Creating Oscillons and Oscillating Kinks in Two Scalar Field Theories

TL;DR

This paper introduces new classes of oscillons and oscillating kinks in a two scalar field system, demonstrating their formation, properties, and potential applications across various physics domains.

Contribution

It presents novel solutions of oscillons and kinks in coupled scalar fields, expanding understanding of nonlinear localized states in 1+1 dimensions.

Findings

Oscillons and kinks with flat-top shapes are generated by varying a control parameter.

Solutions are applicable to condensed matter, cosmology, and high-energy physics.

The study demonstrates the existence of long-lived, localized oscillating states in coupled scalar fields.

Abstract

Oscillons are time-dependent, localized in space, extremely long-lived states in nonlinear scalar-field models, while kinks are topological solitons in one spatial dimension. In the present work, we show new classes of oscillons and oscillating kinks in a system of two nonlinearly coupled scalar fields in $1 + 1$ spatiotemporal dimensions. The solutions contain a control parameter, the variation of which produces oscillons and kinks with a flat-top shape. The model finds applications to condensed matter, cosmology, and high-energy physics.

Equations

$$V(\varphi ,\chi )={\kappa }_1{\varphi }^2+{\kappa }_2{\chi }^2+{\kappa }_3{\varphi }^4+{\kappa }_4{\chi }^4+{\kappa }_5{\varphi }^6$$

$$+{\kappa }_6{\chi }^6+{\kappa }_7{\varphi }^2{\chi }^2+{\kappa }_8{\varphi }^2{\chi }^4+{\kappa }_9{\varphi }^4{\chi }^2,$$

$${\kappa }_1$$

$${\kappa }_5$$

$${\kappa }_8$$

$$V(\varphi ,\chi )=\frac{1}{2}{\chi }^2(\lambda {\chi }^2+3\mu {\varphi }^2+\nu {)}^2+\frac{1}{2}{\varphi }^2(3\lambda {\chi }^2+\mu {\varphi }^2+\nu {)}^2$$

$${\mathcal{M}}_1$$

$${\mathcal{M}}_2$$

$${\mathcal{M}}_3$$

$${\mathcal{M}}_4$$

$${\mathcal{M}}_5$$

$${\mathcal{M}}_6$$

$${\mathcal{M}}_7$$

$${\mathcal{M}}_8$$

$${\mathcal{M}}_9$$

$$\mathcal{L}=\frac{1}{2}({\partial }_{\alpha }\varphi {)}^2+\frac{1}{2}({\partial }_{\alpha }\chi {)}^2-V(\varphi ,\chi ),$$

$$\frac{{\partial }^2\varphi (x,t)}{\partial t^2}-\frac{{\partial }^2\varphi (x,t)}{\partial x^2}+\frac{\partial V(\varphi ,\chi )}{\partial \varphi }$$

$$\frac{{\partial }^2\chi (x,t)}{\partial t^2}-\frac{{\partial }^2\chi (x,t)}{\partial x^2}+\frac{\partial V(\varphi ,\chi )}{\partial \chi }$$

$${\kappa }_{1,2}\to {\kappa }_{1,2}-{\delta }^2{ϵ}^2$$

$$\tau =\omega t,y=ϵx,$$

$$\omega \equiv \sqrt{1-{ϵ}^2}.$$

$$(1-{ϵ}^2)\frac{{\partial }^2\varphi }{\partial {\tau }^2}-{ϵ}^2\frac{{\partial }^2\varphi }{\partial y^2}=-(2({\kappa }_1-{ϵ}^2{\delta }^2)\varphi$$

$$+4{\kappa }_3{\varphi }^3+6{\kappa }_5{\varphi }^5+2{\kappa }_7\varphi {\chi }^2+2k_8\varphi {\chi }^4+4{\kappa }_9{\varphi }^3{\chi }^2),$$

$$(1-{ϵ}^2)\frac{{\partial }^2\chi }{\partial {\tau }^2}-{ϵ}^2\frac{{\partial }^2\chi }{\partial y^2}=-(2({\kappa }_2-{ϵ}^2{\delta }^2)\chi$$

$$+4{\kappa }_4{\chi }^3+6{\kappa }_6{\varphi }^5+2{\kappa }_7{\varphi }^2\chi +2{\kappa }_8{\varphi }^2{\chi }^3+4{\kappa }_9{\varphi }^4\chi ).$$

$$\varphi (y,\tau )$$

$$\chi (y,\tau )$$

$$ϵ\left(\frac{{\partial }^2{\varphi }_1}{\partial {\tau }^2}+2{\kappa }_1{\varphi }_1\right)+{ϵ}^3(\frac{{\partial }^2{\varphi }_3}{\partial {\tau }^2}+2{\kappa }_1^2{\varphi }_3-\frac{{\partial }^2{\varphi }_1}{\partial {\tau }^2}$$

$$-\frac{{\partial }^2{\varphi }_1}{\partial y^2}-2{\delta }^2{\varphi }_1+4{\kappa }_3{\varphi }_1^3+2{\kappa }_7{\varphi }_1{\chi }_1^2)+{\mathcal{O}(ϵ}^5)=0,$$

$$ϵ\left(\frac{{\partial }^2{\chi }_1}{\partial {\tau }^2}+2{\kappa }_2{\chi }_1\right)+{ϵ}^3(\frac{{\partial }^2{\chi }_3}{\partial {\tau }^2}+2{\kappa }_2{\chi }_3-\frac{{\partial }^2{\chi }_1}{\partial {\tau }^2}$$

$$-\frac{{\partial }^2{\chi }_1}{\partial y^2}-2{\delta }^2{\chi }_1+4{\kappa }_4{\chi }_1^3+2{\kappa }_7{\varphi }_1^2{\chi }_1)+{\mathcal{O}(ϵ}^5)=0.$$

$$\frac{{\partial }^2{\varphi }_1}{\partial {\tau }^2}+2{\kappa }_1{\varphi }_1=0,$$

$$\frac{{\partial }^2{\chi }_1}{\partial {\tau }^2}+2{\kappa }_2{\chi }_1=0,$$

$$\frac{{\partial }^2{\varphi }_3}{\partial {\tau }^2}+2{\kappa }_1{\varphi }_3=\frac{{\partial }^2{\varphi }_1}{\partial {\tau }^2}+\frac{{\partial }^2{\varphi }_1}{\partial y^2}$$

$$+2{\delta }^2{\varphi }_1-4{\kappa }_3{\varphi }_1^3-2{\kappa }_7{\varphi }_1{\chi }_1^2,$$

$$\frac{{\partial }^2{\chi }_3}{\partial {\tau }^2}+2{\kappa }_2{\chi }_3=\frac{{\partial }^2{\chi }_1}{\partial {\tau }^2}+\frac{{\partial }^2{\chi }_1}{\partial y^2}$$

$$+2{\delta }^2{\chi }_1-4{\kappa }_4{\chi }_1^3-2{\kappa }_7{\varphi }_1^2{\chi }_1.$$

$${\varphi }_1(y,\tau )$$

$${\chi }_1(y,\tau )$$

$$\frac{{\partial }^2{\varphi }_3}{\partial {\tau }^2}+2k_1{\varphi }_3=$$

$$-\left(2({\kappa }_1-{\delta }^2){\phi +3\kappa }_3{{\phi }^3+\kappa }_7\phi {\sigma }^2-\frac{d^2\phi }{d{y}^2}\right)\cos (\sqrt{2{\kappa }_1}\tau )$$

$$-\phi ({\kappa }_3{\phi }^2-\frac{{\kappa }_7}{2}{\sigma }^2)\cos (3\sqrt{2{\kappa }_1}\tau ),$$

$$\frac{{\partial }^2{\chi }_3}{\partial {\tau }^2}+2k_2{\chi }_3=$$

$$-\left(2({\kappa }_2-{\delta }^2){\sigma +3\kappa }_4{{\sigma }^3+\kappa }_7{\phi }^2\sigma -\frac{d^2\sigma }{d{y}^2}\right)\sin (\sqrt{2{\kappa }_2}\tau )$$

$$+\sigma ({\kappa }_4{\sigma }^2-\frac{{\kappa }_7}{2}{\phi }^2)\sin (3\sqrt{2{\kappa }_2}\tau ).$$

Full text

In the context of the field theory it is quite common the appearance of oscillons, which are time-dependent, localized in space, and extremely long living scalar field solutions which arise from nonlinear interactions. On the other hand, kinks are topological solitons in one spatial dimension. Here, we report the existence of new classes of oscillons and oscillating kinks in a system of two nonlinearly coupled scalar fields in 1 + 1 space-time dimensions. Such model is of interest in applications that include condensate matter, cosmology and high energy physics. We show the existence of diferent kinds of oscillons and oscillating kinks, which are characterized by the presence of an controllable parameter in the solutions. As we are changing the value that parameter the oscillons becomes flat in its top, thereby occupying a significant amount of the spacial distribution. In the case of the oscillating kinks, we obtain that the behavior of the solitons changes drastically, by developing a kind of plateau in its center.

I Introduction

Nonlinear field theories are commonly used in Physics to model phenomena at all energy scales ref1 , in condensed matter ref2 , in quantum solitons ref3 , in cosmology ref4 , in particle physics Bento:2018fmy ; Iguro:2018oou ; Chakrabarty:2019bbj , and in other areas of physics ref5 ; ref6 ; ref7 . One of fundamental reasons for the increasing interest in nonlinear phenomena is the possibility to have multiple degenerate minimum-energy configurations of the system. A good testbed to study the dynamics associated with the presence of degenerated minima are scalar field models whose potential energy has two or more degenerate minima. The present work addresses this topic, focusing in a particular model.

In nonlinear classical-field theories, solitons ref10 ; ref8 are localized field configurations with finite energy density, that keep their shape unaltered after collision with other solitons. These type of configurations have been studied in a wide class of contexts, such as nonlinear lattices ref11 , Skyrmion models ref12 ; ref13 , theories breaking Lorentz invariance ref14 ; ref15 , and the Maxwell-Chern-Simons gauge model ref16 .

The scalar nonlinear field theories allow for various types of configurations and, among then, are spatially localized time-periodic states named oscillons. Oscillons were discovered by Bogolyubsky and Makhankov ref17 and rediscovered a posteriori by Gleiser ref18 . Typically, they have bell-shaped profiles that oscillates sinusoidally in time ref18.1 ; ref18.2 . Further, oscillons are states in non-integrable models that continuously loose energy via radiation losses ref20 but, nevertheless, can be represented by extremely long-lived modes and preserve the self-trapping in the respective models ref21 . These properties are not shared with other classes of solutions of nonlinear scalar field theories that the breathers of the one-dimensional sine-Gordon equations ref19 are an example..

A vast body of results have been published for oscillons. In particular, they have been investigated in the presence of external potentials ref22 ; Copeland:2014qra , in string-inspired cosmological models ref23 , in the context of supersymmetry ref24 , in perturbed sign-Gordon models ref25 , and in higher-derivative field theories ref26 . Oscillons have also been studied in connection with gravitational waves Zhou:2013tsa ; ref27 ; ref28 , Abelian-Higgs models ref30 , in the context of the standard $SU(2)\times U(1)$ theory for electroweak interactions ref31 , and in theories probing violation of the Lorentz and CPT symmetries ref32 ; ref33 .

An interesting scenario where oscillons can play a fundamental role is after cosmic inflation Lozanov:2014zfa ; Lozanov:2016pac ; Lozanov:2016hid ; Lozanov:2018kpk as they act as a source of gravitational waves Lozanov:2019ylm and lead to a reduction in the uncertainties in the predictions for inflationary observables Lozanov:2017hjm .

Oscillons configurations were not studied in detail in models with two or more interacting scalar fields. In this context, it was pointed out in Ref. ref34 that a hybrid inflation model with two real scalar fields, interacting quadratically, can produce oscillon configurations whose lifetimes are much larger than those associated with single-field oscillons. As shown in that work, such configurations persist for at least four cosmological expansion times, accounting for up to $20\%$ of the total energy density of the early Universe.

The oscillons solutions in theories with two scalar fields were also addressed in Ref. ref35 in the framework of the dynamics of the Abelian Higgs model. By using a multiscale expansion method ref36 , a class of oscillons and oscillating kinks was obtained for both Higgs and gauge fields. Moreover, it was argued that similar dynamics occurs in superconductors and in the superfluid phase of atomic Bose-Einstein condensates trapped in optical lattices. Furthermore, in the framework of a similar scenario it was shown ref37 that it is possible to classify oscillons solutions according to their mass ratio and, accordingly, they may correspond to type-I or type-II superconductors. Various fluxon states produced by a system of sine-Gordon equations modelling a triangular configuration of three coupled long Josephson junctions were reported in Ref. Stan . It is also relevant to mention that it was recently shown ref49 ; ref50 ; ref51 that cosmological backgrounds based on models with more than one scalar field accurately comply with constraints inferred from data collected by the Planck satellite.

The aim of the present work is to show that oscillons naturally emerge in a system of two nonlinearly coupled scalar fields in ($1+1$)-dimensional spatiotemporal continuum. We address a class of sixth-degree polynomial potentials, which allows for rich structure of vacua ref38 ; ref39 . This model also plays an important role in the description of topological twistons in polyethylene ref40 ; ref41 ; ref42 and has also been used to investigate topological defects in molecular chains with zig-zag ref43 and helix ref44 structures. Unlike the oscillons and oscillating kinks considered in Refs. ref35 ; ref37 , where solutions were produced by an effective nonlinear Schrödinger equation, herein we use the expansion procedure developed in Ref. ref18.1 to construct two new classes of oscillons and two classes of oscillating kinks. In particular, it will be shown that oscillating-kink solutions include both single- and double-kink modes, and oscillons exhibit both bell-shaped and flat-top profiles.

This paper is organized as follows. In Section 2 we introduce the models with two interacting scalar fields. In Section 3 the method used for constructing oscillons solutions is presented. Then, new classes of oscillons and oscillating kinks are reported in Section 4, and their applications are discussed in Section 5. The work is concluded in Section 6.

II The two-scalar-fields model

Scalar field models with more than one scalar field naturally emerge in the study of many physical systems. Systems with two interacting scalar fields, that we address here, appear as models of networks of BPS (Bogomol’nyi-Prasad-Sommerfeld) and non-BPS defects ref45 , are used to describe fermion localization on degenerate and critical Bloch branes ref45.1 ; ref46 , on the hierarchy problems ref47 , and in field-theory kinks ref48 .

Here, we aim to construct oscillons and related states in a two-component system that is close to one considered in Ref. ref38 . The ($1+1$)-dimensional model includes a sixth-degree polynomial potential,

[TABLE]

where $\phi=\phi(x,t)$ and $\chi=\chi(x,t)$ are real scalar fields, and $\kappa_{i}$ are real coupling constants. Note that with the choice of

[TABLE]

$V(\phi,\chi)$ reproduces the potential energy of the model introduced in ref38 ,

[TABLE]

where $\lambda$, $\mu$ and $\nu$ are real positive coupling constants. This potential has a rich vacuum structure. For instance, for $\nu\lambda<0$ and $\nu\mu<0$ it has the following nine minima:

[TABLE]

where we used the notation $\mathcal{M}_{n}=(\phi,\chi)$ for two components of the $n$-th vacuum state.

The vacuum structure associated with potential (4) is sketched in Fig. 1, that also displays the potential’s profile. One can predict different classical configurations by connecting different vacua.

Previous analysis of this model addressed only static field configurations, revealing several kink-like and lump-like solutions ref38 . Our objective is to investigate solutions of the field equations that are time- and space-dependent.

III Solutions of the coupled field equations

The two-component system is defined by the Lagrangian density,

[TABLE]

with $V(\phi,\chi)$ given by Eq. (4). We use a system of units such that $c=\hbar=1$, the metric is $\eta_{\alpha\beta}=$ diag$(1,-1)$, and the coordinates are $x^{\alpha}=(t,x)$. The classical equations of motion generated by $\mathcal{L}$ are

[TABLE]

A small perturbation will be added to potential (1) as a means for creating time-dependent configurations. A simple way to introduce the perturbation is to replace

[TABLE]

in Eq. (1), where $\epsilon<<1$ is a small parameter, while $\delta$ is a generic coefficient (it is required in this definition as the same small parameter $\epsilon$ is introduced below in another context, see Eqs. (10) and (11) ).

Our main objective is to find very long-lived spatially localized time-dependent configurations, using an appropriate procedure devised in Ref. ref18.1 . As a first step, $x$ and $t$ are rescaled as

[TABLE]

where $\epsilon$ is the same small parameter as in Eq. (9), and

[TABLE]

Such transformations naturally appear in the context of the multiscale expansion method ref36 , which defines several temporal and spatial variables, that are scaled differently and treated as independent ones.

Under the scale transformation (10), Eqs. (7) and Eq. (8), with the potential energy given in Eq. (1), become

[TABLE]

The standard approach used to obtain oscillons configurations is based on the expansion of the scalar fields as powers of $\epsilon$. Given that Eqs. (12) and (13) are odd in the fields, the expansions defining $\phi$ and $\chi$ include only odd powers of $\epsilon$:

[TABLE]

Then, Eqs. (12) and (13) yield

[TABLE]

In the two lowest orders, Eqs. (16) and (17) lead to the following set of coupled nonlinear differential equations:

[TABLE]

Equations (18) and (19) are of the harmonic-oscillator type, hence their solution can be written as

[TABLE]

Functions $\varphi(y)$ and $\sigma(y)$, i.e. the spatial profile of the configurations appearing in Eqs. (22) and (23), can be obtained inserting $\phi_{1}(y,\tau)$ and $\chi_{1}(y,\tau)$ in Eqs. (20) and (21). After straightforward manipulations, one gets

[TABLE]

For field configurations that are periodic in time, the contribution proportional to $\cos(\sqrt{2\kappa_{1}}\tau)$ and $\sin(\sqrt{2\kappa_{2}}\tau)$ in the right-hand side of Eqs. (24) and (25) must vanish, otherwise solutions of the equations produced by these resonant driving terms will generate a term linear in $\tau$, hence the solutions for $\phi_{3}$ and $\chi_{3}$ will be not be time-periodic ones. This condition translates into the equations

[TABLE]

that determine $\varphi(y)$ and $\sigma(y)$. In general, nonlinear equations (26) and (27) are quite difficult to solve. However, as shown in the next section, they can be mapped into first-order linear differential equations whose general solution can be constructed by means of standard methods.

IV Oscillons and oscillating-kink solutions

Now, our goal is to reduce the second-order differential equations (26) and (27) to first-order ones. To this end, we write these equations as

[TABLE]

where

[TABLE]

To decouple the pair of Eqs. (28), we multiply the first and second equation by $d\sigma/dy$ and $d\varphi/dy$, respectively, and take their sum, to arrive at the result

[TABLE]

Here $\alpha_{0}$ is an arbitrary constant, that should be set equal to zero to get solutions that connect different vacua of the system, and $U(\varphi,\sigma)$ is an effective potential that can be written, using a supersymmetric representation for function $U(\varphi,\sigma)$ in terms of a superpotential, $\mathcal{W}(\varphi,\sigma)$, as

[TABLE]

where subscripts stand for derivatives with respect to $\varphi$ and $\sigma$, and the appropriate superpotential is

[TABLE]

with the following definitions

[TABLE]

It follows from Eqs. (34)-(36) that, to produce real solutions, the constants should be subject to the following restrictions:

[TABLE]

The advantage of this transformation is that it is possible to reduce the second-order differential equations (28) to first-order ones,

[TABLE]

Following Ref. ref52 , one obtains from Eq. (38) an equation for $\varphi$ considered as a function of $\sigma$:

[TABLE]

In terms of variable $\rho\equiv\varphi^{2}-1$, Eq. (39) is cast in the form of a linear ODE,

[TABLE]

whose general solution is

[TABLE]

where $c_{0}$ and $d_{1}$ are arbitrary integration constants. Next, these solutions feed the first-order ODE for $\sigma$ in Eq. (38), that become

[TABLE]

As shown in Ref. ref52 , Eq. (43) can be solved analytically, at least, for four different particular combinations of constants. Moreover, requiring that the solutions remain globally finite implies that $c_{0}$ cannot exceed some critical values. Therefore, borrowing the solutions from Ref. ref52 , one can write the corresponding classical field, at order $\mathcal{\epsilon}$, for the following cases:

A1. For $c_{0}<-2$ and $b_{1}=c_{1}$,

[TABLE]

A2. For $b_{1}=4c_{1}$ and $c_{0}<1/16$,

[TABLE]

B1. For $b_{1}=c_{1}$ and $c_{0}=-2$,

[TABLE]

B2. For $b_{1}=4c_{1}~{}$and $c_{0}=1/16$,

[TABLE]

The solutions of types A1 – B2 are illustrated in Figs. 2 and 3 that display the $\phi$ and $\chi$ components for solutions of type $A1$ and a particular set of the parameters, as indicated in the figure caption. The solutions in the $\phi$ component interpolate at $|y|\rightarrow\infty$ between different vacua of the classical-field potential. Their spatial profiles look like kinks in the $\phi^{4}$ theory, hence we call them oscillating kinks. The localized $\chi$ components will be called oscillating lumps.

The displayed examples demonstrate that the system with two scalar fields admits approximate solutions that oscillate with time, with the $\phi$ component keeping the oscillating kink-like profile, while the $\chi$ component features the oscillating lump-shape. Note that in all the families of solutions, the presence of a nonvanishing arbitrary parameter $c_{1}$ is necessary for the emergence of a flat segment in the lump (the $\chi$ field) and double-kink structure in the $\phi$ component.

V Emission of radiation

As shown in the seminal work by Segur and Kruskal kruskal (see also Ref. KM ), one-dimensional oscillons very slowly decay through emission of small-amplitude radiation waves. In this section we compute the outgoing radiation of the oscillons and oscillating kinks, by means of a method similar to the one elaborated by Hertzberg hertz for a single scalar field. Here we extend it for systems with two scalar fields.

The method relies on writing the solutions of the classical field equations in the following form:

[TABLE]

where $\phi_{\mathrm{osc}}$ and $\chi_{\mathrm{sol}}$ are the oscillating-kink and oscillon solutions, respectively, while, $\xi_{\mathrm{rad}}$ and $\zeta_{\mathrm{rad}}$ represents small radiation components, which are generated by linearized equations following from the substitution of the decompositions (53) and (54) in Eqs. (7) and (8):

[TABLE]

where $\square$ stands the usual D’Alembertian operator, $\Gamma_{\xi}\approx 2\kappa_{1}$ and $\Gamma_{\zeta}\approx 2\kappa_{2}$, and the radiations sources are written as

[TABLE]

Following the approach developed in Ref. hertz , we can write the radiation fields in the form

[TABLE]

In the above equation, $j_{\xi}(K,\Omega)$ and $j_{\zeta}(K,\Omega)$ are the Fourier transforms of $j_{\xi}(x,t)$ and $j_{\zeta}(x,t)$. Then, after straightforward manipulations, we obtain

[TABLE]

where we define

[TABLE]

and $\Delta_{1}$ and $\Delta_{2}$ being phase constants.

In Fig. 4 we show how the amplitudes of the outgoing radiation vary in time, being, indeed, very small in comparison with the amplitudes of the underlying oscillating kink and oscillating-lump modes. Furthermore, we also examined how the outgoing radiation vary in time for the configurations corresponding to Eqs. (47) and (48), concluding that the qualitative picture remain the same.

It has been checked by means of long simulations (see below) that the solutions classified above as A1 an A2 are completely stable. At this point, it is important to remark that the physical reason for their stability comes from the fact that when $c_{0}<-2$ and $c_{0}<1/16$ the spatial profiles of the solutions belong to the range of lower configurational entropy (CE) rafael-gleiser-PLB , thereby ensuring the stability of the structures for a long period of time, but not infinitely. On the other hand, the solutions of the B1 type are unstable, and B2 are partially stable. Namely, in solutions B1 both components are unstable, while in B2 the field $\phi$ collapses very fast, while component $\chi$ is a long-standing one. Thus, it is concluded that B1 solutions decay very fast after their creation, and in B2 the two-field configuration rapidly evolves into a long-lived single-field oscillon, while the A1 and A2 species are truly long-lived oscillating patterns.

VI Numerical Results

To check the above analytical results, numerical solutions have been produced for the oscillons and oscillating kinks. To this end, Eqs. (7) and (8) were solved with initial conditions

[TABLE]

which correspond to coefficients $b_{1}=c_{1}$, see Eqs. (35) and (36), and $c_{0}=-2.1$. To illustrate the field configurations produced by the simulations for oscillating kinks and oscillons, in Fig. (5) we plot fields $\phi(x,t)$ and $\chi(x,t)$ at $x=0.5$. The figure, as well as additional numerical results, not shown here, demonstrate good agreement of the analytical solutions with their numerical counterparts.

VII Summary and conclusions

In the present work we have investigated solutions of a system of two nonlinearly coupled scalar fields in the one-dimensional space, with a multiple vacuum structure. The analysis shows that systems of this type has a rich dynamics. In particular, we were able to determine new classes of time-dependent solutions, in the form of oscillatory kinks and lump oscillons. As far as we know, the existence and dynamics of oscillons and kinks in models with two scalar fields were previously addressed only in Ref. ref35 . However, in that work only solutions with states of the same type (kinks or lumps) in both components were considered, while here we investigate solutions combining oscillatory kinks and lumps in the two components. In our case, we have found that some of them are long-lived, but not infinitely stable (types A1 and A2), while others (B1 and B2) are unstable. Despite the spatial interpolation of the classical vacua of the theory, it is not clear if these type of solutions are stable and are representative solutions of different topological sectors for the Manifold of the classical solutions of the field equations. Indeed, the fact that the oscillons being approximate solutions of the field equations also raise the question if their time evolution can lead to a classification of the classical field equations into topological sectors, a problem that we will address in a future work.

The new lump and kink oscillons configurations reported here may include relatively large flat regions, where the fields are essentially different from the vacuum values. Away from the central flat region, the solutions varying rapidly, approaching vacua values. Plotting the respective orbits in the $\left(\phi,\chi\right)$ plane, see Fig. 3, one finds that the fields surround the vacuum point, featuring very small values, which is a typical behavior demonstrated by oscillons.

Our interest in the two-component system considered in this work is that it provides a relatively simple model with a rich vacuum structure, if compared to other theories with complex vacuum structures, such as gauge theories. Further, the model considered in the current work has applications to condensate matter, cosmology and high-energy physics. In particular, in the study of low-dimensional materials similar to graphene, scalar fields naturally appear as models of impurities and defects CastroNeto:2009zz , and also model a carbon structure on the top of which the dynamics of the electrons occurs Oliveira:2010hq .

Acknowledgements.

RACC thanks FAPESP, grants 2016/03276-5 and 2017/26646-5, for financial support. RACC would like to thank Profs. Gyula Fodor and Péter Forgács from Wigner Research Centre for Physics for helpful discussion concerning oscillons. RACC also thanks Tel Aviv University, University of Coimbra and Tokyo University of Science for their hospitality during the development of this work. RACC also thanks to Prof. Giuseppe Mussardo for your considerable discussions about nonlinear field theories. The work of BAM is supported, in part, by the Israel Science Foundation through grant No. 1287/17.