# BPS Skyrme Submodels of The Five-dimensional Skyrme Model

Emir Syahreza Fadhilla1<EMAIL_ADDRESS>Bobby Eka Gunara1
<EMAIL_ADDRESS>(Corresponding author) Ardian Nata Atmaja2
<EMAIL_ADDRESS>1Theoretical High Energy Physics Research Division,
Institut Teknologi Bandung, Jl. Ganesha 10 Bandung 40132, Indonesia.  2
Research Center for Physics, Indonesian Institute of Sciences (LIPI), Kompleks
PUSPIPTEK Serpong, Tangerang 15310, Indonesia.

###### Abstract

In this paper, we search for the BPS skyrmions in some BPS submodels of the
generalized Skyrme model in five-dimensional spacetime using the BPS
Lagrangian method. We focus on the static solutions of the Bogomolny’s
equations and their corresponding energies with topological charge $B>0$ is an
integer. We consider two main cases based on the symmetry of the effective
Lagrangian of the BPS submodels, i.e. the spherically symmetric and non-
spherically symmetric cases. For the spherically symmetric case, we find two
BPS submodels. The first BPS submodels consist of a potential term and a term
proportional to the square of the topological current. The second BPS
submodels consist of only the Skyrme term. The second BPS submodel has BPS
skyrmions with the same topological charge $B>1$, but with different energies,
that we shall call “topological degenerate” BPS skyrmions. It also has the
usual BPS skyrmions with equal energies, if the topological charge is a prime
number. Another interesting feature of the BPS skyrmions, with $B>1$, in this
BPS submodel, is that these BPS skyrmions have non-zero pressures in the
angular direction. For the non-spherically symmetric case, there is only one
BPS submodel, which is similar to the first BPS submodel in the spherically
symmetric case. We find that the BPS skyrmions depend on a constant $k$ and
for a particular value of $k$ we obtain the BPS skyrmions of the first BPS
submodel in the spherically symmetric case. The total static energy and the
topological charge of these BPS skyrmions also depend on this constant. We
also show that all the results found in this paper satisfy the full field
equations of motions of the corresponding BPS submodels.

###### Contents

  1. I Introduction
  2. II Static $O(5)$ Skyrme Model
  3. III BPS Skyrme Submodels in the Spherically Symmetric Lagrangian Density
    1. III.1 BPS Skyrme submodel with $\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$
    2. III.2 BPS Skyrme submodel with $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$
  4. IV BPS Skyrme Submodel in the Non-Spherically Symmetric Lagrangian Density
  5. V Conclusions
  6. A Full Field Equations of Motion
    1. A.1 BPS Skyrme submodel with $\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$
    2. A.2 BPS Skyrme submodel with $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$
      1. A.2.1 Type I potential $V(\phi^{5})$
      2. A.2.2 Type II Potential $V(\phi^{1},\phi^{2})$ and Type III Potential $V(\phi^{3},\phi^{4})$
      3. A.2.3 Type III potential, $V(\phi^{3},\phi^{4})$
  7. B Submodels With Four Derivative Terms
    1. B.1 Submodel with $\lambda_{1}=\lambda_{2}=\lambda_{4}=0$
    2. B.2 Submodel with $\lambda_{2}=\lambda_{3}=\lambda_{4}=0$

##  I Introduction

The Skyrme model was initially designed to be a model of the nucleon Skyrme
(1961, 1962). It has topological soliton solutions, known as skyrmions, which
can be interpreted as baryons. As topological solitons, the skyrmions are
characterized by a conserved topological charge $B$ which is identified as the
baryon number Witten (1983); Dothan and Biedenharn (1987). The energies of
these skyrmions satisfy the Faddeev-Bogomolny lower energy bound $E\geq|B|$
Faddeev (1976). Unfortunately, this bound can not be saturated by skyrmions
with $B\neq 0$ Manton (1987). Under a particular spherically symmetric ansatz,
it has been shown that a unit skyrmion (the nucleon), with $B=1$, has an
energy about 23% higher than its Faddeev-Bogomolny lower bound Adkins  _et
al._ (1983). For other skyrmions with $B>1$, their binding energies are even
higher than the multiple unit skyrmions and hence mostly unstable Manton and
Sutcliffe (2007). Nevertheless, skyrmions are not only important in high
energy and nuclear physics but also play an important role in other areas of
physics from solid-state physics to nonlinear optics Donati  _et al._ (2016);
Khawaja and Stoof (2001); Baskaran (2011); Kiselev  _et al._ (2011); Fukuda
and Žumer (2011).

Several attempts have been made to find (more) stable skyrmions, in
particular, with $B>1$. These may include imposing non-spherically symmetric
ansatzs Braaten  _et al._ (1990); Hitchin  _et al._ (1995); Houghton and
Sutcliffe (1996). Other approaches are by considering additional higher-order
terms other than the Skyrme term. A good mathematical description of skyrmion,
found in Manton (1987), describes the skyrmions as a topologically nontrivial
map from a Riemannian manifold to another. This mathematical description
becomes the key to generalize the Skyrme model by adding the higher-order
term, first done in Marleau (1990), and it has been turned recently into a
combinatoric problem in Gudnason and Nitta (2017). A simple extension to the
Skyrme model is by adding a term proportional to the square of the topological
current density Adkins and Nappi (1985). This additional term turns out to be
responsible for stabilizing the skyrmions Jackson  _et al._ (1985).
Furthermore, BPS skyrmions, which saturate the Faddeev-Bogomolny bound, are
found in a model consist only of this term and a potential term Adam  _et al._
(2010a). Although initially the construction of higher-order term is given for
four-dimensional spacetime, a possible generalization in higher dimensional
spacetime is demonstrated in Brihaye  _et al._ (2017). In that paper, they
only studied unit skyrmions (static and stationary), with $B=1$, since the
main goal was to find and study rotating black holes with Skyrme hair in five-
dimensional spacetime.

The main purpose of this work is to find other possible BPS skyrmions in the
generalized five-dimensional Skyrme model considered in Brihaye  _et al._
(2017). In particular, we will try to find BPS skyrmions, for any integer $B$,
in some (genuine) submodels of the generalized five-dimensional Skyrme model
via the BPS Lagrangian method, which was initially used to find BPS vortices
in three-dimensional spacetime models Atmaja (2017). The method not only
succeeded in re-deriving the known BPS vortices, but it was also able to find
new Bogomolny equations for other BPS vortices Atmaja (2020a). It has also
been used to find the BPS monopoles and dyons in various $SU(2)$ Yang-Mills-
Higgs models in the four-dimensional spacetime Atmaja and Prasetyo (2018);
Atmaja (2020b). The reason for using this method is that it has been
successfully used to find the BPS skyrmions in some (genuine and nongenuine)
submodels of the generalized Skyrme model in the four-dimensional spacetime
Atmaja  _et al._ (2020)111We call genuine submodels of a generalized Skyrme
model for submodels that can be obtained by turning off one or more couplings
in the generalized Skyrme model, while non-genuine ones that cannot.. Two main
cases considered here are the spherically symmetric and non-spherically
symmetric cases, which are defined to be the symmetry of the static energy
density in a particular ansatz.

In section II we describe briefly the Skyrme model and the ansatz that we use
in this paper. Then, in section III we discuss submodels in the spherically
symmetric Lagrangian density which is divided into two subsections, submodel
with $\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$ and submodel with
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$. Submodels in the non-spherically
symmetric Lagrangian are discussed in section IV which is divided into two
subsections, submodel with one derivative term and submodels with four
derivative terms. We conclude the results in section V. Finally, we put some
additional computations that support our results in the Appendix.

##  II Static $O(5)$ Skyrme Model

As we know, the standard static Skyrme Model in $3+1$ dimension is a
$\mathbb{R}^{3}\rightarrow S^{3}$ mapping with the degree equals to one, in
which the dynamical equation minimizes the following action functional

$\mathcal{S}_{4}=-\int
d^{4}x\sqrt{-g}\left[\gamma_{1}g^{ij}~{}Tr(R_{i}R_{j})+\gamma_{2}g^{ij}g^{kl}~{}Tr([R_{i},R_{k}][R_{j},R_{l}])\right]~{},$
(1)

with $R_{i}\equiv(\partial_{i}U)U^{\dagger}$ where
$U=\phi^{0}I_{2}+i\vec{\phi}.\vec{\sigma}$ is an $SU(2)$ valued chiral field
Manton and Sutcliffe (2007); Brihaye  _et al._ (2017) and $i,j,k,l$ are
spacetime indices. $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are Pauli
matrices and $\phi=(\phi^{0},\vec{\phi})$ are real scalar fields that satisfy
the $O(4)$ model condition, $\phi^{a}\phi^{a}=1$. This is the original model
proposed by Skyrme Skyrme (1961, 1962). On the other hand, one can construct
such a functional by introducing strain tensor $D=JJ^{T}$, where $J$ is the
Jacobean matrix of the map Manton (1987). This strain tensor characterizes the
distortion induced by the map. The energy functional of the static Skyrme
model can be constructed from the invariant of the strain tensor, which are
combinations of its eigenvalues. As a result of these arguments we can rewrite
the Lagrangian in (1) as a sum of these two terms, namely

$\displaystyle\phi^{a}_{i}\phi^{a}_{j}g^{ij}~{},$ (2)
$\displaystyle\phi^{a}_{[i}\phi^{b}_{j]}\phi^{a}_{[k}\phi^{b}_{l]}g^{ij}g^{kl}~{},$
(3)

where $\phi^{a}_{i}\equiv\frac{\partial\phi^{a}}{\partial x^{i}}$ (with
$a,b=0,...,3$) and $g^{ij}$ are components of the inverse of the spacetime
metric. This construction of the Skyrme model gives rise to the generalization
for the higher order terms given in Gudnason and Nitta (2017) and the Skyrme
model in five dimensions given in Brihaye  _et al._ (2017).

Suppose we have $D$ eigenvalues from a $d+1$ dimensional Skyrme Model as
$\lambda_{1}^{2},\lambda_{2}^{2},\dots,\lambda_{d}^{2}$. The corresponding
Lagrangian can be written in the most general way as

$L=\gamma_{0}V+\sum_{n=1}^{d}\gamma_{n}L_{n}~{},$ (4)

where $\gamma_{0},\gamma_{n}\in\mathbb{R}$ are coupling constants, $V\equiv
V(\phi)$ is an arbitrary potential, and $L_{n}$ satisfy

$\displaystyle L_{1}$ $\displaystyle\propto$
$\displaystyle\lambda_{1}^{2}+\dots+\lambda_{d}^{2}~{},$ (5) $\displaystyle
L_{2}$ $\displaystyle\propto$
$\displaystyle\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{1}^{2}\lambda_{3}^{2}+\dots+\lambda_{d-1}^{2}\lambda_{d}^{2}~{},$
(6) $\displaystyle\vdots$ $\displaystyle L_{d}$ $\displaystyle\propto$
$\displaystyle\lambda_{1}^{2}\lambda_{2}^{2}\dots\lambda_{d}^{2}~{}.$ (7)

In four-dimensional case ($d=3$), this gives the generalized Skyrme model
which is just the traditional model of Skyrme added with the BPS-Skyrme term
($L_{3}\propto\lambda_{1}^{2}\lambda_{2}^{2}\lambda_{3}^{2}$) proposed in Adam
_et al._ (2010b). By following the prescription given in the four-dimensional
model, we can construct $L$ for arbitrary spacetime dimensions with

$\displaystyle L_{n}$ $\displaystyle=$
$\displaystyle\frac{F^{2n}}{\left(n!\right)^{2}}~{},$ (8) $\displaystyle
F^{2n}$ $\displaystyle\equiv$
$\displaystyle\phi^{a_{1}}_{[i_{1}}\dots\phi^{a_{n}}_{i_{n}]}\phi^{a_{1}}_{[j_{1}}\dots\phi^{a_{n}}_{j_{n}]}g^{i_{1}j_{1}}\dots
g^{i_{n}j_{n}}~{},$ (9)

where $a_{1},...,a_{n}=1,...,d+1$ are internal indices, whereas $i$s and $j$s
are spacetime indices, and $\phi^{a_{1}}_{[i_{1}}\dots\phi^{a_{n}}_{i_{n}]}$
is totally antisymetric tensor Brihaye  _et al._ (2017). Therefore the
Lagrangian of the model can be written in the definition above as

$L=\gamma_{0}V+\sum_{n=1}^{d}\frac{\gamma_{n}}{\left(n!\right)^{2}}\phi^{a_{1}}_{[i_{1}}\dots\phi^{a_{n}}_{i_{n}]}\phi^{a_{1}}_{[j_{1}}\dots\phi^{a_{n}}_{j_{n}]}g^{i_{1}j_{1}}\dots
g^{i_{n}j_{n}}.$ (10)

This is just an $O(d+1)$ model generalization as proposed in Brihaye  _et al._
(2017) if we take the potential to be $V=1-\phi^{d+1}$. As a summary, the
action functional for $d+1$ dimensional Skyrme model takes the form

$\mathcal{S}_{d+1}=-\int
dx^{d+1}\sqrt{-g}\left[\gamma_{0}V+\sum_{n=1}^{d}\frac{\gamma_{n}}{\left(n!\right)^{2}}\phi^{a_{1}}_{[i_{1}}\dots\phi^{a_{n}}_{i_{n}]}\phi^{a_{1}}_{[j_{1}}\dots\phi^{a_{n}}_{j_{n}]}g^{i_{1}j_{1}}\dots
g^{i_{n}j_{n}}\right].$ (11)

Now let us focus on the five-dimensional Skyrme model discussed in Brihaye
_et al._ (2017) which is the main discussion of the paper. In this section we
review shortly about it. In particular, we consider a static case in which the
spacetime metric can be written down in terms of bipolar spherical coordinates

$ds^{2}=-dt^{2}+dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta~{}d\varphi_{1}^{2}+\cos^{2}\theta~{}d\varphi_{2}^{2}\right)~{},$
(12)

where $-\infty<t<\infty$ and $0\leq r<\infty$ are the time and radial
coordinates, respectively, while $0\leq\theta\leq{\pi\over 2}$ is the polar
coordinate and $0\leq\varphi_{1,2}<2\pi$ are the azimuthal coordinates. Then,
the form of the effective Lagrangian density and the energy-momentum tensor
will be cast in terms of these coordinates using (12).

The five-dimensional Skyrme model can be thought of as an $O(5)$ sigma model
of real fields $\phi^{a}$, $a=1,\ldots,5$, satisfying the constraint
$\phi^{a}\phi^{a}=1$ Brihaye  _et al._ (2017). Here, we consider the
generalized Skyrme model with the following Lagrangian density

$L_{eff}=\lambda_{0}V\left(\phi\right)+\lambda_{1}F^{2}+\frac{\lambda_{2}}{4}F^{4}+\frac{\lambda_{3}}{36}F^{6}+\frac{\lambda_{4}}{576}F^{8}~{},$
(13)

where $\lambda_{n}$ is the coupling of the $2n$-th order term, with
$n=0,\ldots,4$. Here $V$ is the potential (zeroth order) term. We choose a
possible static ansatz for the scalar field that satisfies the constraint
$\phi^{a}\phi^{a}=1$, that is

$\displaystyle\phi^{1}+i\phi^{2}$
$\displaystyle=\sin\xi\frac{f}{\sqrt[]{1+f^{2}}}e^{ig_{1}}~{},$ (14a)
$\displaystyle\phi^{3}+i\phi^{4}$
$\displaystyle=\sin\xi\frac{1}{\sqrt[]{1+f^{2}}}~{}e^{ig_{2}}~{},$ (14b)
$\displaystyle\phi^{5}$ $\displaystyle=\cos\xi~{},$ (14c)

where $\xi\equiv\xi(r)$, $f\equiv f(\theta)$, $g_{1}\equiv
g_{1}\left(\varphi_{1}\right)$, and $g_{2}\equiv
g_{2}\left(\varphi_{2}\right)$ are defined to be the effective fields such
that the Lagrangian density (13) can be recast into an effective Lagrangian
density

$L_{eff}=L_{0}+L_{1}+L_{2}+L_{3}+L_{4}~{},$ (15)

where

$L_{0}=\lambda_{0}V\left(\phi\right)~{},$ (16) $\displaystyle
L_{1}=\lambda_{1}\left[\left(\xi^{\prime}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}}\left(\frac{f^{\prime}}{1+f^{2}}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}\sin^{2}\theta~{}}\frac{\left(fg_{1}^{\prime}\right)^{2}}{1+f^{2}}+\frac{\sin^{2}\xi}{r^{2}\cos^{2}\theta~{}}\frac{\left(~{}g_{2}^{\prime}~{}\right)^{2}}{1+f^{2}}\right]~{},$
(17) $\displaystyle L_{2}$ $\displaystyle=$
$\displaystyle\lambda_{2}\left[\frac{\sin^{2}\xi}{r^{2}}\frac{1}{\left(1+f^{2}\right)^{2}}\left(\xi^{\prime}f^{\prime}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}\sin^{2}\theta~{}}\frac{f^{2}}{\left(1+f^{2}\right)}\left(\xi^{\prime}g_{1}^{\prime}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}\cos^{2}\theta~{}}\frac{1}{\left(1+f^{2}\right)}\left(\xi^{\prime}g_{2}^{\prime}\right)^{2}\right.$
(18)
$\displaystyle\qquad\left.+\frac{\sin^{4}\xi}{r^{4}\sin^{2}\theta~{}}\frac{f^{2}}{\left(1+f^{2}\right)^{3}}\left(f^{\prime}g_{1}^{\prime}\right)^{2}+\frac{\sin^{4}\xi}{r^{4}\cos^{2}\theta~{}}\frac{1}{\left(1+f^{2}\right)^{3}}\left(f^{\prime}g_{2}^{\prime}\right)^{2}\right.$
$\displaystyle\qquad\left.+\frac{\sin^{4}\xi}{r^{4}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{2}}\left(g_{1}^{\prime}g_{2}^{\prime}\right)^{2}\right],$
$\displaystyle L_{3}$ $\displaystyle=$
$\displaystyle\lambda_{3}\left[\frac{1}{r^{4}\sin^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{3}}\sin^{4}\xi\left(\xi^{\prime}f^{\prime}g_{1}^{\prime}\right)^{2}+\frac{1}{r^{4}\cos^{2}\theta}\frac{1}{\left(1+f^{2}\right)^{3}}\sin^{4}\xi\left(\xi^{\prime}f^{\prime}g_{2}^{\prime}\right)^{2}\right.$
(19)
$\displaystyle\qquad\left.+\frac{1}{r^{6}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(f^{\prime}g_{1}^{\prime}g_{2}^{\prime}\right)^{2}\right.$
$\displaystyle\qquad\left.+\frac{1}{r^{4}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{2}}\sin^{4}\xi\left(\xi^{\prime}g_{1}^{\prime}g_{2}^{\prime}\right)^{2}\right]~{},$
$L_{4}=\lambda_{4}\left[\frac{1}{r^{6}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(\xi^{\prime}f^{\prime}g_{1}^{\prime}g_{2}^{\prime}\right)^{2}\right]~{}.$
(20)

where $\xi^{\prime}\equiv\frac{d\xi}{dr}$,
$f^{\prime}\equiv\frac{df}{d\theta}$,
$g_{1}^{\prime}\equiv\frac{dg_{1}}{d\varphi_{1}}$, and
$g_{2}^{\prime}\equiv\frac{dg_{2}}{d\varphi_{2}}$ are derivatives of the
effective fields with respect to their arguments. Using the ansatz (14), the
topological charge has a simple form of

$B=-\frac{3}{8\pi^{2}}\int\sin^{3}\xi\frac{f}{\left(1+f^{2}\right)^{2}}d\xi
dfdg_{1}dg_{2}~{},$ (21)

We would also like to point out the expression for the components of the
energy-momentum tensor for this model as follows

$T_{ij}=\lambda_{0}T_{ij}^{(0)}+\lambda_{1}T_{ij}^{(1)}+\lambda_{2}T_{ij}^{(2)}+\lambda_{3}T_{ij}^{(3)}+\lambda_{4}T_{ij}^{(4)}~{},$
(22)

with

$\displaystyle T_{ij}^{(0)}$ $\displaystyle=$ $\displaystyle-g_{ij}V~{},$ (23)
$\displaystyle T_{ij}^{(1)}$ $\displaystyle=$ $\displaystyle
2\phi_{i}^{a}\phi_{j}^{a}-g_{ij}F^{2}~{},$ (24) $\displaystyle T_{ij}^{(2)}$
$\displaystyle=$
$\displaystyle\phi_{ik_{1}}^{ab}\phi_{jk_{2}}^{ab}g^{k_{1}k_{2}}-\frac{1}{4}g_{ij}F^{4}~{},$
(25) $\displaystyle T_{ij}^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\phi_{ik_{1}l_{1}}^{abc}\phi_{jk_{2}l_{2}}^{abc}g^{k_{1}k_{2}}g^{l_{1}l_{2}}-\frac{1}{36}g_{ij}F^{6}~{},$
(26) $\displaystyle T_{ij}^{(4)}$ $\displaystyle=$
$\displaystyle\frac{1}{72}\phi_{ik_{1}l_{1}m_{1}}^{abcd}\phi_{jk_{2}l_{2}m_{2}}^{abcd}g^{k_{1}k_{2}}g^{l_{1}l_{2}}g^{m_{1}m_{2}}-\frac{1}{576}g_{ij}F^{8}~{},$
(27)

that will be useful for our analysis in this paper. It is straightforward to
show that $T_{ij}=0$ for $i\neq j$ is within the choice of ansatz (14) and so
$T_{ij}$ is a diagonal matrix. In general, the spatial components of this
diagonal matrix are not necessarily zero as we shall see for the BPS skyrmions
with $B>1$ in subsection III.1.

##  III BPS Skyrme Submodels in the Spherically Symmetric Lagrangian Density

In the spherically symmetric case, one can simplify the effective Lagrangian
density (15) to depend only on the radial coordinate $r$ by choosing
$f\left(\theta\right)=\tan\theta$ ,
$g_{1}\left(\varphi_{1}\right)=n_{1}\varphi_{1}$ ,
$g_{2}\left(\varphi_{2}\right)=n_{2}\varphi_{2}$, with
$n_{1},n_{2}\in\mathbb{Z}$. It is necessary to impose a boundary condition
$\xi\left(0\right)=m\pi$ with $m\in\mathbb{Z}$ in order for all $\phi$’s to be
unique at the origin. This boundary condition leads to the topological charge
satisfying

$B=\frac{n_{1}n_{2}}{16}\left(9\cos\left(\xi_{\infty}\right)-\cos\left(3\xi_{\infty}\right)\pm
8\right)~{},$ (28)

with $\xi_{\infty}=\xi\left(r\rightarrow\infty\right)$, $\left(+\right)$ sign
is for odd $m$, and $(-)$ for even $m$. Here we are only looking for the BPS
skyrmions with $B\in\mathbb{Z}$ and we can choose to write $B=n_{1}n_{2}$.
Therefore the boundary condition for $\xi\left(r\rightarrow\infty\right)$ must
satisfy the following equations

$9\cos\left(\xi_{\infty}\right)-\cos\left(3\xi_{\infty}\right)\pm 8=16.$ (29)

If $m$ is odd then $\xi_{\infty}$ is equal to even multiples of $\pi$, while
even values of $m$ gives us complex numbers. Here we will use the same
boundary conditions as in Skyrme (1962) for all submodels discussed in this
paper, which are

$\displaystyle\xi\left(0\right)=\pi,\qquad\xi\left(r\rightarrow\infty\right)=0,$
(30)

and searching for the BPS skyrmions with $B>0$ such that $n_{1}$ and $n_{2}$
must have the same positive or negative signs. Here, without loss of
generality, we choose positive sign for both $n_{1}$ and $n_{2}$. Upon
substituting $f$, $g_{1}$, and $g_{2}$ mentioned above, the effective
Lagrangian density (15) is simplified further to

$\displaystyle L_{eff}$ $\displaystyle=$
$\displaystyle\left(\xi^{\prime}\right)^{2}\left[\lambda_{1}+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{3}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{4}\frac{\sin^{6}\xi}{r^{6}}\right]$
(31)
$\displaystyle+\left[\lambda_{0}V+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{1}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{3}\frac{\sin^{6}\xi}{r^{6}}\right].$

Because the derivative terms in the effective Lagrangian density (31) contain
only the first derivative of $\xi$, or $\xi^{\prime}$, the BPS-Lagrangian
method in Atmaja  _et al._ (2020) suggests that the corresponding $L_{BPS}$
should take the following form

$L_{BPS}=\frac{1}{r^{3}}\left(Q_{X}X+Q_{0}\right)~{},$ (32)

with $X\equiv\xi^{\prime}$. In the BPS limit $L_{eff}-L_{BPS}=0$,

$\displaystyle\left[\lambda_{1}+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{3}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{4}\frac{\sin^{6}\xi}{r^{6}}\right]X^{2}-\frac{1}{r^{3}}Q_{X}X$
$\displaystyle+\left[\lambda_{0}V+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{1}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{3}\frac{\sin^{6}\xi}{r^{6}}-\frac{Q_{0}}{r^{3}}\right]=0~{},$

which is a quadratic equation in $X$ and has two solutions

$X_{\pm}=\frac{Q_{X}}{2r^{3}\left[\lambda_{1}+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{3}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{4}\frac{\sin^{6}\xi}{r^{6}}\right]}\pm\sqrt{D}~{}.$
(34)

These two solutions coincide, which is a necessary condition for Bogomolny’s
equations to be valid, if $D=0$,

$\displaystyle\frac{Q_{X}^{2}}{4r^{6}}-\left[\lambda_{1}+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{3}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{4}\frac{\sin^{6}\xi}{r^{6}}\right]$
$\displaystyle\times\left[\lambda_{0}V+\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{1}\frac{\sin^{2}\xi}{r^{2}}+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}\frac{\sin^{4}\xi}{r^{4}}+n_{1}^{2}n_{2}^{2}\lambda_{3}\frac{\sin^{6}\xi}{r^{6}}-\frac{Q_{0}}{r^{3}}\right]=0~{}.$

Since equation (III) must be satisfied for any value of $r$ then we may
consider it as a polynomial equation of (explicit) $r$ in which each of its
“coefficients” are zero. Non-trivial solutions can only be achieved by setting
$Q_{0}=0$. The remaining vanishing “coefficients” are

$\displaystyle\lambda_{3}\lambda_{4}$ $\displaystyle=$ $\displaystyle 0~{},$
(36) $\displaystyle-\lambda_{2}\lambda_{4}$ $\displaystyle=$
$\displaystyle\lambda_{3}^{2}~{},$ (37) $\displaystyle-\lambda_{1}^{2}$
$\displaystyle=$ $\displaystyle V\lambda_{0}\lambda_{2}~{},$ (38)
$\displaystyle-\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)V\lambda_{0}\lambda_{3}$
$\displaystyle=$
$\displaystyle\left(\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)+\left(1+n_{1}^{2}+n_{2}^{2}\right)^{2}\right)\lambda_{1}\lambda_{2}~{},$
(39) $\displaystyle-
n_{1}^{2}n_{2}^{2}\left(1+n_{1}^{2}+n_{2}^{2}\right)\lambda_{1}\lambda_{4}$
$\displaystyle=$
$\displaystyle\left(n_{1}^{2}n_{2}^{2}\left(1+n_{1}^{2}+n_{2}^{2}\right)+\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)^{2}\right)\lambda_{2}\lambda_{3}~{},$
(40)
$\displaystyle\left(1+n_{1}^{2}+n_{2}^{2}\right)\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda_{2}^{2}$
$\displaystyle+$
$\displaystyle\left(\left(1+n_{1}^{2}+n_{2}^{2}\right)\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)+n_{1}^{2}n_{2}^{2}\right)\lambda_{1}\lambda_{3}$
$\displaystyle+n_{1}^{2}n_{2}^{2}V\lambda_{0}\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{Q_{X}^{2}}{4\sin^{6}\xi\left(r\right)}~{}.$ (41)

There are two non-trivial solutions that simultaneously solve these equations.
The first one is

${Q^{2}_{X}\over
4}=\left(1+n_{1}^{2}+n_{2}^{2}\right)\left(n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}\right)\lambda^{2}_{2}\sin^{6}\xi,$
(42)

with $\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$, that, after
substituting them into equation (34), gives us the Bogomolny equation

$\xi^{\prime}=\pm\gamma(n_{1},n_{2})\frac{\sin\xi}{r}~{},$ (43)

where $\gamma(n_{1},n_{2})$ is defined as

$\gamma(n_{1},n_{2})\equiv\sqrt{\frac{n_{1}^{2}n_{2}^{2}+n_{1}^{2}+n_{2}^{2}}{1+n_{1}^{2}+n_{2}^{2}}}~{}.$
(44)

The second one is

${Q^{2}_{X}\over 4}=n_{1}^{2}n_{2}^{2}V\lambda_{0}\lambda_{4}\sin^{6}\xi~{},$
(45)

with $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$, whose corresponding Bogomolny
equation is given by

$\xi^{\prime}=\pm~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}}{n_{1}n_{2}\sin^{3}\xi}\sqrt[]{V(\xi)}~{}.$
(46)

Both cases show the dependence of the resulting Bogomolny’s equation on the
topological charge $B=n_{1}n_{2}$. Let us discuss the features and solutions
of those Bogomolny’s equations in the next subsections.

###  III.1 BPS Skyrme submodel with
$\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$

Solutions to equation (43) are

$\xi=2\tan^{-1}\left[\left(\frac{r}{r_{0}}\right)^{\pm\gamma}\right]~{},$ (47)

with $r_{0}$ is a real integration constant. Here we only pick the $(-)$
signed solutions that satisfy the boundary conditions (30), while the $(+)$
signed solutions are related to the anti-BPS skyrmions, with $B<0$. These
solutions, with $n_{1}=n_{2}=1$ or a unit topological charge $B=1$, coincide
with the solutions studied in Brihaye  _et al._ (2017). The solutions are
smooth functions up to the second-order derivative in the whole domain, thus
everywhere regular. The total static energy density of a BPS skyrmion, with
topological charge $B$, in this model is given by

$\varepsilon_{B}=-T^{0}_{0}=32\lambda_{2}\left(B^{2}+n^{2}+2B\right)\frac{r^{4(\gamma-1)}r_{0}^{4\gamma}}{\left(r^{2\gamma}+r_{0}^{2\gamma}\right)^{4}}~{}.$
(48)

with $n\equiv n_{2}-n_{1}$. One may ask if these BPS Skyrmions saturate the
total energy bound. We can show that they do saturate the energy bound at the
effective level of total energy. For that purpose we write explicitly the
Bogomolny equation (43) as follows

$\sqrt{1+n^{2}+2B}\xi^{\prime}\mp\sqrt{B^{2}+n^{2}+2B}\frac{\sin\xi}{r}=0.$
(49)

In the static case, the total energy is proportional to the action,
$E=\frac{\lambda_{2}}{4}\int\sqrt{-g}~{}d^{4}x~{}F^{4}$. Substituting the
ansatz (14) into (25), the total energy becomes

$\displaystyle E$ $\displaystyle=$ $\displaystyle{\lambda_{2}\over
4}\int\sqrt{-g}d^{4}x\left[(1+n^{2}+2B)\xi^{\prime
2}+(B^{2}+n^{2}+2B)\frac{\sin^{2}\xi}{r^{2}}\right]\frac{\sin^{2}\xi}{r^{2}}$
(50) $\displaystyle=$ $\displaystyle{\lambda_{2}\over
4}\int\sqrt{-g}d^{4}x\left[\left(\sqrt{1+n^{2}+2B}\xi^{\prime}\mp\sqrt{B^{2}+n^{2}+2B}\frac{\sin\xi}{r}\right)^{2}\right.$
$\displaystyle\left.\pm
2\sqrt{(1+n^{2}+2B)(B^{2}+n^{2}+2B)}\frac{\sin\xi}{r}\xi^{\prime}\right]\frac{\sin^{2}\xi}{r^{2}}$
$\displaystyle\geq$ $\displaystyle\pm{\lambda_{2}\over
2}\int\sqrt{-g}d^{4}x\sqrt{(1+n^{2}+2B)(B^{2}+n^{2}+2B)}\frac{\sin^{3}\xi}{r^{3}}\xi^{\prime},$

where the last line is the boundary term. Therefore, the total energy of the
BPS Skyrmions is

$E_{B}=\pm\lambda_{2}\pi^{2}\int_{\xi(\infty)}^{\xi(0)}\sqrt{(1+n^{2}+2B)(B^{2}+n^{2}+2B)}\sin^{3}\xi~{}d\xi.$
(51)

The $T^{1}_{1}$ component of the stress-energy-momentum tensor vanishes while
the remaining diagonal components have the form

$\displaystyle T^{2}_{2}$ $\displaystyle=$ $\displaystyle
2\frac{\sin^{4}\xi}{r^{4}}\frac{(n^{2}+2B)(1-B^{2})}{1+n^{2}+2B}~{},$ (52)
$\displaystyle T^{3}_{3}$ $\displaystyle=$
$\displaystyle\frac{\sin^{4}\xi}{r^{4}}\left[n\sqrt{n^{2}+4B}(1+\gamma^{2})+\frac{(n^{2}+2B)(B^{2}-1)}{1+n^{2}+2B}\right]~{},$
(53) $\displaystyle T^{4}_{4}$ $\displaystyle=$
$\displaystyle\frac{\sin^{4}\xi}{r^{4}}\left[-n\sqrt{n^{2}+4B}(1+\gamma^{2})+\frac{(n^{2}+2B)(B^{2}-1)}{1+n^{2}+2B}\right]~{}.$
(54)

For $B=1$, which also implies $n=0$, all the remaining diagonal components of
the energy-momentum tensor vanish, while they are non-zero for $B>1$ in
general. The latter cases indicate that we might have BPS skyrmions with non-
zero pressures. Using (48), the total static energy of the BPS skyrmion is

$E_{B}=\frac{16}{3}\lambda_{2}\pi^{2}\sqrt{\left(1+n^{2}+2B\right)\left(B^{2}+n^{2}+2B\right)}~{}.$
(55)

This is in agreement with the result found in Brihaye  _et al._ (2017), that
the bound energy is linear in $B$, because
$\frac{\sqrt{(1+n^{2}+2B)(B^{2}+n^{2}+2B)}}{3}\geq B$ and the equality occurs
only for $B=1$. We can see a possible repulsive interaction in this BPS
submodel between $B$ unit-solitons in the multi-soliton solution, with $B>1$,
because the difference between the energy of the multi-soliton solution (55)
and $B$ times the unit-soliton energy, which is
$BE_{1}=16\lambda_{2}\pi^{2}B$, are positive and monotonically increasing. For
example the energy of $B=2$ multi-soliton is
$E_{2}=16\sqrt{6}~{}\lambda_{2}\pi^{2}$ which is higher than the total energy
of the two unit-solitons, $2E_{1}=32\lambda_{2}\pi^{2}$. This repulsive effect
implies the possibility of a soliton creation in the usual collision set-up,
similar to the one demonstrated in Vachaspati (2011).

Furthermore, for some fixed values of $B>1$, there are more than one BPS
skyrmions in which their energies, determined by different values of $n_{1}$
and $n_{2}$, are not equal. As an example for $B=4$ equation (43) has two
different solutions and thus yields two unequal energies:
$E_{4}=32\sqrt{6}\lambda_{2}\pi^{2}$, with $n=2-2=0$, and
$E_{4}=16\sqrt{66}\lambda_{2}\pi^{2}$, with $n=1-4=-3$ and $n=4-1=3$. The
graphical representation for this statement is given in figure 1. We shall
call these types of skyrmions “topological degenerate” BPS skyrmions which are
defined as BPS skyrmions with the same topological charges but different
energies. This feature arises from how many different ways we can assign
integer values on $n_{1}$ and $n_{2}$ resulting in the same value of $B$.
Interestingly, there are also BPS skyrmions without these degenerate
topological numbers in which their topological charges are prime numbers.

Figure 1: Plots for $(-)$ sign solutions of equation (47) with topological
charge $B=1$ (left) and $B=4$ (right). On the right plot, the blue curve is
for solution with $E_{4}=16\sqrt{66}\lambda_{2}\pi^{2}$ and the red curve is
for solution with $E_{4}=32\sqrt{6}\lambda_{2}\pi^{2}$.

The function $\gamma(n_{1},n_{2})$ is plotted in Figure 2 as a function of
$n_{1}$ and $n_{2}$, with $n_{1},n_{2}=1,\dots,10$. The curves with different
colors connect all configurations of $n_{1}$ and $n_{2}$ with the same
topological charge $B$.

Figure 2: Plot for values of $\gamma(n_{1}n_{2})$. The red dots shows
configurations in which the topological charges are prime numbers and the blue
dots are for the topological degenerate configurations.

###  III.2 BPS Skyrme submodel with $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$

In this submodel, solutions of the Bogomolny’s equation (46) depend on the
ratio of the coupling constants, $\frac{\lambda_{0}}{\lambda_{4}}$, and also
on the non-zero potential. In order to find the solutions of the Bogomolny’s
equation (46), we choose the potential to take the following form
$V(\xi)=\left(1-\cos\xi\right)^{q}$, with $q\in\mathbb{R}^{+}$, such that the
Bogomolny’s equation (46) can be rewritten as222Here again we pick the
negative sign solutions that satisfy the boundary conditions (30).

$\sin^{3}\left(\frac{\xi}{2}\right)\cos^{3}\left(\frac{\xi}{2}\right)\xi^{\prime}=-\sqrt{\frac{2^{2(q-3)}\lambda_{0}}{\lambda_{4}}}\frac{r^{3}\sin^{q}\left(\frac{\xi}{2}\right)}{B}.$
(56)

The total static energy for this submodel, related to this form of potential,
is given by

$E_{B}=128\pi^{2}\sqrt{2\lambda_{0}\lambda_{4}}\frac{B}{(q+4)(q+6)}.$ (57)

Now, let us introduce a dimensionless variable
$z=\left(\frac{2^{2(q-6)}\lambda_{0}}{\lambda_{4}}\right)^{\frac{1}{2}}r^{4}$
and a function $u=\frac{\xi}{2}$. Substituting them into equation (56) yields

$\frac{du}{dz}=-\frac{\sin^{q-3}u}{B\cos^{3}u}.$ (58)

A solution to this equation, that satisfies the boundary condition (30) at the
origin, $u={\pi\over 2}$, is given by

$\frac{\sin^{4-q}u}{4-q}-\frac{\sin^{6-q}u}{6-q}=-\frac{z}{B}+\frac{2}{\left(4-q\right)\left(6-q\right)}.$
(59)

Unfortunately, this solution does not satisfy the boundary condition (30) at
infinity, $u=0$, for $0<q<4$. For these values of $q$, the function $u$ is
equal to zero at some finite point $z=z_{cut-off}$ and beyond this point $u$
becomes complex numbers. Therefore, for each value of $q$, we may define a
cut-off point at $z=z_{cut-off}$ such that $\xi(z_{cut-off})=0$, which is
given by

$z_{cut-off}=\begin{cases}\frac{2B}{(4-q)(6-q)}&0<q<4\\\ \infty&q\geq
4\end{cases},$ (60)

Since the constant $\xi=0$ is also a solution of the differential equation
(56), we may then construct the compacton type solutions for $0<q<4$, while
for $q\geq 4$ we have regular solutions. Unfortunately, it is difficult to
write explicitly the function $u$, or $\xi$, from the solution (59) in general
and so we have to solve it numerically. Notice that equation (59) is singular
for $q=4,6$ and so we can not use this equation as their solution. Below we
will find their solutions starting by writing the explicit form of their
Bogomolny’s equations (58).

Let us now consider the case of $q=4$ in which the potential is
$V(\xi)=\left(1-\cos\xi\right)^{4}$. The Bogomolny’s equation (58) is
explicitly written as

$\frac{du}{dz}=-\frac{\sin u}{B\cos^{3}u}$ (61)

The right hand side of equation (61) is zero near infinity,
$z\rightarrow\infty$. This allows $\xi(r)$ to converge at this limit, then
satisfies the boundary condition near the infinity. Solving equation (61) and
imposing the boundary condition at the origin, $u\left(0\right)=\pi/2$, gives

$\ln(\sin u)-\frac{1}{2}\sin^{2}u+\frac{1}{2}=-\frac{z}{B}$ (62)

Unfortunately, equation (62) is a transcendental equation for $u(z)$, and so
we have to use a numerical calculation to find its solution.

Another case is $q=6$ that has the potential
$V(\xi)=\left(1-\cos\xi\right)^{6}$. In this case the Bogomolny’s equation
(58) becomes

$\frac{du}{dz}=-\frac{\sin^{3}u}{B\cos^{3}u}$ (63)

This equation has the same convergence feature as equation (61) with a higher
convergence rate at $z\rightarrow\infty$. Integration on (63) gives the
solution we need. Again by imposing boundary condition at the origin,
$u\left(0\right)=\pi/2$, we get

$-\ln(\sin u)-\frac{1}{2\sin^{2}u}+\frac{1}{2}=-\frac{z}{B}$ (64)

It is easy to observe that equation (64) is also transcendental in $u(z)$ so
we need to use numerical calculation to find its solution. All numerical
calculations are given in Figure 3. We can see that an everywhere regular
solution starts to emerge from $q=4$.

Figure 3: Numerical solutions of equation (58) for $q=1,\dots,8$ respectively
from left to right curves. The red curves are the compacton solutions and the
blue curves are the regular solutions.

##  IV BPS Skyrme Submodel in the Non-Spherically Symmetric Lagrangian Density

In this section, we consider the ansatz (14) without fixing all the effective
fields. It is easy to see that the effective Lagrangian (15) also depends
explicitly on the polar coordinate $\theta$ besides the radial coordinate $r$.
In this case, we use the prescription in Atmaja  _et al._ (2020) as such the
general BPS Lagrangian density has the form

$L_{BPS}=\frac{1}{r^{3}\sin\theta\cos\theta}\left(Q_{0}+\sum_{i=1}^{4}Q_{i}X_{i}+\sum_{i=1}^{3}\sum_{j=i+1}^{4}Q_{ij}X_{i}X_{j}\right)~{},$
(65)

where $X_{i}$ are functions of $\xi^{\prime}(r)$, $f^{\prime}(\theta)$,
$g_{1}^{\prime}(\varphi_{1})$ and $g_{2}^{\prime}(\varphi_{2})$ to obtain a
set of Bogomolny’s equations.

The submodel discussed here is obtained by taking
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$ such that the Lagrangian (15) simply
becomes

$L_{eff}=L_{0}+L_{4}=\lambda_{0}V\left(\phi\right)+\lambda_{4}\left[\frac{1}{r^{6}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(\xi^{\prime}~{}f^{\prime}~{}g_{1}^{\prime}~{}g_{2}^{\prime}\right)^{2}\right]~{}.$
(66)

Taking the $L_{BPS}$ to be

$L_{BPS}=\frac{Q_{X}}{r^{3}\sin\theta\cos\theta}X~{},$ (67)

with $X=\xi^{\prime}f^{\prime}g_{1}^{\prime}g_{2}^{\prime}$, the
$L_{eff}-L_{BPS}=0$ becomes a quadratic equation of $X$,

$\left(\frac{\lambda_{4}}{r^{6}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(r\right)\right)X^{2}-\left(\frac{Q_{X}}{r^{3}\sin\theta\cos\theta}\right)X+\lambda_{0}V\left(\phi\right)=0~{},$
(68)

which has solutions

$X_{\pm}=\left(\frac{Q_{X}\left(1+f^{2}\right)^{4}r^{3}\sin\theta\cos\theta}{2\lambda_{4}f^{2}\sin^{6}\xi\left(r\right)}\right)\left(1\pm\sqrt[]{1-4\left(\frac{\lambda_{4}}{Q_{X}^{2}}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(r\right)\right)\lambda_{0}V}\right)~{}.$
(69)

The two solutions will coincide if

$4\left(\frac{\lambda_{4}}{Q_{X}^{2}}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(r\right)\right)\lambda_{0}V=1~{}$
(70)

which implies

$Q_{X}=\pm
2\left(\frac{f}{\left(1+f^{2}\right)^{2}}\sin^{3}\xi\left(r\right)\right)\sqrt[]{\lambda_{0}\lambda_{4}V}~{}$
(71)

and the corresponding Bogomolny’s equation is

$\xi^{{}^{\prime}}f^{{}^{\prime}}g_{1}^{{}^{\prime}}g_{2}^{{}^{\prime}}=\pm~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{\left(1+f^{2}\right)^{2}r^{3}\sin\theta\cos\theta}{f\sin^{3}\xi}\sqrt[]{V}~{}.$
(72)

The explicit form of $L_{BPS}$ is given by

$L_{BPS}=\pm\frac{2f\sin^{3}\xi\left(r\right)\sqrt[]{\lambda_{0}\lambda_{4}V}}{r^{3}\sin\theta\cos\theta\left(1+f^{2}\right)^{2}}\xi^{{}^{\prime}}\left(r\right)f^{{}^{\prime}}\left(\theta\right)g_{1}^{{}^{\prime}}\left(\varphi_{1}\right)g_{2}^{{}^{\prime}}\left(\varphi_{2}\right).$
(73)

It is easy to show that no constraint equations are coming from the Euler-
Lagrange equations of the above BPS Lagrangian density.

From the Bogomolny’s equation (72), we know that $g_{1}$ and $g_{2}$ have to
be the linear functions $\varphi_{1}$ and $\varphi_{2}$ respectively.
Therefore, similar to the spherically symmetric case, we take

$g_{1}\left(\varphi_{1}\right)=n_{1}~{}\varphi_{1},~{}g_{2}\left(\varphi_{2}\right)=n_{2}~{}\varphi_{2},$
(74)

with $n_{1},n_{2}>0$ are integer constants due to the uniqueness in
coordinates $\varphi_{1,2}\to\varphi_{1,2}+2\pi$. Using the Bogomolny’s
equation (72), the total static energy density $\varepsilon$ of the BPS
skyrmion is given by

$\varepsilon=-T_{0}^{0}=\sqrt[]{\lambda_{0}\lambda_{4}V}\frac{2f\sin^{3}\xi}{\left(1+f^{2}\right)^{2}r^{3}\sin\theta\cos\theta}\xi^{\prime}f^{\prime}g_{1}^{\prime}g_{2}^{\prime},$
(75)

and so the total static energy of the BPS skyrmion is

$\displaystyle E_{B}$ $\displaystyle=$ $\displaystyle\int_{domain-
space}d^{4}x\sqrt{\lambda_{0}\lambda_{4}V}\frac{2f\sin^{3}\xi}{\left(1+f^{2}\right)^{2}}\xi^{\prime}f^{\prime}g_{1}^{\prime}g_{2}^{\prime},$
(76) $\displaystyle=$ $\displaystyle
2\sqrt{\lambda_{0}\lambda_{4}}\int_{target-
space}\sqrt{V}\frac{f\sin^{3}\xi}{\left(1+f^{2}\right)^{2}}d\xi
dfdg_{1}dg_{2},$ $\displaystyle=$
$\displaystyle\frac{16\pi^{2}}{3}B\sqrt{\lambda_{0}\lambda_{4}}\left<\sqrt{V}\right>_{S^{4}}.$

All other diagonal components of the stress-energy-momentum tensor vanish for
this BPS skyrmion. This formula for the total static energy is a
generalization of the total static energy (57), that is for a particular form
of the potential, in the spherically symmetric case.

There are three types of potentials, using the ansatz (14), that are invariant
under global $O(5)$ symmetry. The first one (Type-I) is the spherically
symmetric potential with $V\equiv V(\phi^{5})=V(\cos\xi)$ in which the
Bogomolny’s equation (72) becomes

$\frac{f~{}f^{{}^{\prime}}}{\left(1+f^{2}\right)^{2}\sin\theta\cos\theta}=\pm~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}\sqrt{V(\cos\xi)}}{\xi^{{}^{\prime}}\sin^{3}\xi~{}n_{1}n_{2}}.$
(77)

Using separation variables, we can solve this equation by first solving

$\xi^{\prime}=-~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}}{Bk\sin^{3}\xi}\sqrt{V},$
(78)

with the topological charge is given by $B=~{}n_{1}n_{2}$. The solution to
this equation has been discussed in section III.2 for a particular form of
potential, which is equation (56).

$\frac{f~{}f^{{}^{\prime}}}{\left(1+f^{2}\right)^{2}\sin\theta\cos\theta}=\mp
k~{},$ (79)

with $k>0$ to satisfy boundary conditions for $\xi$. The solution for this
equation is

$f=\pm~{}\sqrt[]{\frac{1}{c\mp k\cos^{2}\theta}-1},$ (80)

Putting this solution to (21) and imposing the choice for topological charge,
$B=n_{1}n_{2}$, yields a constraint $k=\mp 1$ but we need to omit the negative
one because $k>0$. With this constraint, $f$ on its boundaries takes value

$\displaystyle f(0)=\pm\sqrt{\frac{-c}{1+c}}$ (81) $\displaystyle
f(\frac{\pi}{2})=\pm\sqrt{\frac{1-c}{c}}$ (82)

We need $f$ to be real in this case, thus the only possibility is $c=0$ which
lead to

$f=\pm\tan\theta$ (83)

This result is just the spherically symmetric case we have in III.2. The total
static energy of this BPS skyrmion is given by equation (57) for the case of
$V(\cos\xi)=\left(1-\cos\xi\right)^{q}$.

The Type-II potential is of the form $V\equiv
V((\phi^{1})^{2}+(\phi^{2})^{2})=V\left({f^{2}\over
1+f^{2}}\sin^{2}\xi\right)$. As an example we take $V={f^{2}\over
1+f^{2}}\sin^{2}\xi$ such that the Bogomolny’s equation (72) becomes

$\frac{f^{{}^{\prime}}}{\left(1+f^{2}\right)^{3/2}\sin\theta\cos\theta}=\pm~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}}{\xi^{{}^{\prime}}\sin^{2}\xi~{}n_{1}n_{2}}~{}.$
(84)

As previously, we solve it using separation variable as such the differential
equation for $f$ is given by

$\frac{f^{{}^{\prime}}}{\left(1+f^{2}\right)^{\frac{3}{2}}\sin\theta\cos\theta}=\mp
k~{},$ (85)

with $k$ is a positive separation constant. The corresponding solution is

$f=\pm\sqrt{\frac{(c_{1}\pm\frac{k}{2}\cos^{2}\theta)^{2}}{1-(c_{1}\pm\frac{k}{2}\cos^{2}\theta)^{2}}}~{},$
(86)

with $c_{1}$ is a integration constant. Again, by imposing $B=n_{1}n_{2}$
yields a relation $c_{1}=\mp\frac{4+k^{2}}{4k}$. This relation can be used to
evaluate $f$ on its boundary

$\displaystyle
f(0)=\pm\sqrt{\frac{\left(\frac{4-k^{2}}{4k}\right)^{2}}{1-\left(\frac{4-k^{2}}{4k}\right)^{2}}}=\pm\sqrt{\frac{\left(4-k^{2}\right)^{2}}{16k^{2}-\left(4-k^{2}\right)^{2}}}$
(87) $\displaystyle
f(\frac{\pi}{2})=\pm\sqrt{\frac{\left(\frac{4+k^{2}}{4k}\right)^{2}}{1-\left(\frac{4+k^{2}}{4k}\right)^{2}}}=\pm\sqrt{\frac{\left(4+k^{2}\right)^{2}}{-\left(4-k^{2}\right)^{2}}}$
(88)

Because $f$ is real valued, then from (88) we can deduce that $k=2$. Hence the
solution for $f$ is

$f=\pm\frac{\sin^{2}\theta}{\sqrt{1-\sin^{4}\theta}}$ (89)

differential equation for $\xi$ is now

$\sin^{3}\xi~{}\xi^{\prime}=-\sqrt{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}}{Bk}\sin\xi,$
(90)

Solving equation (90) give us a compacton solution

$\xi-\frac{\sin(2\xi)}{2}=-\sqrt{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{4}}{4B}+\pi,$
(91)

in which the cut-off is at $r_{cut-
off}=\left(4\sqrt{\lambda_{4}\over\lambda_{0}}{B}\pi\right)^{1/4}$. The total
static energy for this BPS skyrmion is given by

$E_{B}=\pi^{3}\sqrt{\lambda_{0}\lambda_{4}}B.$ (92)

Another possible potential is of the form $V\equiv
V((\phi^{3})^{2}+(\phi^{4})^{2})=V\left({\sin^{2}\xi\over 1+f^{2}}\right)$
which is the Type-III potential. Again, as an example, we take
$V={\sin^{2}\xi\over 1+f^{2}}$ such that the Bogomolny’s equation (72) becomes

$\frac{ff^{{}^{\prime}}}{\left(1+f^{2}\right)^{3/2}\sin\theta\cos\theta}=-~{}\sqrt[]{\frac{\lambda_{0}}{\lambda_{4}}}\frac{r^{3}}{\xi^{{}^{\prime}}\sin^{2}\xi~{}n_{1}n_{2}}~{}.$
(93)

As previously, solving the first equation for the angular field $f$,

$\frac{ff^{{}^{\prime}}}{\left(1+f^{2}\right)^{3/2}\sin\theta\cos\theta}=\mp
k,$ (94)

with $k$ is a positive constant. The same steps from the previous cases can be
applied to this case yields

$f=\pm\sqrt{\frac{1-(c_{2}\pm\frac{k}{2}\cos^{2}\theta)^{2}}{(c_{2}\pm\frac{k}{2}\cos^{2}\theta)^{2}}}~{},$
(95)

with $c_{2}$ is an integration constant. Again, by imposing $B=n_{1}n_{2}$
yields a relation $c_{2}=\pm\frac{4-k^{2}}{4k}$. Hence the boundary condition
for $f$ is

$\displaystyle
f(0)=\pm\sqrt{\frac{1-\left(\frac{4+k^{2}}{4k}\right)^{2}}{\left(\frac{4+k^{2}}{4k}\right)^{2}}}=\pm\sqrt{\frac{-\left(4-k^{2}\right)^{2}}{\left(4+k^{2}\right)^{2}}}$
(96) $\displaystyle
f(\frac{\pi}{2})=\pm\sqrt{\frac{1-\left(\frac{4-k^{2}}{4k}\right)^{2}}{\left(\frac{4-k^{2}}{4k}\right)^{2}}}=\pm\sqrt{\frac{16k^{2}-\left(4-k^{2}\right)^{2}}{\left(4-k^{2}\right)^{2}}}$
(97)

From (97) we can deduce that $k=2$ which lead to a specific solution of $f$,
namely

$f=\pm\frac{\sqrt{1-\cos^{4}\theta}}{\cos^{2}\theta}$ (98)

The differential equation for $\xi$ for this case is the same as the one from
the previous case which is given in (90), thus the solution is given by (91).
This solution has total static energy given by (92). According to the
prescription in Atmaja  _et al._ (2020), there could be other possible
(genuine) BPS submodels of the effective Lagrangian density (15), which
consist of four derivative terms. However, we found that those submodels do
not have Bogomolny’s equations as shown in the appendix. To sum up this
section, the plot of $\xi$ and positive signed $f$ solutions for the same
total energy of all three types of potentials is given in figure 4.

Figure 4: Plots of solutions for three types of potential with the same total
energy of non-spherically symmetric submodel with only potential and octic
term. Solution for $\xi$ (left) is the same for the second and third type, but
solution for $f$ is different for every type of potential (right).

##  V Conclusions

We have considered some possible (generic) BPS submodels of the generalized
five-dimensional Skyrme Model via the BPS Lagrangian method. The analysis is
mainly focused on finding Bogomolny’s Equations and their solutions, with
topological charge $B$ is a positive integer, and also the total static energy
of these BPS skyrmions under a particular ansatz (14). The two main cases that
have been considered, based on the effective Lagrangian density of the BPS
submodels, are the spherically symmetric and the non-spherically symmetric
cases. Employing the BPS Lagrangian method, we followed the prescription in
Atmaja  _et al._ (2020) for the BPS Lagrangian density. In the spherically
symmetric case, where only the effective field $\xi$ in the ansatz (14) is
undetermined, we found there are only two possible BPS submodels which are the
submodel with $\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$, or namely
the Skyrme term, and the submodel with
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$, the potential term and a term that is
proportional to the square of the topological current.

The first BPS submodel with only the Skyrme term has been studied in Brihaye
_et al._ (2017) for a unit BPS skyrmion solution in which $B=1$. Here we
extended the study to the BPS skyrmion solutions with $B>1$. We found two
distinct features compared to the corresponding four-dimensional BPS Skyrme
model discussed in Harland (2014). The first one is that the BPS skyrmions
exist only if there is no potential which is in contradiction with the four-
dimensional case in which the BPS skyrmions may exist if the potential takes
some specific forms Harland (2014); Atmaja  _et al._ (2020). The second one is
the existence of “topological degenerate” BPS skyrmions defined as BPS
skyrmions with the same topological charge, but different total static
energies; except for the unit BPS skyrmion, $B=1$. Interestingly there are
also BPS skyrmions with the same topological charge and energies only if the
topological charge is the prime number. Furthermore, the BPS skyrmions, with
$B>1$, have non-zero pressures at least in two angular directions while the
unit BPS skyrmion has no pressures. The BPS skyrmions, with $B>1$, have total
static energy that is higher than the sum of $B$ times unit BPS skyrmion which
indicates that these BPS skyrmions are repulsive and unstable. It is
interesting to investigate the dynamics of soliton scattering in this submodel
because the repulsive effect implies the possibility of soliton creation in
the usual collision setup, similar to the one demonstrated in Vachaspati
(2011). We may need to add gravitational force, by incorporating gravity into
the BPS submodel, to overcome the pressures so that these BPS skyrmions, with
$B>1$, could be stable. This will be studied further in the future article.

The second BPS submodel depends on the specific form of the potential term. In
this article, we considered the potential term of the form
$V=(1-\cos\xi)^{q}$, with $q\in\mathbb{R}^{+}$. We found the BPS skyrmions are
compacton solutions for $0<q<4$ and are regular solutions for $q\geq 4$. The
compacton solutions have a finite cut-off radius proportional to $B^{1/4}$.
The total static energy of BPS skyrmions in this submodel is proportional to
the topological charge $B$ similar to the BPS-Skyrme model in four dimensions
Adam  _et al._ (2010b). This may be because both models consist of the
potential term and the term that is proportional to the square of the
topological current, $B^{\mu}B_{\mu}$.

In the non-spherically symmetric case, in which all the effective fields in
the ansatz (14) are undetermined, we found the only possible BPS submodel is
similar to the second BPS submodel in the spherically symmetric case. In this
case, the potential term could depend on the polar coordinate $\theta$, and
hence the effective field $f$ could have general forms which depend on the
symmetry of the potential term. This BPS submodel gives a generalization of
the corresponding spherically symmetric one in which the general form of the
total static energy (76) is similar to the one found in Adam  _et al._
(2010b). The results found in this more general scheme cover the preceding
spherically symmetric case.

###### Acknowledgements.

BEG would like to acknowledge Eugene Radu for email correspondence related to
the paper Brihaye  _et al._ (2017). We also thank M. Satriawan for careful
reading the paper and improving English grammar in the paper. The work in this
paper is supported by Kemendikbudristek-ITB WCR 2021. BEG would also thank ITB
for financial support through Riset ITB 2021.

##  Appendix A Full Field Equations of Motion

This section is devoted to showing that all the results in this paper indeed
satisfy the corresponding full-field equations of motion of the generalized
five-dimensional Skyrme model derived from the Lagrangian (13). Let us first
write down the action related to (13) as

$\mathcal{S}=\int\sqrt{-g}~{}d^{5}x\left[\lambda_{0}V\left(\phi\right)+\lambda_{1}F^{2}+\frac{\lambda_{2}}{4}F^{4}+\frac{\lambda_{3}}{36}F^{6}+\frac{\lambda_{4}}{576}F^{8}\right]~{},$
(99)

with
$F^{2n}=\phi^{a_{1}}_{[i_{1}}\dots\phi^{a_{n}}_{i_{n}]}\phi^{a_{1}}_{[j_{1}}\dots\phi^{a_{n}}_{j_{n}]}g^{i_{1}j_{1}}\dots
g^{i_{n}j_{n}}$ satisfy $O(5)$ model constraint $\phi^{a}\phi^{a}=1$. The full
field equations of motion for this model are given by

$\displaystyle\left(\delta^{gf}-\phi^{g}\phi^{f}\right)\left[2\lambda_{1}\nabla_{n}\phi^{f,n}+2\lambda_{2}\nabla_{n}\left(\phi^{a}_{i}\phi^{a,i}\phi^{f,n}-\phi^{a}_{i}\phi^{f,i}\phi^{a,n}\right)\right.$
$\displaystyle\left.\lambda_{3}\nabla_{n}\left((\phi^{a}_{i}\phi^{a,i})^{2}\phi^{f,n}-\phi^{a}_{i}\phi^{a,j}\phi^{b}_{j}\phi^{b,i}\phi^{f,n}-2\phi^{a}_{i}\phi^{a,i}\phi^{b}_{j}\phi^{f,j}\phi^{b,n}+2\phi^{f}_{i}\phi^{a,i}\phi^{a}_{j}\phi^{b,j}\phi^{b,n}\right)\right.$
$\displaystyle\left.\frac{\lambda_{4}}{72}\nabla_{n}\left(B_{m}\varepsilon^{ijknm}\varepsilon^{abcfe}\phi_{i}^{a}\phi_{j}^{b}\phi_{k}^{c}\phi^{e}\right)\right]=\lambda_{0}\frac{\partial
V(\phi)}{\partial\phi^{f}}~{}.$ (100)

###  A.1 BPS Skyrme submodel with
$\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$

For the genuine submodel discussed in Brihaye  _et al._ (2017) with
$\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$ the Lagrangian becomes

$\displaystyle\mathcal{L}=\frac{\lambda_{2}}{4}F^{4}=\frac{\lambda_{2}}{4}\phi^{ab}_{i_{1}i_{2}}\phi^{ab}_{j_{1}j_{2}}g^{i_{1}j_{1}}g^{i_{2}j_{2}}~{},$
(101)

which leads to the following equations of motion

$\displaystyle(\delta^{da}-\phi^{d}\phi^{a})\left(\phi^{a,k}\partial_{k}(\phi^{b}_{i}\phi^{b,i})+\phi^{b}_{i}\phi^{b,i}\nabla_{k}\phi^{a,k}-\phi^{b,k}\partial_{k}(\phi^{a}_{i}\phi^{b,i})-\phi^{a}_{i}\phi^{b,i}\nabla_{k}\phi^{b,k}\right)=0~{}.$
(102)

Using the spherically symmetric ansatz fields (14) and the bipolar spherical
metric (12), we can simplify some of the terms in (102)

$\displaystyle\phi^{b}_{i}\phi^{b,i}$ $\displaystyle=$
$\displaystyle\xi^{\prime
2}+(1+n_{1}^{2}+n^{2}_{2})\frac{\sin^{2}\xi}{r^{2}}~{},$ (103)
$\displaystyle\phi^{a}\nabla_{k}\phi^{a,k}$ $\displaystyle=$
$\displaystyle-\left(\xi^{\prime
2}+(1+n_{1}^{2}+n^{2}_{2})\frac{\sin^{2}\xi}{r^{2}}\right)~{},$ (104)
$\displaystyle\phi^{a}\phi^{b,k}\partial_{k}(\phi^{a}_{i}\phi^{b,i})$
$\displaystyle=$ $\displaystyle-\left(\xi^{\prime
4}+(1+n_{1}^{4}+n^{4}_{2})\frac{\sin^{4}\xi}{r^{2}}\right)~{},$ (105)
$\displaystyle\phi^{a}\phi^{a}_{i}$ $\displaystyle=$ $\displaystyle
0~{}\text{for}~{}i\in\\{0,1,2,3,4\\}~{},$ (106)
$\displaystyle\phi^{a}_{i}\phi^{a}_{j}$ $\displaystyle=$ $\displaystyle
0~{}\text{for}~{}i\neq j~{},$ (107)

such that each of equations in (102) can be simplified to

$\displaystyle\frac{\sin\xi}{r^{4}}\left(\left(n_{1}^{2}+n_{2}^{2}+1\right)r^{2}\xi^{\prime
2}\sin(2\xi)-\left(n_{1}^{2}+n_{2}^{2}+1\right)r\sin^{2}\xi\left(\xi^{\prime}+3r\xi^{\prime
2}\cot\xi+r\xi^{\prime\prime}\right)\right.$
$\displaystyle\left.+\left(\left(n_{2}^{2}+1\right)n_{1}^{2}+n_{2}^{2}\right)\sin(2\xi)\sin^{2}\xi\right)=0~{},$
(108)

multiplied by a non-zero multiplication factor. Suppose we have the solution
of $\xi$ which satisfies the first order equation
$\xi^{\prime}=\gamma\frac{\sin\xi}{r}$ with $\gamma$ is an arbitrary nonzero
real constant. Substituting this first-order equation, equation (A.1) becomes

$-\frac{32\left(\gamma^{2}+\left(\gamma^{2}-1\right)n_{2}^{2}+n_{1}^{2}\left(\gamma^{2}-n_{2}^{2}-1\right)\right)r^{4\gamma-4}\left(r^{2\gamma}-1\right)}{\left(r^{2\gamma}+1\right)^{5}}=0~{},$
(109)

that must be valid for all $r$ and so we have

$\gamma^{2}(n_{1}^{2}+n_{2}^{2}+1)-(n_{1}^{2}+n_{2}^{2}+n_{1}^{2}n_{2}^{2})=0~{}.$
(110)

This leads to (44) which implies that the Bogomolny equation (43) satisfies
the full field equations of motion of the BPS submodel.

###  A.2 BPS Skyrme submodel with $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$

The next genuine submodel with octic and potential terms has
$\lambda_{1}=\lambda_{2}=\lambda_{3}=0$ whose Lagrangian is given by

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{4}}{576}F^{8}+\lambda_{0}V$ (111)
$\displaystyle=$
$\displaystyle\frac{\lambda_{4}}{576}\phi^{abcd}_{i_{1}i_{2}i_{3}i_{4}}\phi^{abcd}_{j_{1}j_{2}j_{3}j_{4}}g^{i_{1}j_{1}}g^{i_{2}j_{2}}g^{i_{3}j_{3}}g^{i_{4}j_{4}}+\lambda_{0}V~{}.$

To simplify the calculation we use the fact that the octic term is the norm of
the topological current (before the normalization)

$B^{m}=\varepsilon^{ijklm}\varepsilon^{abcde}\phi^{a}_{i}\phi^{b}_{j}\phi^{c}_{k}\phi^{d}_{l}\phi^{e}~{},$
(112)

satisfy $F^{8}=B_{m}B^{m}$. For the non-spherically symmetric cases in the
submodel with the octic term and the potential term, we have the equation of
motion as

$\displaystyle\left(\delta^{gf}-\phi^{g}\phi^{f}\right)\left[\frac{2\lambda_{4}}{576}\nabla_{n}\left(B_{m}\frac{\partial
B^{m}}{\partial\phi^{f}_{n}}\right)-\lambda_{0}\frac{\partial
V(\phi)}{\partial\phi^{f}}\right]=0~{},$ (113)

which can be further simplified to

$\left(\delta^{gf}-\phi^{g}\phi^{f}\right)\left[\frac{8\lambda_{4}}{576}\nabla_{n}\left(B_{m}\varepsilon^{ijknm}\varepsilon^{abcfe}\phi_{i}^{a}\phi_{j}^{b}\phi_{k}^{c}\phi^{e}\right)-\lambda_{0}\frac{\partial
V(\phi)}{\partial\phi^{f}}\right]=0~{}.$ (114)

Next, we consider three types of potential, that is, $V(\phi^{5})$,
$V(\phi^{1},\phi^{2})$, and $V(\phi^{3},\phi^{4})$.

####  A.2.1 Type I potential $V(\phi^{5})$

First, we consider the case in which the potential is chosen to be
$V(\phi^{5})=V(\xi(r))$. Upon substituting the ansatz (14) and the Bogomolny
equation (79) for $f$ into the full field equations (114), we obtain that each
of equation (114) can be simplified as

$\frac{2\lambda_{4}k^{2}n_{1}^{2}n_{2}^{2}\sin^{7}\xi\left(3\xi^{\prime}\left(r\xi^{\prime}\cot\xi-1\right)+r\xi^{\prime\prime}\right)}{r^{7}}+\lambda_{0}\frac{\partial
V}{\partial\phi^{5}}\sin^{2}\xi=0~{}.$ (115)

multiplied by a nonzero multiplication factor. Substituting further the
Bogomolny equation (56) for $\xi$ with general potential $V(\xi(r))$, that is,
$\xi^{\prime}=-\sqrt{\lambda_{0}\over\lambda_{4}}{r^{3}\sqrt{V}\over
n_{1}n_{2}k\sin^{3}\xi}$, into equation (115) gives us a trivial result.
Therefore we may conclude that the Bogomolny equations (79) and (56) satisfy
the full field equations (114).

####  A.2.2 Type II Potential $V(\phi^{1},\phi^{2})$ and Type III Potential
$V(\phi^{3},\phi^{4})$

The second type of potential is chosen explicitly to be
$V(\phi^{1},\phi^{2})=V((\phi^{1})^{2}+(\phi^{2})^{2})={f^{2}\over
1+f^{2}}\sin^{2}\xi$. Upon substituting the ansatz (14) and the Bogomolny
equation (85) for $f$ into equations (114), we get a simplified equation

$\lambda_{0}r^{7}\cos\xi-\lambda_{4}k^{2}n_{1}^{2}n_{2}^{2}\sin^{4}\xi\left(-3\xi^{\prime}\sin\xi+3r\left(\xi^{\prime}\right)^{2}\cos\xi+r\xi^{\prime\prime}\sin\xi\right)=0~{}$
(116)

for each of equations (114) with a non-zero multiplication factor. The next
step is to substitute the Bogomolny equation (90) for $\xi$ into equation
(116) which give us another trivial result. Therefore again the Bogomolny
equations (85) and (90) satisfy the full field equations (114).

####  A.2.3 Type III potential, $V(\phi^{3},\phi^{4})$

The third type of potential is chosen explicitly to be
$V(\phi^{3},\phi^{4})=V((\phi^{3})^{2}+(\phi^{4})^{2})={1\over
1+f^{2}}\sin^{2}\xi$. Again substituting the ansatz (14) and the Bogomolny
equation (94) for $f$ into equations (114) implies the same simplified
equation (116) which is also turn out to be trivial after substituting further
the Bogomolny equation (90) for $\xi$. Therefore also the Bogomolny equations
(94) and (90) satisfy the the full field equations (114).

##  Appendix B Submodels With Four Derivative Terms

The possible BPS submodels with four number of derivative terms can be
obtained by setting $\lambda_{1}=\lambda_{2}=\lambda_{4}=0$ or
$\lambda_{2}=\lambda_{3}=\lambda_{4}=0$ in the effective Lagrangian density
(15). Both of them are possible because we can express all derivative terms in
four $X_{i}$’s. On the other hand, the BPS Lagrangian method does not apply to
submodel with only quartic terms which have
$\lambda_{0}=\lambda_{1}=\lambda_{3}=\lambda_{4}=0$ in this choice of ansatz
because we cannot transform the derivative terms into four $X_{i}$’s.

###  B.1 Submodel with $\lambda_{1}=\lambda_{2}=\lambda_{4}=0$

Taking $\lambda_{1}=\lambda_{2}=\lambda_{4}=0$, the effective Lagrangian
density (15) becomes

$\displaystyle L_{eff}$ $\displaystyle=$ $\displaystyle L_{0}+L_{3}$
$\displaystyle=$
$\displaystyle\lambda_{0}V\left(\phi\right)+\lambda_{3}\left[\frac{1}{r^{4}\sin^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{3}}\sin^{4}\xi\left(\xi^{\prime}f^{\prime}g_{1}^{\prime}\right)^{2}+\frac{1}{r^{4}\cos^{2}\theta}\frac{1}{\left(1+f^{2}\right)^{3}}\sin^{4}\xi\left(\xi^{\prime}f^{\prime}g_{2}^{\prime}\right)^{2}\right.$
$\displaystyle\left.+\frac{1}{r^{6}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{4}}\sin^{6}\xi\left(f^{\prime}g_{1}^{\prime}g_{2}^{\prime}\right)^{2}+\frac{1}{r^{4}\sin^{2}\theta\cos^{2}\theta}\frac{f^{2}}{\left(1+f^{2}\right)^{2}}\sin^{4}\xi\left(\xi^{\prime}g_{1}^{\prime}g_{2}^{\prime}\right)^{2}\right].$

A suitable form of the BPS Lagrangian is given by (65) with
$X_{1}=\xi^{\prime}g_{1}^{\prime}g_{2}^{\prime}$ ,
$X_{2}=f^{\prime}g_{1}^{\prime}g_{2}^{\prime}$ ,
$X_{3}=\xi^{\prime}f^{\prime}g_{2}^{\prime}$ , and
$X_{4}=\xi^{\prime}f^{\prime}g_{1}^{\prime}$. We set $L_{eff}-L_{BPS}=0$ and
then solve it as a quadratic equation of $X_{1},X_{2},X_{3},$ and $X_{4}$
subsequently. Demanding each $X_{i}$’s solutions to be equal we then get an
algebraic equation that does not contain any derivative of the effective
fields and may explicitly depend on the coordinates, namely $r$ and $\theta$.
Rewriting this equation as a polynomial equation of (explicit) $r$, we then
set each of its “coefficients” to be zero such that we obtain the following
equations

$\displaystyle-
Q_{0}Q_{14}^{2}Q_{23}^{2}f^{8}-Q_{0}Q_{13}^{2}Q_{24}^{2}f^{8}+2Q_{1}Q_{3}Q_{13}Q_{24}^{2}f^{8}-Q_{0}Q_{12}^{2}Q_{34}^{2}f^{8}$
$\displaystyle+2Q_{3}^{2}Q_{12}Q_{14}Q_{24}f^{8}-2Q_{1}Q_{3}Q_{14}Q_{23}Q_{24}f^{8}+2Q_{0}Q_{13}Q_{14}Q_{23}Q_{24}f^{8}$
$\displaystyle+2Q_{0}Q_{12}Q_{14}Q_{23}Q_{34}f^{8}-2Q_{1}Q_{3}Q_{12}Q_{24}Q_{34}f^{8}+2Q_{0}Q_{12}Q_{13}Q_{24}Q_{34}f^{8}$
$\displaystyle+2Q_{1}^{2}Q_{23}Q_{24}Q_{34}f^{8}-4Q_{0}Q_{14}^{2}Q_{23}^{2}f^{6}-4Q_{0}Q_{13}^{2}Q_{24}^{2}f^{6}+8Q_{1}Q_{3}Q_{13}Q_{24}^{2}f^{6}-4Q_{0}Q_{12}^{2}Q_{34}^{2}f^{6}$
$\displaystyle+8Q_{3}^{2}Q_{12}Q_{14}Q_{24}f^{6}-8Q_{1}Q_{3}Q_{14}Q_{23}Q_{24}f^{6}+8Q_{0}Q_{13}Q_{14}Q_{23}Q_{24}f^{6}$
$\displaystyle+8Q_{0}Q_{12}Q_{14}Q_{23}Q_{34}f^{6}-8Q_{1}Q_{3}Q_{12}Q_{24}Q_{34}f^{6}$
$\displaystyle+8Q_{0}Q_{12}Q_{13}Q_{24}Q_{34}f^{6}+8Q_{1}^{2}Q_{23}Q_{24}Q_{34}f^{6}-6Q_{0}Q_{14}^{2}Q_{23}^{2}f^{4}-6Q_{0}Q_{13}^{2}Q_{24}^{2}f^{4}$
$\displaystyle+12Q_{1}Q_{3}Q_{13}Q_{24}^{2}f^{4}-6Q_{0}Q_{12}^{2}Q_{34}^{2}f^{4}+12Q_{3}^{2}Q_{12}Q_{14}Q_{24}f^{4}-12Q_{1}Q_{3}Q_{14}Q_{23}Q_{24}f^{4}$
$\displaystyle+12Q_{0}Q_{13}Q_{14}Q_{23}Q_{24}f^{4}+12Q_{0}Q_{12}Q_{14}Q_{23}Q_{34}f^{4}-12Q_{1}Q_{3}Q_{12}Q_{24}Q_{34}f^{4}$
$\displaystyle+12Q_{0}Q_{12}Q_{13}Q_{24}Q_{34}f^{4}+12Q_{1}^{2}Q_{23}Q_{24}Q_{34}f^{4}-4Q_{0}Q_{14}^{2}Q_{23}^{2}f^{2}$
$\displaystyle-4Q_{0}Q_{13}^{2}Q_{24}^{2}f^{2}+8Q_{1}Q_{3}Q_{13}Q_{24}^{2}f^{2}-4Q_{0}Q_{12}^{2}Q_{34}^{2}f^{2}+8Q_{3}^{2}Q_{12}Q_{14}Q_{24}f^{2}$
$\displaystyle-8Q_{1}Q_{3}Q_{14}Q_{23}Q_{24}f^{2}+8Q_{0}Q_{13}Q_{14}Q_{23}Q_{24}f^{2}+8Q_{0}Q_{12}Q_{14}Q_{23}Q_{34}f^{2}$
$\displaystyle-8Q_{1}Q_{3}Q_{12}Q_{24}Q_{34}f^{2}+8Q_{0}Q_{12}Q_{13}Q_{24}Q_{34}f^{2}+8Q_{1}^{2}Q_{23}Q_{24}Q_{34}f^{2}$
$\displaystyle-2V\sin^{6}(\xi)Q_{13}Q_{14}Q_{34}\lambda_{0}\lambda_{3}f^{2}-Q_{0}Q_{14}^{2}Q_{23}^{2}-Q_{0}Q_{13}^{2}Q_{24}^{2}$
$\displaystyle+2Q_{1}Q_{3}Q_{13}Q_{24}^{2}-Q_{0}Q_{12}^{2}Q_{34}^{2}+2\left(f^{2}+1\right)^{4}Q_{4}^{2}Q_{12}Q_{13}Q_{23}+2Q_{3}^{2}Q_{12}Q_{14}Q_{24}$
$\displaystyle-2Q_{1}Q_{3}Q_{14}Q_{23}Q_{24}+2Q_{0}Q_{13}Q_{14}Q_{23}Q_{24}+2\left(f^{2}+1\right)^{4}Q_{2}^{2}Q_{13}Q_{14}Q_{34}$
$\displaystyle+2Q_{0}Q_{12}Q_{14}Q_{23}Q_{34}-2Q_{1}Q_{3}Q_{12}Q_{24}Q_{34}+2Q_{0}Q_{12}Q_{13}Q_{24}Q_{34}$
$\displaystyle+2Q_{1}^{2}Q_{23}Q_{24}Q_{34}+2\left(f^{2}+1\right)^{4}Q_{4}\left(\left(Q_{2}Q_{13}-Q_{1}Q_{23}\right)\right.$
$\displaystyle\left.\left(\right.Q_{13}Q_{24}-Q_{14}Q_{23}\right)-Q_{12}\left(Q_{2}Q_{13}+Q_{1}Q_{23}\right)Q_{34}+Q_{3}Q_{12}\left(\right.-Q_{14}Q_{23}-Q_{13}Q_{24}$
$\displaystyle+Q_{12}Q_{34}\left.\right)\left.\right)+2\left(f^{2}+1\right)^{4}Q_{2}\left(Q_{3}Q_{14}\left(Q_{14}Q_{23}-Q_{13}Q_{24}\right.\right.$
$\displaystyle\left.\left.-Q_{12}Q_{34}\right)+Q_{1}Q_{34}\left(-Q_{14}Q_{23}-Q_{13}Q_{24}+Q_{12}Q_{34}\right)\right)=0~{},$
(118) $\displaystyle
4\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{12}+8\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{12}+\csc^{2}(\theta)Q_{2}^{2}Q_{13}^{2}f^{10}$
$\displaystyle+\csc^{2}(\theta)Q_{1}^{2}Q_{23}^{2}f^{10}+20\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{10}-2\csc^{2}(\theta)Q_{1}Q_{2}Q_{13}Q_{23}f^{10}$
$\displaystyle+2\csc^{2}(\theta)Q_{0}Q_{12}Q_{13}Q_{23}f^{10}+40\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{10}$
$\displaystyle+4\csc^{2}(\theta)Q_{2}^{2}Q_{13}^{2}f^{8}+\sec^{2}(\theta)Q_{2}^{2}Q_{14}^{2}f^{8}+4\csc^{2}(\theta)Q_{1}^{2}Q_{23}^{2}f^{8}$
$\displaystyle+\sec^{2}(\theta)Q_{1}^{2}Q_{24}^{2}f^{8}+40\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{8}-8\csc^{2}(\theta)Q_{1}Q_{2}Q_{13}Q_{23}f^{8}$
$\displaystyle+8\csc^{2}(\theta)Q_{0}Q_{12}Q_{13}Q_{23}f^{8}-2\sec^{2}(\theta)Q_{1}Q_{2}Q_{14}Q_{24}f^{8}$
$\displaystyle+2\sec^{2}(\theta)Q_{0}Q_{12}Q_{14}Q_{24}f^{8}+80\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{8}+6\csc^{2}(\theta)Q_{2}^{2}Q_{13}^{2}f^{6}$
$\displaystyle+4\sec^{2}(\theta)Q_{2}^{2}Q_{14}^{2}f^{6}+6\csc^{2}(\theta)Q_{1}^{2}Q_{23}^{2}f^{6}+4\sec^{2}(\theta)Q_{1}^{2}Q_{24}^{2}f^{6}$
$\displaystyle+40\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{6}-12\csc^{2}(\theta)Q_{1}Q_{2}Q_{13}Q_{23}f^{6}$
$\displaystyle+12\csc^{2}(\theta)Q_{0}Q_{12}Q_{13}Q_{23}f^{6}-8\sec^{2}(\theta)Q_{1}Q_{2}Q_{14}Q_{24}f^{6}$
$\displaystyle+8\sec^{2}(\theta)Q_{0}Q_{12}Q_{14}Q_{24}f^{6}+80\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{6}$
$\displaystyle+4\csc^{2}(\theta)Q_{2}^{2}Q_{13}^{2}f^{4}+6\sec^{2}(\theta)Q_{2}^{2}Q_{14}^{2}f^{4}+4\csc^{2}(\theta)Q_{1}^{2}Q_{23}^{2}f^{4}$
$\displaystyle+6\sec^{2}(\theta)Q_{1}^{2}Q_{24}^{2}f^{4}+20\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{4}-8\csc^{2}(\theta)Q_{1}Q_{2}Q_{13}Q_{23}f^{4}$
$\displaystyle+8\csc^{2}(\theta)Q_{0}Q_{12}Q_{13}Q_{23}f^{4}-12\sec^{2}(\theta)Q_{1}Q_{2}Q_{14}Q_{24}f^{4}$
$\displaystyle+12\sec^{2}(\theta)Q_{0}Q_{12}Q_{14}Q_{24}f^{4}+40\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{4}+\csc^{2}(\theta)Q_{2}^{2}Q_{13}^{2}f^{2}$
$\displaystyle+4\sec^{2}(\theta)Q_{2}^{2}Q_{14}^{2}f^{2}+\csc^{2}(\theta)Q_{1}^{2}Q_{23}^{2}f^{2}+4\sec^{2}(\theta)Q_{1}^{2}Q_{24}^{2}f^{2}$
$\displaystyle+4\csc^{2}(2\theta)Q_{2}^{2}Q_{34}^{2}f^{2}-2\csc^{2}(\theta)Q_{1}Q_{2}Q_{13}Q_{23}f^{2}+2\csc^{2}(\theta)Q_{0}Q_{12}Q_{13}Q_{23}f^{2}$
$\displaystyle-8\sec^{2}(\theta)Q_{1}Q_{2}Q_{14}Q_{24}f^{2}+8\sec^{2}(\theta)Q_{0}Q_{12}Q_{14}Q_{24}f^{2}$
$\displaystyle+\left(f^{2}+1\right)^{4}\csc^{2}(\theta)Q_{3}^{2}\left(Q_{12}^{2}+\left(f^{2}+1\right)\sec^{2}(\theta)Q_{24}^{2}\right)f^{2}$
$\displaystyle+8\csc^{2}(2\theta)Q_{0}Q_{23}Q_{24}Q_{34}f^{2}-2\left(f^{2}+1\right)^{4}\csc^{2}(\theta)Q_{3}$
$\displaystyle\left(\left(f^{2}+1\right)Q_{24}\left(Q_{4}Q_{23}+Q_{2}Q_{34}\right)\sec^{2}(\theta)+Q_{12}\left(Q_{2}Q_{13}+Q_{1}Q_{23}\right)\right)$
$\displaystyle
f^{2}-V\sin^{6}(\xi)\left(\right.\left(\csc^{2}(\theta)Q_{13}^{2}+4\left(f^{2}+1\right)\csc^{2}(2\theta)Q_{34}^{2}\right)f^{2}$
$\displaystyle+\sec^{2}(\theta)Q_{14}^{2}\left.\right)\lambda_{0}\lambda_{3}f^{2}+\sec^{2}(\theta)Q_{2}^{2}Q_{14}^{2}+\sec^{2}(\theta)Q_{1}^{2}Q_{24}^{2}$
$\displaystyle+\left(f^{2}+1\right)^{4}Q_{4}^{2}\left(\sec^{2}(\theta)Q_{12}^{2}+4f^{2}\left(f^{2}+1\right)\csc^{2}(2\theta)Q_{23}^{2}\right)$
$\displaystyle-2\sec^{2}(\theta)Q_{1}Q_{2}Q_{14}Q_{24}+2\sec^{2}(\theta)Q_{0}Q_{12}Q_{14}Q_{24}$
$\displaystyle+2\left(f^{2}+1\right)^{4}Q_{4}\left(-4f^{2}\left(f^{2}+1\right)Q_{2}Q_{23}Q_{34}\csc^{2}(2\theta)-\sec^{2}(\theta)Q_{12}\left(Q_{2}Q_{14}+Q_{1}Q_{24}\right)\right)=0~{},$
$\displaystyle
f^{2}\left(f^{2}+1\right)Q_{23}\left(2Q_{2}Q_{3}-Q_{0}Q_{23}\right)\csc^{2}(\theta)$
$\displaystyle+\left(f^{2}+1\right)Q_{24}\left(2Q_{2}Q_{4}-Q_{0}Q_{24}\right)\sec^{2}(\theta)-Q_{0}Q_{12}^{2}+2Q_{1}Q_{2}Q_{12}$
$\displaystyle-\left(f^{2}+1\right)^{4}f^{2}\lambda_{3}^{2}Q_{2}^{2}\sin^{6}(\xi)+f^{16}Q_{1}^{2}Q_{34}^{2}+2f^{16}Q_{0}Q_{13}Q_{14}Q_{34}$
$\displaystyle+8f^{14}Q_{1}^{2}Q_{34}^{2}+16f^{14}Q_{0}Q_{13}Q_{14}Q_{34}+28f^{12}Q_{1}^{2}Q_{34}^{2}+56f^{12}Q_{0}Q_{13}Q_{14}Q_{34}$
$\displaystyle+56f^{10}Q_{1}^{2}Q_{34}^{2}+112f^{10}Q_{0}Q_{13}Q_{14}Q_{34}+70f^{8}Q_{1}^{2}Q_{34}^{2}+140f^{8}Q_{0}Q_{13}Q_{14}Q_{34}$
$\displaystyle+56f^{6}Q_{1}^{2}Q_{34}^{2}+112f^{6}Q_{0}Q_{13}Q_{14}Q_{34}+28f^{4}Q_{1}^{2}Q_{34}^{2}+56f^{4}Q_{0}Q_{13}Q_{14}Q_{34}$
$\displaystyle+8f^{2}Q_{1}^{2}Q_{34}^{2}+16f^{2}Q_{0}Q_{13}Q_{14}Q_{34}+\left(f^{2}+1\right)^{8}Q_{4}^{2}Q_{13}^{2}+\left(f^{2}+1\right)^{8}Q_{3}^{2}Q_{14}^{2}$
$\displaystyle-2\left(f^{2}+1\right)^{8}Q_{1}Q_{3}Q_{14}Q_{34}-2\left(f^{2}+1\right)^{8}Q_{4}Q_{13}\left(Q_{3}Q_{14}+Q_{1}Q_{34}\right)$
$\displaystyle+f^{4}\lambda_{0}\lambda_{3}^{3}V\sin^{12}(\xi)+Q_{1}^{2}Q_{34}^{2}+2Q_{0}Q_{13}Q_{14}Q_{34}=0~{},$
(120) $\displaystyle
Q_{14}^{2}Q_{23}^{2}-2Q_{14}\left(Q_{13}Q_{24}+Q_{12}Q_{34}\right)Q_{23}+\left(Q_{13}Q_{24}-Q_{12}Q_{34}\right)^{2}=0~{},$
(121) $\displaystyle
2Q_{12}\left(f^{2}Q_{13}Q_{23}\cos^{2}(\theta)+Q_{14}Q_{24}\sin^{2}(\theta)\right)+2\left(f^{2}+1\right)f^{2}Q_{23}Q_{24}Q_{34}=0~{},$
(122) $\displaystyle
Q_{12}^{2}+(1+f^{2})(f^{2}\frac{Q_{23}^{2}}{\sin^{2}\theta}+\frac{Q_{24}^{2}}{\cos^{2}\theta})=0~{},$
(123) $\displaystyle
4Q_{1}\left(f^{2}Q_{3}Q_{13}\cos^{2}(\theta)+Q_{4}Q_{14}\sin^{2}(\theta)\right)-2Q_{0}\left(f^{2}\left(\left(f^{2}+1\right)Q_{34}^{2}+Q_{13}^{2}\cos^{2}(\theta)\right)\right.$
$\displaystyle\left.+Q_{14}^{2}\sin^{2}(\theta)\right)+4\left(f^{2}+1\right)f^{2}Q_{3}Q_{4}Q_{34}=0~{},$
(124) $\displaystyle
Q_{1}^{2}+(1+f^{2})(f^{2}\frac{Q_{3}^{2}}{\sin^{2}\theta}+\frac{Q_{4}^{2}}{\cos^{2}\theta})=0~{},$
(125) $\displaystyle Q_{0}=0~{},$ (126)

To solve those equations, we may further rewrite them as polynomial equations
of $\sin\theta$ and set all their “coefficients” to be zero. Then, we have

$Q_{1}=Q_{2}=Q_{3}=Q_{4}=Q_{12}=Q_{23}=Q_{24}=Q_{34}=Q_{13}=Q_{14}=0~{},$
(127)

which eventually leads to trivial solutions. Therefore this submodel does not
have Bogomolny’s equation and so there is no BPS skyrmion.

###  B.2 Submodel with $\lambda_{2}=\lambda_{3}=\lambda_{4}=0$

In this submodel, the effective Lagrangian density (15) becomes

$\displaystyle L_{eff}$ $\displaystyle=$ $\displaystyle L_{0}+L_{1}$
$\displaystyle=$
$\displaystyle\lambda_{0}V\left(\phi\right)+\lambda_{1}\left[\left(\xi^{\prime}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}}\left(\frac{f^{\prime}}{1+f^{2}}\right)^{2}+\frac{\sin^{2}\xi}{r^{2}\sin^{2}\theta~{}}\frac{\left(fg_{1}^{\prime}\right)^{2}}{1+f^{2}}+\frac{\sin^{2}\xi}{r^{2}\cos^{2}\theta~{}}\frac{\left(~{}g_{2}^{\prime}~{}\right)^{2}}{1+f^{2}}\right]$

Again, a suitable form of the BPS Lagrangian is given by (65) with
$X_{1}=\xi^{\prime}$ , $X_{2}=f^{\prime}$ , $X_{3}=g_{1}^{\prime}$ and
$X_{4}=g_{2}^{\prime}$. Setting $L_{eff}-L_{BPS}=0$ and solving it as a
polynomial equation of (explicit) $r$, we obtain one of the “coefficient”
equations is

$\lambda_{1}^{4}\lambda_{0}Vf^{2}\sin^{6}\xi=0~{}.$ (129)

Here the only possible solution is $\lambda_{0}=0$, otherwise the submodel is
non-dynamical, such that the remaining equations, which are written as
polynomials in $\sin\theta$, become

$\displaystyle(1+f^{2})f^{2}Q_{2}^{2}+\sin^{2}\theta
Q_{3}^{2}+f^{2}\cos^{2}\theta Q_{4}^{2}$ $\displaystyle=$ $\displaystyle
0~{},$ (130) $\displaystyle(1+f^{2})Q_{2}(\sec^{2}\theta
Q_{3}Q_{23}+f^{2}\csc^{2}\theta Q_{4}Q_{24})+Q_{3}Q_{4}Q_{34}$
$\displaystyle=$ $\displaystyle 0~{},$ (131) $\displaystyle\sin^{2}\theta
Q_{1}Q_{13}+f^{2}((1+f^{2})Q_{2}Q_{12}+\cos^{2}\theta Q_{4}Q_{14})$
$\displaystyle=$ $\displaystyle 0~{},$ (132)
$\displaystyle(1+f^{2})\sec^{2}\theta(Q_{2}Q_{13}-Q_{1}Q_{23})^{2}$
$\displaystyle+f^{2}(1+f^{2})\csc^{2}\theta(Q_{4}^{2}Q_{12}^{2}+(Q_{2}Q_{14}-Q_{1}Q_{24})^{2}-2Q_{4}Q_{12}(Q_{2}Q_{14}+Q_{1}Q_{24}))$
$\displaystyle-2Q_{3}((1+f^{2})\sec^{2}\theta
Q_{12}(Q_{2}Q_{13}+Q_{1}Q_{23})+Q_{14}(Q_{4}Q_{13}+Q_{1}Q_{34}))$
$\displaystyle+Q_{3}^{2}((1+f^{2})\sec^{2}\theta
Q_{12}^{2}+Q_{14}^{2})+(Q_{4}Q_{13}-Q_{1}Q_{3}4)^{2}$ $\displaystyle=$
$\displaystyle 0~{}.$ (133)

Taking

$Q_{2}=Q_{3}=Q_{4}=Q_{23}=Q_{24}=Q_{34}=0$ (134)

we obtain that it could be either $Q_{1}=0$ or $\lambda_{1}=0$. This fact
leads to a conclusion that this submodel cannot satisfy the BPS limit which
shows the non-existence of Bogomolny’s equation.

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