# Formalization of Transform Methods in Higher-order Logic: A Survey

Muhammad Ahmed and Adnan Rashid

###### Abstract

Most of the engineering and physical systems are generally characterized by
differential and difference equations based on their continuous-time and
discrete-time dynamics, respectively. Moreover, these dynamical models are
analyzed using transform methods to prove various properties of these systems,
such as, transfer function, frequency response and stability, and to find out
solutions of the differential/difference equations. The conventional
techniques for performing the transform methods based analysis have been
unable to provide an accurate analysis of these systems. Therefore, higher-
order-logic theorem proving, a formal method, has been used for accurately
analyzing systems based on transform methods. In this paper, we survey
developments for transform methods based analysis in various higher-order-
logic theorem provers and overview the corresponding real world case studies
from the avionics, medicine and transportation domains that have been analyzed
based on these developments.

##  1 Introduction

The engineering and physical systems exhibiting dynamical behaviours are
generally modeled using differential [Y+13] and difference [KP01] equations
based on their corresponding dynamics that can be continuous-time and
discrete-time, respectively. To analyze these models capturing the dynamics of
such systems, transform methods, such as, the Laplace transform [BMVdBVdV03],
the Fourier transform [BB86], the Discrete Fourier Transform (DFT) [Sun01] and
the $z$-Transform [OBS01], have been widely used. These transform methods
facilitate finding out solutions of these differential and difference
equations based models and analyzing their various important properties, such
as, transfer function, frequency response and stability, as shown in Figure 1.

Figure 1: Transform Methods

The continuous-time dynamical behaviour of a system is generally modeled using
differential equations. These differential equations-based models are not
easier to analyze in time-domain, especially, for the case of larger systems
($n$-order differential equations for larger $n$). Transform methods, which
include the Laplace and the Fourier transforms have been widely used to
analyze these differential equations. These methods involve a transformation
of the time-domain model to its corresponding frequency domain representation
($s$-domain for the Laplace and $\omega$-domain for the Fourier transform),
which converts the differential equations involving integrals and
differentials to their corresponding algebraic equations having multiplication
and division operators, as shown in Figure 1. These equations are further
solved to analyze various properties of the system, such as, transfer function
and frequency response, and to obtain solutions in frequency domain
[BMVdBVdV03, BB86]. Finally, the inverse transforms (the inverse Laplace and
the inverse Fourier) are applied to obtain the time-domain solutions of the
differential equations-based models.

Similarly, the dynamics of a discrete-time system are captured using
difference equations. These difference equations based models are analyzed
using the discrete-time transform methods, which include the Discrete Fourier
Transform (DFT) and the $z$-transform. These transform methods perform a
conversion of the discrete time model to its corresponding frequency domain
representation ($z$-domain for the $z$-transform and $\omega$ for the DFT), as
shown in Figure 1. These representations are further solved to analyze various
properties of the given system, such as, transfer function and frequency
response, and to obtain solutions in frequency domain [Sun01, OBS01]. Finally,
the inverse transforms are applied to obtain the time-domain solutions of the
difference equations based models.

Conventionally, the transform methods based analysis has been performed using
paper-and-pencil based proofs and computer based symbolic and numerical
techniques. However, these methods suffer from their inherent limitations and
thus compromise the accuracy of the analysis [DPV13]. Formal methods [HT15],
in particular, higher-order-logic theorem proving [Har09] can overcome the
limitations of the conventional methods and thus provides an accurate
transform methods based analysis of the systems. It has been widely used for
the transform methods based analysis of the engineering and physical systems.
These transform methods have been formalized using various higher-order-logic
theorem provers, such as, HOL Light [Har96], HOL4 [SN08], Coq [BC13] and
Isabelle [Pau94]. Moreover, these formalizations have been used for analyzing
various safety-critical systems, such as, linear analog circuits, automobile
suspension systems, a drug therapy model, unmanned aerial vehicles, synthetic
biological circuits, power converters and digital filters. In this paper, we
survey these contributions regarding formalization of transform methods that
have been done using higher-order-logic theorem proving.

The rest of the paper is organized as follows: Section 2 provides the
developments of the Laplace and the Fourier transforms in various theorem
provers and their associated analysis of the continuous-time systems. We
provide the formalization of the DFT and the $z$-transform and the associated
analysis of the discrete-time systems in Section 3. Section 4 presents a
discussion about the features and availability of the transform methods based
analysis and its comparison in different theorem provers. Finally, Section 5
concludes the paper.

##  2 Transform Methods for Analyzing Continuous-time Systems

The Laplace and the Fourier transforms have been formalized using various
higher-order-logic theorem provers. Moreover, these formalizations have been
used for formally analyzing many engineering and physical systems. Taqdees et
al. [TH13] formalized the Laplace transform using multivariate calculus
theories of HOL Light. This formalization mainly includes the formal
definition of the Laplace transform and the formal verification of its various
properties, such as, linearity, frequency shifting, first-order
differentiation, higher-order differentiation and integration in time-domain.
Moreover, the authors used their formalization of the Laplace transform for
formally verifying the transfer function of a Linear Transfer Converter (LTC)
circuit. Next, the authors extended their framework by providing a support to
formally reason about the linear analog circuits, such as first-order and
second-order Sallen-Key low-pass filters by formalizing the system governing
laws such as Kirchhoff’s Current Law (KCL) and Kirchhoff’s Voltage Law (KVL)
using HOL Light [TH17]. Later, Rashid et al. [RH17] proposed a new
formalization of the Laplace transform based on the notion of sets and
verified some more properties of the Laplace transform, such as, time
shifting, time scaling, the Laplace transform a $n$-order differential
equation and uniqueness [RH18]. The authors also formally verified the Laplace
transform of some commonly used functions, such as, exponential function, sine
and cosine functions. Finally, they used their proposed formalization of the
Laplace transform for analyzing the control system of an Unmanned Free-
swimming Submersible (UFSS) vehicle [RH17] and 4-$\pi$ soft error cross talk
model [RH18]. Similarly, the formalization of the Laplace transform has been
used for formally analyzing the unmanned aerial vehicles [ARH20b] and
synthetic biological circuits [ARH20a, RH+20].

Wang et al. [WC17] provided a formalization of the Laplace transform using the
Coq theorem prover. The authors formally defined the Laplace transform and
formally verified a few of its classical properties, such as, linearity,
frequency shifting and differentiation in time domain. Moreover, they applied
their proposed formalization for formally verifying a flight control system
using Coq. Similarly, Immler [Imm21] formalized the Laplace transform using
the Isabelle theorem prover. The authors mainly verified some of the classical
properties of the Laplace transform, such as, linearity, frequency shifting,
uniqueness, differentiation and integration in time-domain. Gang et al.
[GCnY+14] used HOL4 for the formalization of the Laplace transform. The
authors formally modeled the Laplace transform and verified some of its
classical properties, such as, linearity, frequency shifting, differentiation
and integration in time domain. Next, they used their proposed formalization
for formally verifying the transfer function of a motor.

Table 1: Continuous-time Transform Methods in Various Theorem Provers Transform Methods | Theorem Prover | Formalization Details/ Verified Properties
---|---|---
| Coq | Linearity, frequency shifting, differential [WC17]
| HOL Light |  | Linearity, frequency shifting, integration, time shifting,
---
time scaling, first-order and higher-order
differentiations in time domain [TH13, RH18]
| Isabelle |  | Linearity, frequency shifting, integration,
---
first-order and higher-order differentiations
in time domain, integration [Imm21]
Laplace Transform | HOL4 |  | Linearity, frequency shifting, differentiation and
---
integration in time domain [GCnY+14]
| HOL Light |  | Linearity, frequency shifting, modulation, time reversal,
---
first-order and higher-order differentiations
in time domain [RH19]
Fourier Transform | HOL4 |  | Linearity, time reversal, frequency shifting, differentiation,
---
and integration in time domain [GZS+20]

Rashid et al. [RH16] formalized the Fourier transform using the HOL Light
theorem prover. The authors provided a formal definition of the Fourier
transform and formally verified its various classical properties, such as,
linearity, frequency shifting, modulation, time reversal, first-order and
higher-order differentiations in time-domain. Furthermore, they formally
verified the Fourier transform of some commonly used functions, such as,
exponential, sine and cosine functions. Moreover, they used their proposed
formalization for formally analyzing an Automobile Suspension System (ASS), an
audio equalizer, a drug therapy model and a MEMs accelerometer [RH19].
Similarly, Guan et al. [GZS+20] formalized the Fourier transform using HOL4.
The authors provided a formal definition of the Fourier transform and formally
verified its various properties, such as, linearity, time reversal, frequency
shifting, differentiation and integration in time domain. Moreover, they used
their proposed formalization for formally verifying the frequency response of
a RLC circuit. A summary of the formalizations of the Laplace and the Fourier
transforms in various theorem provers and the systems that have been formally
analyzed using these transform methods can be found in Tables 1 and 2

Table 2: Analysis of Continuous-time Systems using Transform Methods Transform Methods | Theorem Provers | Applications
---|---|---
| Coq | Flight control system [WC17]
| HOL Light |  | LTC [TH13], UFSS vehicle [RH17],
---
4-$\pi$ soft error cross talk model [RH18],
Sallen-Key low-pass filter [TH17],
unmanned aerial vehicles [ARH20b],
synthetic biological circuits [ARH20a, RH+20]
| Isabelle | No application
Laplace Transform | HOL4 | Motor [GCnY+14]
| HOL Light |  | Automobile suspension system [RH16],
---
a drug therapy model [RH19],
an audio equalizer [RH19],
MEMs accelerometer [RH19]
Fourier Transform | HOL4 | RLC circuit [GZS+20]

##  3 Transform Methods for Analyzing Discrete-time Systems

The DFT and $z$-Transform have been formalized using various higher-order-
logic theorem provers. Moreover, these formalizations have been used for
formally analyzing many discrete-time systems. Siddique et al. [SMT14]
formalized $z$-transform using the HOL Light theorem prover. The authors
provided a formal definition of the $z$-transform and formally verified its
various properties, such as, linearity, time shifting and scaling in
$z$-domain. Moreover, they used their proposed formalization for the formal
analysis of Infinite Impulse Response (IIR) Digital Signal Processing (DSP)
filter. Later, the authors extended their proposed framework by providing the
formal verification of some more properties, such as, time scaling, complex
conjugate and a formal support for the inverse z-transform and used it for
formally analyzing a switched-capacitor interleaved DC-DC voltage doubler
[SMT18].

Table 3: Discrete-time Transform Methods in Various Theorem Provers Transform Methods | Theorem Prover | Formalization Details/ Verified Properties
---|---|---
DFT | HOL4 |  | Implicit periodicity, linearity, symmetry, frequency shifting,
---
time shifting, convolution [SZG+15]
| HOL4 |  | FFT, inverse FFT [AT04]
---
FFT | Coq |  | FFT [Cap01]
---
$z$-Transform | HOL Light |  | linearity, time shifting and scaling in $z$-domain,
---
time scaling, complex conjugation, inverse $z$-transform [SMT18]

Shi et al. [SZG+15] proposed a formalization of DFT using the HOL4 theorem
prover. The authors presented a formal definition of the DFT and formally
verified its various properties, such as, implicit periodicity, linearity,
symmetry, frequency shifting, time sifting and convolution. Moreover, the
authors used their proposed formalization for formally verifying Fast Fourier
Transform (FFT) and cosine frequency shifting. Capretta et al. [Cap01]
formally verified the FFT using the Coq theorem prover. Similarly, Akbarpour
et al. [AT04] provided a formal specification and verification of the FFT at
different abstraction levels using the HOL4 theorem prover. A summary of the
formalization of the transform methods for discrete-time systems in various
theorem provers and their associated applications can be found in Tables 3 and
4.

Table 4: Analysis of Discrete-time Systems using Transform Methods Transform methods | Theorem Provers | Formalization Details / Verified Properties
---|---|---
DFT | HOL4 | FFT, cosine frequency shifting [SZG+15]
| HOL4 | FFT at different levels of abstraction and inverse FFT [AT04]
FFT | Coq | FFT [Cap01]
Z-Transform | HOL Light |  | IIR DSP filter [SMT14], switched-capacitor
---
interleaved DC-DC voltage doubler [SMT18]

##  4 Theorem Proving Support for Transform Methods based Analysis

Figure 2 depicts the formal libraries of transform methods that are available
in various higher-order-logic theorem provers for performing the analysis of
the engineering and physical systems. For example, the Laplace transform is
available in most of the theorem provers. Moreover, its formal library in the
HOL Light theorem prover is quite dense and has been frequently used for
analyzing various safety-critical systems as given in Table 2. Similarly, the
$z$-transform is only available in HOL Light. Similarly, HOL4 contains both
the continuous-time and discrete-time Fourier transforms. Moreover, no
transform methods is available in the PVS theorem prover.

Figure 2: Support for Transform Methods in Higher-order-logic Theorem Provers

##  5 Conclusion

Transform methods are widely used for analyzing the engineering and physical
systems exhibiting dynamical behaviours. Due to the safety-critical nature of
these system, their accurate analysis is of utmost importance. This paper
surveys the transform methods that have been formalized using different
theorem provers by highlighting various safety-critical systems that have been
formally analyzed based on transform methods. In this regard, only HOL Light
theorem prover contains most number of transform methods, such as, the Laplace
and the Fourier transforms, and the $z$-transform. Similarly, PVS contains no
transform methods library and we need to develop these libraries that can be
used for performing transform methods based analysis using PVS.

## References

  * [ARH20a] Sa’ed Abed, Adnan Rashid, and Osman Hasan.  Formal Analysis of the Biological Circuits using Higher-order-logic Theorem Proving.  In ACM Symposium on Applied Computing, pages 3–7, 2020.
  * [ARH20b] Sa’ed Abed, Adnan Rashid, and Osman Hasan.  Formal Analysis of Unmanned Aerial Vehicles Using Higher-Order-Logic Theorem Proving.  Journal of Aerospace Information Systems, 17(9):481–495, 2020.
  * [AT04] Behzad Akbarpour and S Tahar.  Verification of the Fast Fourier Transform using HOL Theorem Proving.  Techincal Report, Concordia University, Montreal-Canada, 2004.
  * [BB86] Ronald Newbold Bracewell and Ronald N Bracewell.  The Fourier Transform and its Applications, volume 31999.  McGraw-Hill, 1986.
  * [BC13] Yves Bertot and Pierre Castéran.  Interactive Theorem Proving and Program Development: Coq’Art: the Calculus of Inductive Constructions.  Springer Science & Business Media, 2013.
  * [BMVdBVdV03] R. J. Beerends, H. G. Morsche, J. C. Van den Berg, and E. M. Van de Vrie.  Fourier and Laplace Transforms.  Cambridge University Press, 2003.
  * [Cap01] Venanzio Capretta.  Certifying the Fast Fourier Transform with Coq.  In Theorem Proving in Higher Order Logics, volume 2152 of LNCS, pages 154–168. Springer, 2001.
  * [DPV13] Antonio J Durán, Mario Pérez, and Juan L Varona.  The Misfortunes of a Mathematicians’ Trio using Computer Algebra Systems: Can We Trust?  CoRR, abs/1312.3270, 2013.
  * [GCnY+14] Zhao Gang, Zhao Chun-na, Guan Yong, Lv Xing-li, and Li Xiao-juan.  Formalization of Laplace Transform Calculus in HOL4.  Journal of Chinese Computer Systems, 35(9):2177–2181, 2014.
  * [GZS+20] Yong Guan, Jie Zhang, Zhiping Shi, Yi Wang, and Yongdong Li.  Formalization of Continuous Fourier Transform in Verifying Applications for Dependable Cyber-physical Systems.  Journal of Systems Architecture, 106:101707, 2020.
  * [Har96] John Harrison.  HOL Light: A Tutorial Introduction.  In Formal Methods in Computer-Aided Design, volume 1166 of LNCS, pages 265–269. Springer, 1996.
  * [Har09] John Harrison.  Handbook of Practical Logic and Automated Reasoning.  Cambridge University Press, 2009.
  * [HT15] Osman Hasan and Sofiène Tahar.  Formal Verification Methods.  Encyclopedia of Information Science and Technology, IGI Global Pub, pages 7162–7170, 2015.
  * [Imm21] Fabian Immler.  Laplace Transform.  2021\.
  * [KP01] Walter G Kelley and Allan C Peterson.  Difference Equations: an Introduction with Applications.  Academic Press, 2001.
  * [OBS01] Alan V Oppenheim, John R Buck, and Ronald W Schafer.  Discrete-time Signal Processing.  Prentice Hall, 2001.
  * [Pau94] Lawrence C Paulson.  Isabelle: A Generic Theorem Prover, volume 828.  Springer Science & Business Media, 1994.
  * [RH16] Adnan Rashid and Osman Hasan.  On the Formalization of Fourier Transform in Higher-order Logic.  In Interactive Theorem Proving, volume 9807 of LNCS, pages 483–490. Springer, 2016.
  * [RH17] Adnan Rashid and Osman Hasan.  Formal Analysis of Linear Control Systems using Theorem Proving.  In Formal Engineering Methods, pages 345–361. Springer, 2017.
  * [RH18] Adnan Rashid and Osman Hasan.  Formalization of Lerch’s Theorem using HOL Light.  Journal of Applied Logics, 5(8):1623–1652i, 2018.
  * [RH19] Adnan Rashid and Osman Hasan.  Formal Analysis of Continuous-time Systems using Fourier Transform.  Journal of Symbolic Computation, 90:65–88, 2019.
  * [RH+20] Adnan Rashid, Osman Hasan, et al.  Formal Reasoning about Synthetic Biology using Higher-order-logic Theorem Proving.  IET Systems Biology, 14(5):271–283, 2020.
  * [SMT14] Umair Siddique, Mohamed Yousri Mahmoud, and Sofiene Tahar.  On the Formalization of $z$-Transform in HOL.  In Interactive Theorem Proving, volume 8558 of LNCS, pages 483–498. Springer, 2014.
  * [SMT18] Umair Siddique, Mohamed Yousri Mahmoud, and Sofiène Tahar.  Formal Analysis of Discrete-Time Systems using $z$-Transform.  Journal of Applied Logics, 5(4):875–906, 2018.
  * [SN08] Konrad Slind and Michael Norrish.  A Brief Overview of HOL4.  In Theorem Proving in Higher Order Logics, pages 28–32. Springer, 2008.
  * [Sun01] Duraisamy Sundararajan.  The Discrete Fourier Transform: Theory, Algorithms and Applications.  World Scientific, 2001.
  * [SZG+15] Zhiping Shi, Yupeng Zhang, Yong Guan, Liming Li, and Jie Zhang.  The Formalization of Discrete Fourier Transform in HOL.  Mathematical Problems in Engineering, 2015, 2015.
  * [TH13] Syeda Hira Taqdees and O. Hasan.  Formalization of Laplace Transform Using the Multivariable Calculus Theory of HOL-Light.  In Logic for Programming, Artificial Intelligence, and Reasoning, volume 8312 of LNCS, pages 744–758. Springer, 2013.
  * [TH17] Syeda Hira Taqdees and Osman Hasan.  Formally Verifying Transfer Functions of Linear Analog Circuits.  IEEE Design & Test, 34(5):30–37, 2017.
  * [WC17] Yifei Wang and Gang Chen.  Formalization of Laplace Transform in Coq.  In Dependable Systems and Their Applications, pages 13–21. IEEE, 2017.
  * [Y+13] Xin-She Yang et al.  Mathematical Modeling with Multidisciplinary Applications.  Wiley, 2013.


# Multimagnetic Monopoles

D. Bazeia  Departamento de Física, Universidade Federal da Paraíba, 58051-970
João Pessoa, PB, Brazil  M. A. Liao  Departamento de Física, Universidade
Federal da Paraíba, 58051-970 João Pessoa, PB, Brazil  M. A. Marques
Departamento de Biotecnologia, Universidade Federal da Paraíba, 58051-900 João
Pessoa, PB, Brazil  Departamento de Física, Universidade Federal da Paraíba,
58051-970 João Pessoa, PB, Brazil

###### Abstract

In this work we investigate the presence of magnetic monopoles that engender
multimagnetic structures, which arise from an appropriate extension of the
$\rm{SU(2)}$ gauge group. The investigation is based on a modified
relativistic theory that contain several gauge and matter fields, leading to a
Bogomol’nyi bound and thus to a first order framework, from which stable
multimagnetic solutions can be constructed. We illustrate our findings with
several examples of stable magnetic monopoles with multimagnetic properties.

##  I Introduction

Magnetic monopole has been an important object of investigation in theoretical
physics ever since Dirac first showed that it can be incorporated into the
framework of electromagnetism in a simple and elegant way Dirac ; DiracII . Of
particular importance is the demonstration, also given by Dirac, that their
mere existence could, when combined with the postulates of quantum mechanics,
provide a natural explanation for the quantization of electric (and magnetic)
charge.

Despite this theoretical triumph, lack of experimental evidence for the
existence of monopoles led Dirac’s theory to be regarded as a mathematical
curiosity. Almost fifty years later, ’t Hooft thooft  and Polyakov polyakov
independently found a topological monopole solution for a Yang-Mills-Higgs
theory; see also Refs. ps ; bogo ; AA . The ’t Hooft-Polyakov monopole
presents a magnetic charge which is directly proportional to a topological
invariant called the winding number, or topological degree BB ; manton .
Perhaps even more important than the solution itself was the realization that
magnetic monopoles were an unavoidable consequence of spontaneous symmetry
breaking in grand unification theories thooft . The so called monopole
problem, regarding the apparent absence of these particles in our universe
despite the theoretical predictions of their production in the early universe
Einhorn  was one of the main motivations for the development of inflationary
universe models Guth ; inflation .

The study of magnetic monopoles has further evolved by the introduction of
models with enhanced gauge symmetry Witten ; Shifman . This enhancement allows
for more complex monopoles, which now emerge from a much richer topology and
present internal structure. Works internal ; smallandhollow ; bimag  have been
developed along these lines, and we have also conducted a similar
investigation in (2,1) dimensions, which led to vortices possessing a novel
internal structure, which is reflected by the magnetic field associated with
these solutions multilayered . In this sense, we believe that the present
study, in which we deal with models where the $\rm SU(2)$ symmetry is enhanced
to describe products of several $\rm SU(2)$ factors will also induce interest
in planar systems, following the lines of multilayered . In the case of
vortices, enhancement of the basic $\rm U(1)$ symmetry to a product of $N$
factors, $\rm U(1)\times U(1)\times\cdots\times U(1)$, can be directly
connected to the addition of several order parameters in the system. The
interest here can be extended to vortices in superconductors described by the
Ginzburg-Landau theory and is directly connected with the study of two- and
three-component systems that appeared in Refs. prl2 ; prl22 ; prl3 . The
subject is also of interest to the case of the multi-component Gross-
Pitaevskii equations that describe the mean-field dynamics of spin-1 and
spin-2 Bose-Einstein condensates; see, e.g., Ref. PRep  and references therein
for more information on these issues. Moreover, the idea can also be very
naturally extended to kinks in the real line, in direct connection with the
work multikink , in which the $\rm Z_{2}$ symmetry is enhanced to $\rm
Z_{2}\times Z_{2}$, and can be used in applications similar to the very recent
one, described in apply , where geometrically constrained kinks contribute to
modify the behavior of fermions, leading to situations of practical interest
for the constructions of electronic devices at the nanometric scale. There are
other more involved situations, in which the gauge group can appear as a
product of the form ${\rm SU(N_{1})\times SU(N_{2})\times\cdots\times
SU(N_{p})}$; this is the case, for instance, in the Atiyah, Drinfeld, Hitchin,
and Manin construction of multi-instantons with product group gauge theories
that arise from D3-branes; see, e.g., instanton  and references therein.

In this work we will build upon the investigation conducted in bimag  to
construct multimagnetic structures, which emerge from the superposition of
magnetic shells, each of which is the result of spontaneous symmetry breaking
in a $\rm{SU(2)}$ subgroup of the full gauge group, under whose
transformations our models will be taken to be invariant. Research from the
last decade has brought to light a great variety of interesting applications
for bimagnetic structures, ranging from engineering to biomedicine Estrader ;
EstraderII ; magnetoresistance ; Lee . One particularly interesting
application lies in the possibility of using the exchange bias present on
multilayers of magnetic material for beating the superparamagnetic limit,
which normally constrains the miniaturization of magnetic recording devices
superpara ; superparaII . Such multilayered or onion-like structures have been
investigated in relation to nanoparticles Salazar ; Catala  and, as
multimagnetic structures expand upon the bimagnetic models studied earlier, so
we hope that the systems investigated here may be useful to expand on the
aforementioned applications. Another interesting property of our models lie in
the presence of a Bogomol’nyi-Prasad-Sommerfield (BPS) limit, in which the
problem is reduced to the solution of a system of first order differential
equations. Solutions of this type are global minima of the energy, and are
therefore expected to be stable under small fluctuations.

Magnetic monopoles have also been found in spin ice systems SI1 ; SI2  and
more recently, the direct observation of static and dynamics of emergent
magnetic monopoles in a chiral magnet has been reported in Ref. PRL20 . They
are also object of theoretical and experimental study in heavy-ion collisions
and in neutron stars PRL17 , and also in the international research
collaboration MoEDAL running at CERN with the prime goal of searching for the
magnetic monopole MO . These studies motivate that we investigate new models
of magnetic monopoles in high energy physics, hoping to contribute with the
construction of models that allow the addition of internal structure, bringing
novel properties and/or features at the fundamental level.

In order to disclose the investigation, we organize the present work as
follows: In Sec. II we introduce the generalized Lagrangian that defines our
models, and derive the second order equations that follow from it. We then
investigate the conditions under which a Bogomol’nyi bound is attainable, and
examine the first order equations which must be satisfied by BPS solutions. We
then proceed, in Sec. III and IV, to solve some examples which engender the
sought-after multimagnetic structures, and investigate their properties.
Finally, in Sec. V, we discuss our results and add some perspectives of future
works in the subject.

##  II General Procedure

We work in four dimensional Minkowski spacetime with metric $(-,+,+,+)$, and
consider a class of non-abelian models with the Lagrangian density of the form

$\displaystyle{\mathcal{L}}$
$\displaystyle=-\sum_{n=1}^{N}\frac{P^{(n)}(\\{|\phi|\\})}{4}F^{(n)a}_{\mu\nu}F^{(n)a\mu\nu}$
(1) $\displaystyle\hskip
11.38109pt-\sum_{n=1}^{N}\frac{M^{(n)}(\\{|\phi|\\})}{2}D^{(n)}_{\mu}\phi^{(n)a}D^{{(n)}\mu}\phi^{{(n)}a}$
$\displaystyle\hskip 11.38109pt-V(\\{|\phi|\\}),$

where $N$ is a positive integer, $(\\{|\phi|\\})$ stands for the set
$(|\phi^{(1)}|,|\phi^{(2)}|,...,|\phi^{(N)}|)$,
$D^{(k)}_{\mu}\phi^{(k)a}=\partial_{\mu}\phi^{(k)a}+g^{(k)}\varepsilon^{abc}A^{(k)b}_{\mu}\phi^{(k)c}$
is the action of covariant derivative in the $k$-th sector with coupling
$g^{(k)}$ and
$F^{(k)a}_{\mu\nu}=\partial_{\mu}A^{(k)a}_{\nu}-\partial_{\nu}A^{(k)a}_{\mu}+g^{(k)}\varepsilon^{abc}A^{(k)b}_{\mu}A^{(k)c}_{\nu}$
is the $k$-th field strength. The potential is denoted by $V(\\{|\phi|\\})$.

This Lagrangian is invariant under the action of the product of $N$
$\rm{SU}_{(k)}(2)$ groups, in the form
$\rm{SU}_{(1)}(2)\times...\times\rm{SU}_{(N)}(2)$. Each scalar field is a
$\rm{SU(2)}$-valued quantity, given by $\phi^{(n)}=\phi^{(n)a}T^{a}$, where
$T^{a}=-i\sigma^{a}/2$ are the generators of SU(2), given in terms of the
three Pauli matrices $\sigma^{a}$. Each $\phi^{(n)}$ is charged only in the
respective $\rm{SU}_{(n)}(2)$ subgroup, and is thus coupled to a gauge field
$A^{(n)}_{\mu}=A^{(n)a}_{\mu}T^{a}$. The condition $\langle
T^{a},T^{b}\rangle=-2{\rm{Tr}}(T^{a}T^{b})=\delta^{ab}$ may be used to define
an inner product in SU(2), under which one obtains the norm
$|\phi^{(n)}|=\sqrt{\phi^{(n)a}\phi^{(n)a}}$. Throughout this work, the
Einstein summation convention is always implied for Lorentz and SU(2) internal
indices, but not for the remaining ones, which simply label the $\rm{SU}(2)$
subgroups and are always enclosed in parentheses.

The functions $P^{(n)}$ and $M^{(n)}$ are nonnegative and may, in principle,
depend on all the scalar fields $\\{|\phi|\\}$, respecting the gauge symmetry,
but we shall shortly restrict their forms, aimed to implement some
possibilities capable of giving interesting stable minimum energy
configurations. We generically require that these functions take finite values
when the field configurations approach the vacuum manifold, which will be
topologically non trivial, in order to allow for finite energy topological
solutions. red The solutions can be classified according to a set of $N$
integers, which, in analogy to the standard monopole, we may relate to
quantized “magnetic” charges emerging from the unbroken $\rm{U(1)}$ symmetry
from each $\rm{SU(2)}$ submanifold.

The field equations derived from (1) for the $k$-th sector are

$\displaystyle
D^{(k)}_{\mu}\left(M^{(k)}D^{(k)\mu}\phi^{(k)a}\right)=\frac{1}{4}\sum_{n=1}^{N}\frac{\partial
P^{(n)}}{\partial\phi^{(k)a}}F^{(n)b}_{\mu\nu}F^{(n)b\mu\nu}$
$\displaystyle+\frac{1}{2}\sum_{n=1}^{N}\frac{\partial
M^{(n)}}{\partial\phi^{(k)a}}D^{(n)}_{\mu}\phi^{(n)b}D^{{(n)}\mu}\phi^{{(n)}b}+\frac{\partial
V}{\partial\phi^{(k)a}},$ (2a) $\displaystyle
D^{(k)}_{\mu}\left(P^{(k)}F^{(k)a\mu\nu}\right)=g^{(k)}M^{(k)}\,\varepsilon^{abc}\phi^{(k)b}D^{(k)\nu}\phi^{(k)c},$
(2b)

where the covariant derivatives also act in the following form
$D^{(n)}_{\mu}F^{(n)a\mu\nu}=\partial_{\mu}F^{(n)a\mu\nu}+g^{(n)}\varepsilon^{abc}A^{(n)b}_{\mu}F^{(n)c\mu\nu}$.

Let us now consider static solutions, which may be obtained as time
independent fields which obey equations of motion in a gauge such that
$A_{0}^{(n)}=0$ (leading to the non Abelian analogues of Gauss’ law to be
trivially solved). The energy functional in this case is given by

$\begin{split}E=\sum_{n=1}^{N}&\int
d^{3}x\left[\frac{1}{2}M^{(n)}|D_{k}^{(n)}\phi^{(n)}|^{2}+\frac{1}{2}P^{(n)}|B_{k}^{(n)}|^{2}\right.\\\
&\left.+V(\\{|\phi|\\})\right],\end{split}$ (3)

where $B_{k}^{(n)}\equiv\varepsilon_{ijk}F^{ij(n)}/2$ and the previously
defined SU(2) norm has been used. We may now follow bogo  to find a first
order framework. This can be achieved by setting $M^{(n)}=1/P^{(n)}$ for each
$n$ and taking the limit $V(\\{|\phi|\\})\to 0$ in (3). This leads to

$\begin{split}E=&\sum_{n=1}^{N}\int
d^{3}x\;P^{(n)}\;\left|B_{k}^{(n)}\pm\frac{D_{k}^{(n)}\phi^{(n)}}{P^{(n)}}\right|^{2}\\\
&+4\pi\sum_{n=1}^{N}\frac{v^{(n)}}{g^{(n)}}|Q_{t}^{(n)}|,\end{split}$ (4)

where $Q_{t}^{(n)}\in\mathbb{Z}$ is the topological charge associated with the
$(\phi^{(n)},A_{\mu}^{(n)})$ solutions, defined in the same way as in the
standard $\rm{SU}(2)$ Yang-Mills-Higgs theory manton . The second term in the
right hand side is recognized as the summation of the fluxes arising from the
magnetic fields in each sector. The observation that the first integral in (4)
is non-negative leads to the Bogomol’nyi bound

$E\geq 4\pi\sum_{n=1}^{N}\frac{v^{(n)}}{g^{(n)}}|Q_{t}^{(n)}|.$ (5)

The equality in (5) is attained if and only if the first order equations

$B_{k}^{(n)}=\mp\frac{D_{k}^{(n)}\phi^{(n)}}{P^{(n)}}$ (6)

are satisfied for each $n$. By direct derivation it is straightforward to show
that solutions of (6) always solve the second order equations. Deviations from
the static case give rise to quadratic terms in $F_{j0}^{(n)}$ and
$\partial_{0}\phi^{(n)}$ which must be added to the integral in (3). Such
terms may only increase the energy, and thus configurations that solve (6) are
global minima of this functional.

We shall henceforth concern ourselves with spherically symmetric static
solutions. Since we are choosing $A_{0}^{(n)}=0$, as is well known, these
restrictions allow us to write the fields in the form thooft ; polyakov

$\displaystyle\phi^{(n)a}$ $\displaystyle=\frac{x_{a}}{r}H^{(n)}(r),$ (7a)
$\displaystyle A_{i}^{(n)a}$
$\displaystyle=\varepsilon_{aib}\frac{x_{b}}{g^{(n)}r^{2}}(1-K^{(n)}(r)),$
(7b)

subject to the conditions

$\displaystyle H^{(n)}(0)$ $\displaystyle=0,$ $\displaystyle K^{(n)}(0)$
$\displaystyle=1,$ (8) $\displaystyle H^{(n)}(\infty)$ $\displaystyle\to\pm
v^{(n)},$ $\displaystyle K^{(n)}(\infty)$ $\displaystyle\to 0.$

These requirements ensure that the ensuing energy is finite, while also
enforcing symmetry breaking and thus resulting in topological solutions. The
form of this Ansatz implies $|Q_{t}^{(n)}|=1$ for all $n$, so that the
magnetic charge in each subsystem achieves the smallest non-zero value allowed
by the quantization condition. In the standard magnetic monopole the vacuum
manifold is the 2-sphere $S^{2}$, but in the present model it is the product
of $N$ 2-sphere, being $S^{2}\times S^{2}\times\cdots\times S^{2}$. As one
knows, the second homotopy group of $S^{2}$ is the group of the integers, $Z$;
however, we also know that the second homotopy group of the product of $N$
2-sphere $S^{2}$ is the product of $N$ second homotopy group of the 2-sphere
$S^{2}$, so we have $Z\times Z\times\cdots\times Z$, that is, a set of $N$
integers. In this sense, the classification by a set of $N$ integers exhausts
the finite energy solutions, including in particular the spherically symmetric
solutions that we will describe below. Under the above assumptions, the second
order equations take the form

$\begin{split}&\frac{1}{r^{2}}\left(r^{2}M^{(n)}{H^{(n)}}^{\prime}\right)^{\prime}-\frac{2M^{(n)}H^{(n)}{K^{(n)}}^{2}}{r^{2}}\\\
&-\frac{1}{2}\sum_{k=1}^{N}\frac{\partial P^{(k)}}{\partial
H^{(n)}}\left(\frac{2{{K^{(k)}}^{\prime}}^{2}}{{g^{(k)}}^{2}r^{2}}-\frac{(1-{K^{(k)}}^{2})^{2}}{{g^{(k)}}^{2}r^{4}}\right)\\\
&-\frac{1}{2}\sum_{k=1}^{N}\frac{\partial M^{(k)}}{\partial
H^{(n)}}\left({{H^{(k)}}^{\prime}}^{2}+\frac{2{H^{(k)}}^{2}{K^{(k)}}^{2}}{r^{2}}\right)\\\
&-\frac{\partial V}{\partial H^{(n)}}=0,\end{split}$ (9a) and
$\begin{split}&r^{2}\left(P^{(n)}{K^{(n)}}^{\prime}\right)^{\prime}-K^{(n)}\left\\{\left(M^{(n)}{g^{(n)}}^{2}r^{2}{H^{(n)}}^{2}\right)\right.\\\
&\left.+\left(P^{(n)}\,(1-{K^{(n)}}^{2})\right)\right\\}=0,\end{split}$ (9b)

where the prime denotes differentiation with respect to the radial coordinate.
These equations constitute a set of $2N$ coupled and nonlinear ordinary
differential equations. This system is very hard to solve, even when we choose
simple couplings among the $N$ sectors with ${\rm SU(2)}$ symmetry to complete
the model. For this reason, let us substitute the Ansatz into (6) and take the
same assumptions used in that derivation to find the first order equations in
the presence of spherical symmetry. For each $n$, there result two independent
equations, obtained by projecting (6) into directions parallel and orthogonal
to $\phi^{(n)}$, so that the energy is minimized by configurations that solve
the $N$ pairs of first order equations

$\displaystyle{H^{(n)}}^{\prime}=\pm
P^{(n)}\frac{1-{K^{(n)}}^{2}}{g^{(n)}r^{2}},$ (10a)
$\displaystyle{K^{(n)}}^{\prime}=\mp g^{(n)}\frac{H^{(n)}K^{(n)}}{P^{(n)}},$
(10b)

for $n=1,...,N$. These configurations saturate the Bogomol’nyi bound, with
energy

$E=4\pi\sum_{k=1}^{N}\frac{v^{(k)}}{g^{(k)}},$ (11)

so that each $\rm{SU(2)}$ subsystem contributes with an additive factor $4\pi
v^{(k)}/g^{(k)}$ to the total energy. This, of course, is a reflection of the
fact that these solutions have unity topological charges in each sector. We
also note that the energy does not depend on any of the $P^{(n)}$, so the
Bogomol’nyi bound only depends on the asymptotic values of the fields, which
always approach the trivial vacuum solutions when the boundary conditions (8)
are satisfied.

We may choose the functions $P^{(n)}$ in a way that makes it easier to solve
the first order equations while still keeping the subsystems coupled in a non-
trivial way. In order to achieve this, we can consider several distinct
possibilities, some of them will be considered below to illustrate the general
situation. Before doing this, however, let us notice that the energy density
for solutions of the first order equations can be written in the form

$\rho(r)=\sum_{n=1}^{N}\rho_{n}(r),$ (12)

where

$\begin{split}\rho_{n}(r)=&\frac{2P^{(n)}{K^{(n)}}^{\prime
2}}{(g^{(n)}r)^{2}}+\frac{{H^{(n)}}^{\prime 2}}{P^{(n)}},\\\
=&\frac{P^{(n)}(1-K^{(n)2})^{2}}{(g^{(n)}r^{2})^{2}}+\frac{2(H^{(n)}K^{(n)})^{2}}{r^{2}P^{(n)}},\end{split}$
(13)

which represents the energy density for the $n$-th sector. In this sense,
since we are interested in constructing multimagnetic monopoles, which are
localized structures composed of several layers with distinct magnetic
profiles, we then concentrate on the case

$P^{(n)}(\\{|\phi|\\})=\prod_{k=1}^{N}\left(\frac{1}{|\phi^{(k)}|}\right)^{\alpha_{k}^{n}},$
(14)

where the real and non-negative quantities $\alpha_{k}^{n}$ identify the set
of parameters that adds interaction among the $N$ subsystems. Each particular
choice of the set $\\{\alpha^{n}_{k}\\}$ describe a specific model, with the
understanding that the zeroes of $\\{\alpha^{n}_{k}\\}$ remove the dependence
of $P^{(n)}$ on the $k$-th scalar field. As we will show below, the specific
values of the non vanishing parameters contribute to identify the maximum and
width of the energy density associated to the corresponding magnetic shell in
the multimagnetic structure.

##  III MODELS WITH N=3

Let us first illustrate the aforementioned procedure with a few examples, in
which the Lagrangian is of the form (1), with gauge group
$\rm{SU}(2)\times\rm{SU}(2)\times\rm{SU}(2)$. For simplicity, let us take
$g^{(1)}=g^{(2)}=g^{(3)}=1$ and $v^{(1)}=v^{(2)}=v^{(3)}=1$ in our
calculations and, since the upper and lower signs in equations (10) are
related by changing $H\to-H$, we then only consider the upper signs in the
investigation that follows.

###  III.1 First Model

As a first example, we shall consider a model with a standard monopole core,
meaning $P^{(1)}=1$. According to the form suggested in Eq. (14), we are
sending $\alpha_{1}^{1}$, $\alpha_{2}^{1}$ and $\alpha_{3}^{1}$ to zero. The
$k=1$ equations in (10) are thus

$\displaystyle H^{(1)^{\prime}}$ $\displaystyle=\frac{1-K^{(1)}}{r^{2}},$
(15a) $\displaystyle K^{(1)^{\prime}}$ $\displaystyle=-H^{(1)}K^{(1)},$ (15b)

which are simply the Bogomol’nyi equations for the ’t Hooft-Polyakov monopole,
for which an analytic solution has long been known ps :

$\displaystyle H^{(1)}(r)=\coth(r)-\frac{1}{r},$ $\displaystyle
K^{(1)}(r)=r\;\rm{csch}(r).$ (16)

The energy density for this solution can be found from (13) to be

$\rho_{1}(r)=\frac{[r^{2}-\sinh^{2}(r)]^{2}}{r^{4}\sinh^{4}(r)}+\frac{2(r\coth(r)-1)^{2}}{r^{2}\sinh^{2}(r)}.$
(17)

We may now proceed as in bimag  and introduce a shell structure by the choices
$P^{(2)}=|\phi^{(1)}|^{-\alpha}$ and $P^{(3)}=|\phi^{(2)}|^{-\beta}$, where
$\alpha=\alpha^{2}_{1}$ and $\beta=\alpha^{3}_{2}$ are now positive integers,
with all the other parameters vanishing. With these choices, we are led to the
equations

$\displaystyle
H^{(2)^{\prime}}=\left(\frac{1}{r\coth(r)-1}\right)^{\alpha}\frac{1-{K^{(2)}}^{2}}{r^{2-\alpha}},$
(18a) $\displaystyle
K^{(2)^{\prime}}=-H^{(2)}K^{(2)}\left(\coth(r)-\frac{1}{r}\right)^{\alpha}.$
(18b)

and

$\displaystyle
H^{(3)^{\prime}}=\left(\frac{1}{H^{(2)}}\right)^{\beta}\frac{1-{K^{(3)}}^{2}}{r^{2}},$
(19a) $\displaystyle
K^{(3)^{\prime}}=-H^{(3)}K^{(3)}\left(H^{(2)}\right)^{\beta}.$ (19b)

Note that we have inserted the solution (16) for $H^{(1)}(r)$ in equations
(18), resulting in a system of two first order equations which can now be
solved for the two unknown functions $H^{(2)}$ and $K^{(2)}$. We have not been
able to obtain $H^{(2)}$ and $K^{(2)}$ in closed form, but the problem is
amenable to numerical analysis.

Once the full equations are solved numerically, we may plug the result
$H^{(2)}$ in (19) in order to solve these equations. We have done this for the
case $\alpha=3$ and $\beta=10$, and present the results in Figs. 1, 2 and 3.
One sees that in all cases these functions present a behavior that is
qualitatively similar to that of the standard ’t Hooft-Polyakov Monopole. The
contributions $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ to the energy density can
be calculated from (13). Their behavior are represented in Figs. 1, 2 and 3.
The last two contributions are strikingly different from the standard monopole
solution (which has energy density equal to what we here have called
$\rho_{1}$), presenting a maximum for $r\neq 0$.

Figure 1: In the left panel, we show the solutions $H^{(1)}(r)$ (ascending
line) and $K^{(1)}(r)$ (descending line) of (15). Their analytical form is
given by (16). In the right panel, we show the contribution $\rho_{1}$ to the
energy density, given by (17).

Figure 2: In the left panel, we show the solutions $H^{(2)}(r)$ (ascending
line) and $K^{(2)}(r)$ (descending line) of (18) for $\alpha=3$. In the right
panel, we show the contribution $\rho_{2}$, calculated from (13), relative to
this sector.

Figure 3: In the left panel, we show the solutions $H^{(3)}(r)$ (ascending
line) and $K^{(3)}(r)$ (descending line) of (19) for $\alpha=3$ and
$\beta=10$. In the right panel, we show the contribution $\rho_{3}$,
calculated from (13), relative to this sector. Here, $\epsilon=0.0024$.

To help us understand the above results, we depict in Fig. 4 a planar section
of the energy density, in which the contributions $\rho_{1}$, $\rho_{2}$ and
$\rho_{3}$ have been plotted simultaneously, with the colors red, blue and
green, respectively. We see how a novel structure, comprised by a standard
monopole core and two shells, arises from the superposition of the different
subsystems. The thickness of the shells and the holes associated to them are
controlled by the choice of the parameters $\alpha$ and $\beta$, and the
maxima of $\rho_{1}(r)$, $\rho_{2}(r)$ and $\rho_{3}(r)$ are attained at
$r=0$, $r\approx 2.6$ and $r\approx 7.1$, respectively.

The total energy could be calculated by direct integration of
$\rho=\rho_{1}+\rho_{2}+\rho_{3}$, but these solutions solve the first order
equations, so we may simply apply (11) to find $E=12\pi$.

Figure 4: Planar section, passing through the center, of the energy density,
in which the contributions $\rho_{1}$ (red), $\rho_{2}$ (blue) and $\rho_{3}$
(green) calculated in subsection III.1 are shown, giving rise to a
multimagnetic structure. We depict the case $\alpha=3$ and $\beta=10$.

###  III.2 Second Model

As our second example, we abandon the standard core of the previous one, and
take $P^{(1)}=|\phi^{(1)}|^{-\alpha}$, implying a self-interaction, as well as
$P^{(2)}=|\phi^{(1)}|^{-\beta}$ and $P^{(3)}=|\phi^{(2)}|^{-\gamma}$. This
means that we are now using $\alpha=\alpha^{1}_{1}$, $\beta=\alpha^{2}_{1}$,
and $\gamma=\alpha^{3}_{2}$, with all other parameters being zero. This
combination of parameters results in a triple shell structure distinct of the
previous one, now with a hollow center and almost empty shells in between the
red and blue and the blue and green portions of the multimagnetic structure.
In this example, the system of first order equations takes the form

$\displaystyle
H^{(1)^{\prime}}=\left(\frac{1}{H^{(1)}}\right)^{\alpha}\frac{1-{K^{(1)}}^{2}}{r^{2}},$
(20a) $\displaystyle
K^{(1)^{\prime}}=-H^{(1)}K^{(1)}\left(H^{(1)}\right)^{\alpha},$ (20b)

and

$\displaystyle
H^{(2)^{\prime}}=\left(\frac{1}{H^{(1)}}\right)^{\beta}\frac{1-{K^{(2)}}^{2}}{r^{2}},$
(21a) $\displaystyle
K^{(2)^{\prime}}=-H^{(2)}K^{(2)}\left(H^{(1)}\right)^{\beta},$ (21b)

and

$\displaystyle
H^{(3)^{\prime}}=\left(\frac{1}{H^{(2)}}\right)^{\gamma}\frac{1-{K^{(3)}}^{2}}{r^{2}},$
(22a) $\displaystyle
K^{(3)^{\prime}}=-H^{(3)}K^{(3)}\left(H^{(2)}\right)^{\gamma}.$ (22b)

Closed form solutions have not been discovered for any of the functions in
this example, but we may solve these equations numerically. Once again, we
find that the $H^{(n)}$ and $K^{(n)}$ present profiles that are qualitatively
similar to those of the standard monopole solution. These solutions, as well
as the individual contributions to the energy density, are represented in
Figs. 5, 6, 7, and 8, for $\alpha=3$, $\beta=10$ and $\gamma=30$, to produce
the desired result, well distinct from the previous case; compare Figs. 4 and
8. The total energy is again equal to $12\pi$.

Figure 5: In the left panel, we show the solutions $H^{(1)}(r)$ (ascending
line) and $K^{(1)}(r)$ (descending line) of (20) for $\alpha=1$. In the right
panel, we show the contribution $\rho_{1}$, calculated from (13), relative to
this sector.

Figure 6: In the left panel, we show the solutions $H^{(2)}(r)$ (ascending
line) and $K^{(2)}(r)$ (descending line) of (21) for $\alpha=1,$ and
$\beta=10$. In the right panel, we show the contribution $\rho_{2}$,
calculated from (13), relative to this sector. Here we have $\epsilon=0.0027$.

Figure 7: In the left panel, we show the solutions $H^{(3)}(r)$ (ascending
line) and $K^{(3)}(r)$ (descending line) of (22) for $\alpha=1,\beta=10$ and
$\gamma=30$. In the right panel, we show the contribution $\rho_{3}$,
calculated from (13), relative to this sector. Here, $\epsilon=0.00022$.
Figure 8: Planar section, passing through the center, of the energy density,
in which the contributions $\rho_{1}$ (red), $\rho_{2}$ (blue) and $\rho_{3}$
(green) calculated in subsection III.2 are shown, giving rise to a
multimagnetic structure. We depict the case $\alpha=1$, $\beta=10$,
$\gamma=30$.

###  III.3 Third Model

We may also consider another possibility, which can be achieved by taking
$P^{(2)}$ and $P^{(3)}$ as functions of $|\phi^{(1)}|$ alone. We proceed in a
way similar to the previous two cases, but now we consider $P^{(1)}=1$,
$P^{(2)}=|\phi^{(1)}|^{-\alpha}$ and $P^{(3)}=|\phi^{(1)}|^{-\beta}$. This
leads to equations (15) and (18), coupled to the additional pair of equations

$\displaystyle
H^{(3)^{\prime}}=\left(\frac{1}{H^{(1)}}\right)^{\beta}\frac{1-{K^{(3)}}^{2}}{r^{2}},$
(23a) $\displaystyle
K^{(3)^{\prime}}=-H^{(3)}K^{(3)}\left(H^{(1)}\right)^{\beta}.$ (23b)

This modification does not change the four other equations, the functions
$H^{(1)},K^{(1)},H^{(2)}$ and $K^{(2)}$ are the same discussed in subsection
III.1. The substitution $H^{(1)}(r)=\coth(r)-1/r$ in (23) gives rise to a
system of ODEs that can be solved for $H^{(3)}$ and $K^{(3)}$. Their features
are shown in Fig. 9, for $\beta=12$. The multimagnetic structure that arises
from the solution of the equations of motion is depicted in Fig. 10, showing
another way to localize the red, blue and green shells of the triple
configuration; compare the three Figs. 4, 8 and 10.

Figure 9: In the left panel, we show the solutions $H^{(3)}(r)$ (ascending
line) and $K^{(3)}(r)$ (descending line) of (23) for $\beta=12$. In the right
panel, we show the contribution $\rho_{3}$, calculated from (13), relative to
this sector. Here, $\epsilon=0.0025$. Figure 10: Planar section, passing
through the center, of the energy density, in which the contributions
$\rho_{1}$ (red), $\rho_{2}$ (blue) and $\rho_{3}$ (green) calculated in
subsection III.3 are shown, giving rise to a multimagnetic structure. We
depict the case $\alpha=3$ and $\beta=12$.

###  III.4 Fourth Model

As a final example of the $N=3$ class of models, let us consider another
example, in which $P^{(1)}=1$, $P^{(2)}=|\phi^{(1)}|^{-\alpha}$ and
$P^{(3)}=(|\phi^{(1)}||\phi^{(2)}|)^{-\beta}$. This model is different from
the previous examples in that both the $\phi^{(1)}$ and $\phi^{(2)}$ fields
appear explicitly in the third sector, with first order equations that are
given by

$\displaystyle
H^{(3)^{\prime}}=\left(\frac{1}{H^{(1)}H^{(2)}}\right)^{\beta}\frac{1-{K^{(3)}}^{2}}{r^{2}},$
(24a) $\displaystyle
K^{(3)^{\prime}}=-H^{(3)}K^{(3)}\left(H^{(1)}H^{(2)}\right)^{\beta}.$ (24b)

These equations are coupled to (15) and (18), and we can plug in the results
for $H^{(1)}$ and $H^{(2)}$ to solve (24). This has been done for the case
$\alpha=3$ and $\beta=10$. The solutions of (24) are displayed in Fig. 11 and
the multimagnetic structure associated to this case is shown in Fig. 12.

##  IV MODELS WITH N=4

Let us now investigate another system, resulting in the group
$\rm{SU(2)}\times\rm{SU(2)}\times\rm{SU(2)}\times\rm{SU(2)}$. This requires
that we have four pairs of first order differential equations. The simplest
way to achieve this is by adding a fourth pair of fields to the systems
described in the previous section, and we investigate some distinct
possibilities below.

Figure 11: In the left panel, we show the solutions $H^{(3)}(r)$ (ascending
line) and $K^{(3)}(r)$ (descending line) of (24) for $\alpha=3$ and
$\beta=10$. In the right panel, we show the contribution $\rho_{3}$,
calculated from (13), relative to this sector. Here, $\epsilon=0.0036$. Figure
12: Planar section, passing through the center, of the energy density, in
which the contributions $\rho_{1}$ (red), $\rho_{2}$ (blue) and $\rho_{3}$
(green) calculated in subsection III.4 are shown, giving rise to a
multimagnetic structure. We depict the case $\alpha=3$ and $\beta=10$.

###  IV.1 First Model

As a first example, we take $P^{(1)}$, $P^{(2)}$ and $P^{(3)}$ as in our very
first model of the previous Section and, additionally, choose
$P^{(4)}=|\phi^{(3)}|^{-\gamma}$. We are then led to equations (15), (18) and
(19), with the additional pair of first order equations

$\displaystyle
H^{(4)^{\prime}}=\left(\frac{1}{H^{(3)}}\right)^{\gamma}\frac{1-{K^{(4)}}^{2}}{r^{2}},$
(25a) $\displaystyle
K^{(4)^{\prime}}=-H^{(4)}K^{(4)}\left(H^{(3)}\right)^{\gamma}.$ (25b)

Since a solution for $H^{(3)}$ has already been found numerically in
subsection III.1, we may plug this result in the equations above, and then
solve them numerically. The result for the case $\alpha=3$, $\beta=10$,
$\gamma=20$, along with the contribution $\rho_{4}$ to the energy density, is
shown in Fig. 13. We see that $\rho_{4}$ attains its maximum at $r\simeq
13.9$, and contributes to the energy with another $4\pi$ factor, raising its
value to $16\pi$.

Figure 13: In the left panel, we show the graph of the functions $H^{(4)}$
(ascending line) and $K^{(4)}$ (descending line) calculated from (25) which,
together with the functions obtained in subsection III.1, form a solution of
the equations of motion. In the right panel, we show the contribution
$\rho_{4}$, calculated from (13), relative to this sector. Both plots
correspond to the choices $\alpha=3$, $\beta=10$, $\gamma=20$ of the model’s
parameters. Here, $\epsilon=0.0005$.

Moreover, we depict in Fig. 14 the multimagnetic structure which results from
the superposition of the four subsystems considered, with the black color used
to describe the energy density $\rho_{4}$ of the new ${\rm SU(2)}$ symmetry
added to the system; compare with Fig. 4.

Figure 14: Planar section, passing through the center, of the four
contributions $\rho_{1}$ (red), $\rho_{2}$ (blue), $\rho_{3}$ (green) and
$\rho_{4}$ (black) relative to the model presented in subsection IV.1.

###  IV.2 Second Model

As another example, we present a solution with four shells. To achieve this,
we take $P^{1}$, $P^{2}$ and $P^{3}$ as in subsection IV.1, as well as
$P^{(4)}=|\phi^{(3)}|^{-\zeta}$. This case results in a system of equations
comprised by (20), (21) and (22), which are now coupled with the additional
pair of equations

$\displaystyle
H^{(4)^{\prime}}=\left(\frac{1}{H^{(3)}}\right)^{\zeta}\frac{1-{K^{(4)}}^{2}}{r^{2}},$
(26a) $\displaystyle
K^{(4)^{\prime}}=-H^{(4)}K^{(3)}\left(H^{(3)}\right)^{\zeta}.$ (26b)

The equations have again been solved through numerical procedures, and the
results can be seen in Fig. 15 below. The energy density $\rho_{4}(r)$ has a
peak at $r\approx 8.2$. We then present in Fig. 16 the multimagnetic structure
that arises from the superposition of the four subsystems considered.

Figure 15: In the left panel, we show the graph of the functions $H^{(4)}$ and
$K^{(4)}$ which solve (26) and, together with the functions calculated in
subsection III.2, form a solution of the equations of motion. In the right
panel, we show the contribution $\rho_{4}$ to the energy density. Both plots
correspond to the choices $\alpha=1$, $\beta=10$, $\gamma=30$, $\zeta=8$ of
the model’s parameters. Here, $\epsilon=0.0006$. Figure 16: Planar section,
passing through the center, of the four contributions $\rho_{1}$ (red),
$\rho_{2}$ (blue), $\rho_{3}$ (yellow) and $\rho_{4}$ (green) to the energy
density relative to the model presented in subsection IV.2.

In the case with $N=4$ there are several other possibilities, similar to the
case of $N=3$ which we have investigated above.

##  V DISCUSSION

In this work we studied the presence of magnetic monopoles in a relativistic
model with a modified Lagrangian that is invariant under $N$ copies of the
$\rm{SU(2)}$ symmetry, that is,
$\rm{SU_{(1)}(2)}\times\rm{SU_{(2)}(2)}\times\cdots\times\rm{SU_{(N)}(2)}$. We
developed a general investigation leading to the presence of first order
differential equations that solve the equations of motion and ensure the
presence of minimum energy configurations. We then discussed some examples
with $\rm N=3$ and $\rm N=4$, with shells on top of a standard monopole core
and in other cases, giving rise to novel onionlike structures. We believe that
these multimagnetic structures may find applications in several areas of
nonlinear science. In particular, the solutions generalize the bimagnetic
structures considered before in bimag , and they may find applications in
several different scenarios in condensed matter Estrader ; EstraderII ; PRL20
; nano .

The distinct multimagnetic structures that we have described in the present
work may respond differently under the action of an external magnetic field,
for instance, so they may be identified via a magnetic probe and, in this
sense, be of application in several distinct scenarios of current interest in
nonlinear physics in general. Instead of the replication of the $\rm{SU(2)}$
symmetry, we can also chose other models with gauge groups of the form
$G_{1}\times...\times G_{N}$, where the $G_{k}$ are themselves non Abelian
gauge groups. One particularly interesting example is found for the gauge
group $\rm{SO(10)}$. This particular grand unification theory has been studied
for quite some time SO(10) ; SO(10)II  and is considered a relatively simple
and promising unification model GUT ; SO10 . This gauge group may undertake
symmetry breaking to
$\rm{SU(3)}\times\rm{SU(2)}\times\rm{SU(2)}\times\rm{U(1)}$ as an intermediary
step before reduction to the gauge group of the standard model gonzalo . Thus,
we could take, for example, an $\rm{SU(3)}$ core with two shells provided by
the symmetry breaking from the $\rm{SU(2)}\times\rm{SU(2)}$ subgroup. Although
mathematically more challenging, this case is, in principle, a straightforward
extension of the examples provided here. In the same way, we could also
consider the SU(3) and SU(5) gauge groups, since they also support monopoles
PRD ; PRL  and may be used with distinct motivations. Another line of current
interest concerns the study of the magnetic monopoles found in the present
work interacting with gravity, to see how they change the standard scenario of
black holes in magnetic monopoles Wei . As we can see, the underlying
principle leading to the solutions studied in this work may lead to a wide
range of multimagnetic monopoles, differing by the gauge group and by the many
possible choices for the functions that modify the standard model. We hope
that the above results will foster new investigation on magnetic monopoles.

The main results of the present work motivate us to think of diminishing the
spatial dimension, investigating, for instance, vortices in planar systems. In
this case, the basic $\rm U(1)$ symmetry can be enhanced to become $\rm
U_{1}(1)\times U_{2}(1)\times\cdots\times U_{N}(1)$, bringing into action a
system with $\rm N$ order parameters. This study can be developed in direct
connection with the works multilayered ; prl2 ; prl22 ; prl3 , and is
presently under consideration. It is also of interest to the case of multi-
component Gross-Pitaevskii equations to describe solutions in the case of
spin-1 and spin-2 Bose-Einstein condensates; see, e.g., Ref. PRep  and
references therein. We can also go further down and consider lineal systems,
and study kinks in one spatial dimension. Here we can follow the lines of
multikink , where we studied kinks in a model with $\rm Z_{2}\times Z_{2}$
symmetry. Extension to the case of the $\rm Z_{2}\times
Z_{2}\times\cdots\times Z_{2}$ symmetry can follow the present investigation,
and is also under consideration. We hope to report on some of these issues in
the near future.

Another line of investigation is related to the recent study in Ref. ahep ,
where the authors found magnetic monopole solution in a multivector boson
theory in which a replication of the basic SU(2) symmetry is also considered.
The construction of the magnetic monopole described in ahep  is based on the
dimensional deconstruction mechanism DD  and the Higgsless theory HL , and
this is different from the procedure which we described in the present work,
where we focus on stable minimum energy configurations with multimagnetic
properties. We would also like to add that the multimagnetic structures which
appeared in this work may be constructed under the guidance of the recent
advances on the study of multishelled hollow micro-nanostructures with triple
or more shells; see, e.g., Refs. AAA ; BBB  and references therein. In these
investigations, the main motivation included potential applications mainly in
the fields of energy conversion and storage, sensors, photocatalysis, and drug
delivery, but the above results may foster investigations concerning the
presence of shells with distinct magnetic properties.

###### Acknowledgements.

The work is supported by the Brazilian agencies Coordenação de Aperfeiçoamento
de Pessoal de Nível Superior (CAPES), grant No 88887.485504/2020-00 (MAL),
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), grants
No. 303469/2019-6 (DB) and No. 404913/2018-0 (DB), and by Paraiba State
Research Foundation (FAPESQ-PB) grant No. 0015/2019.

## References

  * (1) P. A. M. Dirac, Proc. R. Soc. Lond. A 133, 60 (1931).
  * (2) P. A. M. Dirac, Phys. Rev. 74, 817 (1948).
  * (3) G. ’ t Hooft, Nucl. Phys. B 79, 276 (1974).
  * (4) A. M. Polyakov, JETP Lett. 20, 194 (1974).
  * (5) M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. 35, 760 (1975).
  * (6) E. B. Bogomol’nyi, Sov. J. Nucl. Phys. 24, 449 (1976).
  * (7) J. Arafune, P. G. O. Freund, and C. J. Goebel, J. Math Phys. 16, 433 (1975).
  * (8) P. Rossi, Phys. Reports 86, 317 (1982).
  * (9) N. Manton and P. Sutcliffe, Topological Solitons, Cambridge University Press (2004).
  * (10) B. M. Einhorn, D. L. Stein, Phys. Rev. D 21, 3295 (1980).
  * (11) A. H. Guth, Phys. Rev. D 23, 347 (1981).
  * (12) K. A. Olive, D. Seckel, Primordial Inflation and the Monopole Problem, J.L. Stone (eds), Monopole 83 111, Springer (1984).
  * (13) E. Witten, Nucl. Phys. B 249, 557 (1985).
  * (14) M. Shifman, Phys. Rev. D 87, 025025 (2013).
  * (15) D. Bazeia, M. A. Marques, and R. Menezes, Phys. Rev. D 97, 105024 (2018).
  * (16) D. Bazeia, M. A. Marques, and Gonzalo J. Olmo, Phys. Rev. D 98, 025017 (2018).
  * (17) D. Bazeia, M. A. Marques, and R. Menezes, Phys. Rev. D 98, 065003 (2018).
  * (18) D. Bazeia, M. A. Liao, M. A. Marques, and R. Menezes Phys. Rev. Research 1, 033053 (2019).
  * (19) E. Babaev, J. Jaykka and M. Speight, Phys. Rev. Lett. 103, 237002 (2009).
  * (20) E. Babaev, J. Carlström, and M. Speight, Phys. Rev. Lett. 105, 067003 (2010).
  * (21) J. Garaud, J. Carlström, and E. Babaev, Phys. Rev. Lett. 107, 197001 (2011).
  * (22) Y. Kawaguchi and M. Ueda, Phys. Reports 520, 253 (2012).
  * (23) D. Bazeia, M. A. Liao, and M. A. Marques, Eur. Phys. J. Plus 135, 383 (2020).
  * (24) D. Bazeia, A. Mohammadi, and D. C. Moreira, Phys Rev. D 103, 025003 (2021).
  * (25) T. J. Hollowood and V. V. Khoze, Nucl. Phys. B 575, 78 (2000).
  * (26) M. Estrader et al., Nat. Commun. 4, 2960 (2013).
  * (27) A. López-Ortega, M. Estrader, G. Salazar-Alvarez, A. G. Roca, and J. Nogués, Phys. Rep. 553, 1 (2015).
  * (28) I. Groza, R. Morel, A. Brenac, C. Beigne, C. and L. Notin IEEE Trans. Magn. 47, 3355 (2011).
  * (29) J.-H. Lee, J.-T. Jang, J.-S. Choi, S. H. Moon, S.-H. Noh, J.-W. Kim, J.-G. Kim, I.-S. Kim, K. I. Park, and J. Cheon, Nat. Nanotechnol. 6, 418 (2011).
  * (30) V. Skumryev, S. Stoyanov, Y. Zhang, G. Hadjipanayis, D. Givord and J. Nogués, Nature, 423, 850 (2003).
  * (31) R. F. L. Evans, R. Yanes, O. Mryasov, R. W. Chantrell and O. Chubykalo-Fesenko, EPL, 88, 57004 (2009).
  * (32) G. Salazar-Alvarez et al., J. Am. Chem. Soc. 133, 16738 (2011).
  * (33) L. Catala, D. Brinzei, Y. Prado, A. Gloter, O. Stéphan, G. Rogez and T. Mallah, Angew. Chem., Int. Ed. 48, 183 (2008).
  * (34) C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature 451, 42 (2008).
  * (35) T. Fennell et al., Science 326, 415 (2009).
  * (36) N. Kanazawa et al., Phys. Rev. Lett. 125, 137202 (2020).
  * (37) O. Gould and A. Rajantie, Phys. Rev. Lett. 119, 241601 (2017).
  * (38) B. Acharya et al. Int. J. Mod. Phys. A 29, 1430050 (2014).
  * (39) F. A. Cardona et al., RSC Adv. 6, 77558 (2016).
  * (40) H. Fritzsch and P. Minkowski, Ann. Phys. (N.Y.) 93, 193 (1975).
  * (41) H. Georgi, AIP Conf. Proc. 23, 575 (1975).
  * (42) T. Fukuyama, Int. J. Mod. Phys. A 28, 1330008 (2013).
  * (43) G. G. Ross, Grand Unified Theories (Addison-Wesley, 1985).
  * (44) F. F. Deppisch, T. E. Gonzalo and L. Graf, Phys. Rev. D 96, 055003 (2017).
  * (45) P. Irwin, Phys. Rev. D 56, 5200 (1997).
  * (46) L. Pogosian, D. A. Steer, and T. Vachaspaty, Phys. Rev. Lett. 90, 061801 (2003).
  * (47) K. Lee, V. P. Nair, and E. J. Weinberg, Phys. Rev. D 45, 2751 (1992).
  * (48) K. Kobayashi, N. Kan, and K. Shiraishi, Adv. High Energy Phys. 2018, 2396275 (2018).
  * (49) N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Rev. Lett. 86, 4757 (2001).
  * (50) C. Csáki, C. Grojean, L. Pilo, and J. Terning, Phys. Rev. Lett. 92, 101802 (2004).
  * (51) X. Y. Lai, J. Li, B. A. Korgel, Z. H. Dong, Z. M. Li, F. B. Su, J. Du, and D. Wang, Angew. Chem. Int. Ed. 50, 2738 (2011).
  * (52) J. Qi et al., Chem. Soc. Rev. 44, 6749 (2015).