Samuel [

By introducing a dimensional reduction procedure to

In recent years, investigations about topological defects in the presence of Lorentz symmetry violation have been addressed in the literature [

In fact, Lorentz invariance is the most fundamental symmetry of the standard model of particle physics and it has been very well verified in several experiments. But, it is important to remark that we cannot be sure that this, or any other, symmetry is exact. Their consequences shall be verified through experimental data. This affirmation is encouraged due to the fact that there exist some experimental tests of the Lorentz invariance being carried in low energies, in other words, energies smaller than 14 Tev. Thus, from this fact, we can suspect that at high energies the Lorentz invariance could not be preserved. As an example, in the string theory there is a possibility that we could be living in a universe which is governed by noncommutative coordinates [

Furthermore, in a cosmological scenario, the occurrence of high energy cosmic rays above the Greisen-Zatsepin-Kuzmin (GZK) cutoff [

The impact of Lorentz violation on the cosmological scenario is very important, because several of its weaknesses could be easily explained by the Lorentz violation. For instance, it was shown by Bekenstein [

In the inflationary scenario with Lorentz violation, Kanno and Soda [

Here, it is convenient to us to emphasize that the inflation is the fundamental ingredient to solve both the horizons as the flatness problems of the standard model of the very early universe. Approximately

At this point, it is important to remark that the postinflationary universe is governed by real scalar fields where nonlinear interactions are present. Thus, it was shown in [

Thus, in this work we are interested in answering the following issues. Can oscillons and breathers exist in scenarios with Lorentz violation symmetry? If oscillons and breathers exist in these scenarios, how is their profile changed? Furthermore, what happens with the lifetime of the oscillons?

Therefore, in this paper, we will show that oscillons and breathers can be found in Lorentz violation scenarios; our study is performed by using Lorentz violation theories rigorously derived in the literature [

This paper is organized as follows. In Section

In this section, we present a scalar field theory in a

As a simple example, those authors showed for

Now, returning to (

In this case, the Minkowski metric is modified from

Clearly, as a final product, the LV and Lorentz invariant Lagrangians have the same equation of motion. The fundamental difference between these two equations comes from the fact that the new variables

In the Lagrangian density (

In general

In a recent work, Anacleto et al. [

However, we can find systems with Lorentz symmetry break which has an additional scalar field [

In the above Lagrangian density, we have a different coefficient correcting the metric, but the coefficients for Lorentz violation cannot be removed from the Lagrangian density using variables or fields redefinitions and observable effects of the Lorentz symmetry break can be detected in the above theory. Therefore, theories with fewer fields and fewer interactions allow more redefinitions and observable effects.

In this section, we will work in a

At this point, it is important to remark that the Lagrangian density clearly has not manifest covariance. Furthermore, it is possible to observe that the covariance is recovered by choosing

Now, from the above, we can easily construct the corresponding Hamiltonian density:

Let us now see how the Poincarè algebra is modified in this scenario. The idea of the present analysis is to see how the Poincaré invariance is broken. In other words, verify how this scenario has the Lorentz symmetry violated. Therefore, for this we write down the three Poincarè generators, the Hamiltonian

With this, we can calculate the commutation relations of the Poincarè group. Thus, after straightforward calculations of the usual commutation relations, it is not difficult to conclude that

From the above relations, we can see that the Poincarè algebra is not closed, since the usual commutations are not recovered. As a consequence, in this scenario we have one violation of the Lorentz symmetry. However, it is possible to recover the complete commutation relations by taking

At this point we can verify that, for the cases

However, in the above case, we can recover the usual Poincarè algebra using the rescale

In summary, the Lagrangian density (

In this section, we will study the equation of motion in the presence of the scenario with Lorentz violation of the previous section. Here, our aim is to study the case in the

Here, if one applies the transformation involving the Lorentz boost in the above equation of motion, one gets

Following the above demonstration, we can see clearly that this equation is not invariant under boost transformations. For instance, we can conclude that the possibilities [

Note that there is no modification of the equations; in other words, the possibilities [

In order to solve analytically the differential equation (

Note that the rotation angle

From now on we will use the above equation to describe the profile of oscillons and breathers. It is of great importance to remark that the above equation has all the information about the violation of the Lorentz symmetry. In fact, the field

Now, we study the case of a scalar field theory which supports usual oscillons in the presence of Lorentz violating scenarios. The profile of the usual oscillons is one in which the spatial structure is localized in the space and, in the most cases, is governed by a function of the type

Since our primordial interest is to find periodic and localized solutions, it is useful, as usual in the study of the oscillons, to introduce the following scale transformations in

Now we are in a position to investigate the usual oscillons. But it is important to remark that the fundamental point is that here we have the effects of the Lorentz symmetry breaking. We can see this by inspecting the above equation of motion, which is carrying information about the terms of the Lorentz breaking through the variables

Next we expand

Note that the above expansion has only odd powers of

Therefore, the solution of (

Here we call attention to the fact that the solution must be smooth at the origin and vanishing when

In order to find the solution of

Solving the above equation, we find a term which is linear in the time-like variable

At this point, one can verify that the above equation can be integrated to give

As one can see, up to the order

The profile of the above solution is plotted in Figure

Profile of the usual oscillons in

Moreover, one can note that if one wants to recover the Lorentz symmetry, it is necessary to impose that

Some years ago, a new class of oscillons, which is characterized by a kind of plateau at its top, was presented by Amin and Shirokoff [

Now, we begin a direct attack to the problem of finding the flat-top oscillons. Likewise to the procedure presented in [

So, we are in a position to investigate the so-called flat-top oscillons. But it is important to remark that the fundamental point is that all the effects of the Lorentz symmetry breaking are present implicitly in the classical field. Of course, it is possible to recover the original equation of motion presented by Amin and Shirokoff [

Let us go further on our search for flat-top oscillons. For this, we expand

If we substitute the above expansion of the scalar field into the equation of motion (

Therefore, the solution of (

In order to find the solution of

Since that the solution of the function

At this point, one can verify that the above equation has the same profile of the equation presented in [

Now, we must solve (

From this, it follows that

As one can see, up to the order

The profile of the above solution is plotted in Figure

Profile of the flat-top oscillons in

Typical profile of the flat-top oscillon. The left-hand figure corresponds to the case with Lorentz breaking symmetry and the right-hand figure to the one with Lorentz symmetry.

There one can note that the energy density becomes more localized close to the origin when the Lorentz breaking increases.

We will now construct the profile of a breather in a

The sine-Gordon model is invariant under

The above equation can be solved by the inverse-scattering method [

Profile of the breathers

Density plot of a Breather. Solution with Lorentz symmetry breaking (left) and to the one Lorentz symmetry (right).

An important characteristic of the oscillons is its radiation emission. In a seminal work by Segur and Kruskal [

Thus, in this section, we describe the outgoing radiation in scenarios with Lorentz violation symmetry. Here, we will establish a method in

From the above equation, it is possible to find the solution for

As

We can use the Fourier transform for solving differential equation (

Then, we have the corresponding solution

From the above approach, it is possible to find the radiation field for the oscillons. As a consequence of the method, the oscillons expansion must be truncated.

In this subsection, we will study the outgoing radiation of the usual oscillons in a Lorentz violation scenario. In this case, the oscillon expansion truncated in order

As an example, we will consider

Substituting (

Thus, for

On expression (

In Figure

Amplitude of the outgoing radiation determined by the Fourier transform. The left-hand figure corresponds to the case with Lorentz breaking symmetry and the right-hand to the case with Lorentz symmetry.

We will now present the outgoing radiation by the flat-top oscillons in Lorentz violation scenario. Here, the associated oscillon expansion truncated in

Substituting the above expansion in (

From the above expression, we see that there is an outgoing radiation which has its amplitude described by the integral

We can make use of the above generalization to calculate the amplitude of radiation of the flat-top oscillons in Lorentz violation scenario. For instance, for

In this case, we see that the amplitude of the outgoing radiation changes with the parameters

We have seen in Section

In order to decouple the Lagrangian density (

After straightforward computations, one can conclude that

It is important to note that applying the rotations in the fields, the Lagrangian density was decoupled into two independent Lagrangians

We can see that all the preceding approaches and results can be used here to find the fields

As we are working in

In this case, we have

Now it is quite clear why the Lagrangian density (

Now, by using the approaches described in Section

Fortunately, we can find periodical solutions for the fields

In the above solutions,

As above asserted, the original scalar fields

It is important to remark that the resulting solutions do not present merely algebraic relation between

An important question concerns the stability of the solutions; given that each field

In this work we have investigated the so-called flat-top oscillons in the case of Lorentz breaking scenarios. We have shown that the Lorentz violation symmetry is responsible for the appearance of a kind of deformation of the configuration. On the order hand, from inspection of the results coming from the flat-top oscillons in

Moreover, it is important to highlight that the bounds in Lorentz violation theories in the standard model are very small and are compatible with the stability observed for the oscillons here introduced. On the other hand, observable effects of these oscillons in the real world are possible, for instance, in a cosmological context. In that case, the life time of these oscillons can be decisive in the generation of coherent structures after cosmic inflation [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank Professor D. Dalmazi for useful discussions. R. A. C. Correa thanks P. H. R. S. Moraes for discussions regarding cosmology. R. A. C. Correa also thanks Alan Kostelecky for helpful discussions about Lorentz breaking symmetry. The authors also thank CNPq and CAPES for partial financial support. Furthermore, the authors also are grateful to the anonymous referees for the comments that lead to improving the results and conclusions of this work.

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