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We study configurations in one-dimensional scalar field theory, which are time-dependent, localized in space, and extremely long-lived, called oscillons. How the action of changing the minimum value of the field configuration representing the oscillon affects its behavior is investigated. We find that one of the consequences of this procedure is the appearance of a pair of oscillon-like structures presenting different amplitudes and frequencies of oscillation. We also compare our analytical results to numerical ones, showing excellent agreement.

The presence of topologically stable configurations is an important feature of a large number of interesting nonlinear models. 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 by charge conservation is the

Oscillons are quite general configurations and are found in the Abelian-Higgs

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

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

On the other hand, in a thermal background [

Research involving oscillons has also been done in a stellar scenario [

Thus, in this work, we introduce novel configurations in one-dimensional scalar field theories, which are time-dependent, localized in space, and extremely long-lived like the oscillons. This is done through the investigation of how displacement of the oscillons’ position in the field potential affects their features.

This paper is organized as follows. In Section

In this work, we study a real scalar field theory in

In order to introduce the idea we are pursuing, we will analyze the case of the symmetric

The profile of this potential is illustrated in Figure

Profile of the potential of the model on analysis with

This kind of potential presents kink-like configurations interpolating adjacent vacua. Here, however, we are looking for time-dependent field configurations which are localized in the space.

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

From the above equation, it is possible to find the usual oscillons which are localized in the central vacuum

However, in this work, we are interested in analyzing how some simple displacement in the above expansion may affect the configuration and stability of the oscillons. Then, we are searching for a small amplitude solution where we expand the scalar field

Note that the above expansion differs from the usual treatment of oscillons, since we have an additive term, which corresponds to translation in the field. Furthermore, we can recover the usual expansion setting

It is important to remark that our approach is general and can be applied to different nonlinear field theories in order to investigate oscillons configurations. A special case is that one given by choosing

In this section, we will derive the profile of the proposed oscillon type configurations using the expansion given by (

We note that the procedure of performing a small amplitude expansion shows that the scalar field solution

Here, it is necessary to impose that

Let us now look for the solution of (

Looking at the second equation of (

Similarly, from solutions (

Our primordial goal is to get configurations which are periodical in time. Then, if we solve the above partial differential equation in the presented form, we will have a term linear in

At this point, it is necessary to impose that

Now, coming back to (

Thus, using the condition that

From the above results, as one can see, up to order

We can note that the fundamental difference of our solution, compared to the usual oscillon, is given by the presence of

Another important and interesting result that arises from our solution is the emergence of a new “phantom oscillon” after a certain threshold value of

In Figure

Oscillon (thin line) and its phantom (dashed line) for

We also emphasize that, in the phantom zone, the effective frequency

Effective frequency of the oscillon (bottom line) and its phantom (top line) for

An important characteristic of the oscillon is its radiation emission, responsible for its unavoidable decaying. In a seminal work by Segur and Kruskal [

Thus, in this section, we compute the outgoing radiation of these oscillon type configurations. Here, we will use a method in

Since

At this point, we can obtain from (

It is important to remark that, in order to solve (

Radiation power for

In this section, in order to check our analytical results, we will compute the numerical solutions for the oscillon profile. In this way, to analyze the oscillon configuration numerically, we use the initial configuration in the form

In general, oscillons are not an exact solution for the scalar field. Thus, it is convenient to begin with the above initial configuration for evolution of the numerical solution. Another important condition is given by

Now, for evolution of the numerical solution of the field equation (

Analytical and numerical results. The figure is a comparison of the field at

In this work, we have presented novel oscillon-like configurations which we call phantom oscillons. We have found that displacement of the minimum value of the field configuration representing the oscillon affects its behavior. In this case, the procedure of introducing those displacement instances act as a kind of source of new oscillons; in fact, this leads us to think about the possibility of the appearance of a higher number of additional oscillons when one deals with a field potential containing a bigger number of degenerate vacua. This possibility is under analysis and hopefully will be reported in a future work. Moreover, it can be observed in Figure

The authors declare that there are no competing interests regarding the publication of this paper.

The authors thank CNPq and CAPES for partial financial support. R. A. C. Correa also thanks Marcelo Gleiser for the helpful discussions and for the valuable remarks about oscillons.

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