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In this work, we apply the so-called BPS method in order to obtain topological defects for a complex scalar field Lagrangian introduced by Trullinger and Subbaswamy. The BPS approach led us to compute new analytical solutions for this model. In our investigation, we found analytical configurations which satisfy the BPS first-order differential equations but do not obey the equations of motion of the model. Such defects were named nonphysical ones. In order to recover the physical meaning of these defects, we proposed a procedure which can transform them into BPS states of new scalar field models. The new models here founded were applied in the context of hybrid cosmological scenarios, where we derived cosmological parameters compatible with the observed Universe. Such a methodology opens a new window to connect different two scalar fields systems and can be implemented in several distinct applications such as Bloch Branes, Lorentz and Symmetry Breaking Scenarios, Q-Balls, Oscillons, Cosmological Contexts, and Condensed Matter Systems.

Topological defects are present in several scenarios of physics, covering areas like braneworld models, quintessence cosmological approaches, condensed matter, among others [

A well-established method to determine defect-like solutions is the so-called BPS method, proposed by Bogomol’ny, Prasad, and Sommerfeld [

At the end of the seventies, Trullinger and Subbaswamy found topological solutions (or defects) which satisfy the equations of motion related to the following Lagrangian density

In this paper we are going to investigate the model introduced in [

In order to show the potential of our methodology, we apply the derived models in the context of hybrid cosmological scenarios. Since the seminal work of Kinney in this subject [

The ideas behind this investigation are divided into the following sections: Section

In this section, we briefly review some generalities proposed by Trullinger and Subbaswamy [

Then, by substituting the above relation into (

Let us work with the following redefinitions:

Now, the scalar fields and the variable

Looking at the previous results, it is natural to think that (

Note that, the potential (

In the literature about defects, one-dimensional solutions which connect two distinct vacua are called kinks; besides one-dimensional solutions related to only one vacuum are named lumps. In two scalar field models, these one-dimensional defects are combined to construct an orbit in the field space. Therefore, the

In [

An advantage of the BPS method is that it simplifies considerably the integration process of equations of motion, and it also yields to new sets of analytical solutions which satisfy the BPS first-order differential equations. In this section, we will show that most part of the BPS solutions from [

From now on, we will be dealing with the problem of obtaining a BPS bound for the model under investigation. The first step to implement the BPS method to this context consists in rewrite (

Furthermore, the total energy for the fields configurations is such that

Thus, we can see that

The above solutions with a general value of

In the left panel we present the critical solutions

This graphic shows the transition between critical and subcritical solutions. The solutions

Another set of analytical solutions can be determined if we take

In this section, we will answer the questions presented above. Then, we are going to construct effective two-field models with these analytical solutions and their correspondent orbit equation. The effective model is obtained via the application of an extension method developed in [

Let us apply the deformation function (and its inverse), to rewrite the first-order differential equations for

These graphics unveil upside down views of effective potential

These graphics unveil upside down views of

An interesting application of the new analytical models found in the last section is in the context of hybrid inflation. There, the standard Einstein-Hilbert Lagrangian is coupled with a two real scalar fields Lagrangian density. This procedure is adopted in order to describe a Universe passing through different inflationary eras and dominated by dark energy for later values of time. Let us implement such a formalism using the action

By minimizing the action (

The last ingredients enable us to find that

In order to derive analytical cosmological models, we use the first-order formalism, which is based on the constraint

Thus, by taking

By minimizing the action (

After these generalities, we are ready to apply our model in such a cosmological scenario. The analytical solutions which are going to satisfy the first-order equations (

Time evolution of the analytical Hubble parameter derived from our two-field model. There is a small step close to

With the Hubble parameter in hands we are able to determine

Time evolution of the analytical EoS parameter derived from our two-field model. The picture was depicted with

In this section, we study in detail the behavior of the cosmological parameters found with the new class of models determined in Section

After the first inflationary era,

It is really interesting that our analytical model is able to describe all the different eras expected from the inflationary theory; besides, we also point that these special behaviors are related to the value of the

In this paper, we studied the model proposed by Trullinger and Subbaswamy in the BPS perspective. We were able to generalize the class of solutions introduced in [

Moreover, the models derived in the last section were built with two kink-like solutions whose asymptotic behaviors correspond to the vacua values of the potential

As a matter of applicability of our methodology, we used the results obtained in Section

Furthermore, it is important to highlight that the approach applied in the present work can be very powerful to investigate different subjects, such as the generation of coherent structures after cosmic inflation [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors would like to thank Capes and CNPq (Brazilian agencies) for financial support. RACC is partially supported by FAPESP (Foundation for Support to Research of the State of São Paulo) under Grants nos. 2016/03276-5 and 2017/26646-5.