aainstitutetext: Mathematics and Physics department, Roma Tre, Via della Vasca
Navale 84, Rome, Italybbinstitutetext: INFN Roma Tre Section, Physics
department, Via della Vasca Navale 84, Rome, Italy

#  Axialgravisolitons at infinite corners

Federico Manzoni<EMAIL_ADDRESS>ORCID ID: 0000-0002-9979-6154

###### Abstract

Gravitational solitons (gravisolitons) are particular exact solutions of
Einstein field equation in vacuum build on a given background solution. Their
interpretation and importance is not yet fully clear and in this work we
investigate the asymptotically behaviour of axially symmetric gravisolitons
given minimal assumptions on the background metric. We develop an explicit
systematic asymptotically expansion for the $N$-axialsoliton solution; we
compute the leading order of the asymptotic killing vectors and we make a link
with the corner symmetry proposal.

##  1 Introduction

Solitons were first observed in 1834 by the Scottish engineer John Scott
Russell in the Union Canal. After the discovery of soliton solutions in
hydrodynamic equations, such as KdV equation, the study of these solutions led
to the development of new powerful techniques with the aim of having
systematic procedures for the construction of soliton solutions. One of these
techniques is the spectral transform, or in a better way, the inverse
scattering method that was introduced in 1967 by Gardner, Greene, Miura and
Kruskal grenne  to solve the problem at the initial values of the KdV and then
extended to other situations of interest, for example as did by Zakharov and
Shabat in 1972 shabat  with the non-linear Schrödinger equation. The extension
of the spectral transform in the realm of Einstein gravity happened in 1978 by
Belinskii and Zakharov 1978ZhETF..75.1953B  and two years later Alekseev
proposed the extension to the Maxwell-Einstein theory 1980JETPL..32..277A .

Spectral transform method is essentially based on the possibility of writing
the non-linear equation under consideration as a condition of integrability of
an associated matrix system of linear differential equations, the so-called
the Lax pair of the problem. Once the appropriate Lax pair has been
determined, the problem is divided into two parts: the direct problem and the
inverse problem. Let us highlights the most important steps:

  1. 1.

find the Lax pair, that is two linear operators $L$ and $M$ such that
$Lv=\lambda v$ and $\partial_{t}v=Mv$. It is extremely important that the
eigenvalue $\lambda$ be independent of time (isospectality). Necessary and
sufficient conditions for this to occur is given by the so-called Lax
equation, $\partial_{t}L+LM-ML=0$. After finding the appropriate Lax pair it
should be the case that Lax equation recovers the original non-linear PDE.
This step is the direct problem;

  2. 2.

determine the time evolution of the eigenfunctions associated to each
eigenvalue $\lambda$, the normalization constants and the reflection
coefficient; these form the so-called scattering data. This time evolution is
given by a system of linear ordinary differential equations.

  3. 3.

solving the Gelfand–Levitan–Marchenko (GLM) integral equation, a linear
integral equation, to obtain the final solution of the original nonlinear PDE.
All the scattering data is required in order to do this. If the reflection
coefficient is zero, the process becomes much easier. This step is the inverse
problem.

As can be seen the spectral transform method can be thought as a non-linear
analogue, and in some sense generalization, of the Fourier transform.

Soliton solutions emerge when the reflection coefficient vanish, in that case
the GLM integral equation reduces to an algebraic system. The discrete
elements $\lambda_{k}$ of the spectrum are intimately related to solitons,
indeed given a background solution, i.e. the simplest one, of our problem we
can define a new solution

$\psi=\chi\psi_{b}$ (1)

where $\psi_{b}$ is called generating matrix and

$\chi=\mathcal{I}+\sum_{k}\frac{O_{k}}{\lambda-\lambda_{k}}$ (2)

is called the dressing matrix; $O_{k}$ are operators do not depend on the
spectral parameter. Note that the dressing metrix has pole exactly on the
value of the discrete eigenvalues and a $N$ pole dressing matrix can generate
a $N$-soliton solution.

In this work we focus on gravitational solitons, also known as gravisolitons.
These are particular exact solution of the Einstein, or Einstein-Maxwell,
field equations that can be constructed by the use of the inverse scattering
method known, respectively as Belinski–Zakharov (BZ) transform
1978ZhETF..75.1953B  and Belinski-Zakharov-Alekseev (BZA) transform
1980JETPL..32..277A . Their interpretation is not fully clear but black holes
are special cases of gravitational solitons. despite the term ’soliton’ being
used to describe gravisolitons, their behavior is very different from other
solitons; in particular, gravisolitons do not preserve their amplitude and
shape in time and in scattering processes. Recently Asselmeyer_Maluga_2020
has been proposed, with some consistency tests, that gravisolitons have many
properties of dark matter, such as no interaction with electromagnetic field
but act on matter via gravitation. Moreover the authors showed that the
gravitational lensing effect of gravisolitons agreed with the lensing effect
of usual matter and that they have the same equation of state $w=0$ as matter
has. Moreover, in mcnamara2021gravitational , the author shows how
gravitational solitons that can be considered as topological, in the sense of
some a non-trivial homotophy group, naturally carry charges beyond those of
any local excitation of the quantum fields. Therefore, there are symmetries of
the EFT which are broken as soon as the topology of space-time is allowed to
fluctuate, even without any additional UV degrees of freedom. The author finds
that the effect of gravitational solitons is to break the non-invertible
symmetry to the maximal group-like subsymmetry and if we further demand that
remaining group-like symmetry is broken by additional degrees of freedom, we
find a complete spectrum. Hence, the completeness hypothesis follows from the
absence of group-like symmetries once gravitational solitons are taken into
account.
We develop a systematic expansion for the $N$-soliton metrics with the aim to
investigate their asymptotic symmetries. The first example of asymptotic
symmetry in gravity id due to the pioneering work of Bondi, Metzner, Van der
Burg and Sachs Bondi:1962px ; Sachs:1962wk ; PhysRev.128.2851 ;
Compere:2018aar  that find the group of asymptotic physical symmetries, even
on empty Minkowski space- time, is not Poincaré but rather the infinite-
dimensional Bondi–van der Burg–Metzner–Sachs (BMS) group. However, asynptotic
symmetries find place not only in gravity but also in other gauge theories
Strominger:2017aeh ; Francia:2018jtb ; Campoleoni:2018uib ; Pasterski:2015zua
, like standard Maxwell and Yang-Mills theories and more exotic gauge theories
like $p$-forms and mixed symmetry tensors Henneaux:1986ht ; Afshar:2018apx ;
Esmaeili:2020eua ; manz . Thanks to the asymptotic expansion we developed, we
can find the leading order Killing vectors and we make a link with the corner
proposal which is a novel way to deal with quantum gravity Speranza:2017gxd ;
Geiller:2017whh ; Donnelly:2020xgu ; Ciambelli:2021vnn ; Ciambelli:2021nmv ;
Canepa:2022uii ; ciambelli2023asymptotic .

##  2 Axially symmetric gravitational solitons

Let us discuss axially symmetric gravitational solitons Belinski:2001ph
,Manzoni:2021dij . First of all, we need to taken into account an axisymmetric
stationary background metric, which therefore admits two Killing fields: one
is space-like $m=\partial_{\phi}$ and the other is time-like $k=\partial_{t}$.
We will adopt the notation $(x^{0},x^{1},x^{2},x^{3})=(t,\phi,\rho,z)$ for the
coordinates and comma notation for derivatives.
Since we are at your disposal two Killing fields we can eliminate the
dependence on one spatial and one temporal variables, hence metric components
depend only on $\rho$ and $z$. However, we can simplify further invoking
diffeomorphisms invariance of General Relativity. In four dimensional space-
time this symmetry allows us to impose four conditions on the metric without
loss of generality: we will impose that

$(g_{b})_{22}=(g_{b})_{33},\ (g_{b})_{02}=0,\ (g_{b})_{12}=0,\
(g_{b})_{23}=0.$ (3)

However, to use spectral transform method we need one more simplification and
since we exhausted our freedom to manipulate the metric without loss of
generality we must impose a physical conditions $(g_{b})_{03}=(g_{b})_{13}=0$;
in the and we can write the metric in the form

$ds^{2}=(g_{b})_{22}(d\rho^{2}+dz^{2})+(g_{b})_{ij}dx^{i}dx^{j},$ (4)

where, $i,j=0,1$ and where $g_{22}$ and $g_{ij}$ depend only on $x^{2}=\rho$
and $x^{3}=z$. For stationary metrics of the type 4 it can be imposed, without
loss of generality, that the $2\times 2$ block $(g_{b})_{ij}$ has
$det((g_{b})_{ij})=-\rho^{2}$.

Einstein field equations in vacuum for metric 4 can be rewritten as

$\displaystyle(\rho g_{,\rho}g^{-1})_{,\rho}+(\rho g_{,z}g^{-1})_{,z}=0,$ (5)
$\displaystyle(ln(g_{22}))_{,\rho}=-\frac{1}{\rho}+\frac{1}{4\rho}(U^{2}-V^{2})\
,\ (ln(g_{22}))_{,z}=\frac{1}{2\rho}UV;$

where

$U:=\rho g_{,\rho}g^{-1},\ V:=\rho g_{,z}g^{-1}$

The $N$ solitons axisymmetric solution is

$ds^{2}=g_{22}^{(f)}(d\rho^{2}+dz^{2})+g_{ij}^{(f)}dx^{i}dx^{j}$ (6)

with

$\displaystyle
g_{22}^{(f)}=16C(g_{b})_{22}\rho^{-\frac{N^{2}}{2}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}^{N+1}\bigg{(}\prod_{k>l=1}^{N}\frac{1}{(\lambda_{k}-\lambda_{l})^{2}}\bigg{)}det(\Gamma);$
(7) $\displaystyle
g_{ij}^{(f)}=\pm\frac{(g_{b})_{ij}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}=\pm\frac{\big{(}(g_{b})_{ij}-\sum_{k,l=1}^{N}D_{kl}\lambda_{k}^{-1}\lambda_{l}^{-1}L_{i}^{(k)}L_{j}^{(l)}\big{)}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)},$

where

$\displaystyle D_{kl}:=(\Gamma^{-1})_{kl};$ (8)
$\displaystyle\Gamma_{kl}:=m^{(k)}_{i}(g_{b})_{ij}m_{j}^{(l)}(\rho^{2}+\lambda_{k}\lambda_{l})^{-1};$
$\displaystyle L_{i}^{(k)}:=m_{j}^{(k)}(g_{b})_{ji};$ $\displaystyle
m_{i}^{(k)}:=m_{2j}^{(k)}(\psi_{b}^{-1}(\lambda_{k},\rho,z))_{ji}$

where $m_{2j}^{(k)}$ are arbitrary constants and $\psi^{-1}(\lambda,\rho,z)$
is the inverse of the generating matrix solution of the PDE coupled
system111To obtain the background generating matrix just replace the matrices
$\hat{U}$ and $\hat{V}$ with their background counterpart, that is
$U_{b}:=\rho g_{b,\rho}g_{b}^{-1}$ and $V_{b}:=\rho g_{b,z}g_{b}^{-1}$.

$\displaystyle\bigg{(}\partial_{z}-\frac{2\lambda^{2}}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\bigg{)}\psi=\frac{\rho
V-\lambda U}{\lambda^{2}+\rho^{2}}\psi\ ,\
\bigg{(}\partial_{\rho}+\frac{2\lambda\rho}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\bigg{)}\psi=\frac{\rho
V+\lambda U}{\lambda^{2}+\rho^{2}}\psi.$ (9)

The generating matrix satisfy the relation

$g(\rho,z)={\psi}(0,\rho,z);$ (10)

therefore

${\psi}(\lambda,\rho,z)=g(\rho,z)+\sum_{n}\lambda^{n}F(\rho,z).$ (11)

Furthermore, $C$ is an arbitrary constant but with a sign such that we have
$g_{22}^{(f)}\geq 0$; the $+$ or $-$ sign in front of the second of the 7 must
be chosen appropriately in order to have the correct signature of the metric.
Solitons of the form 6 are called stationary solitons.

Equations that determine the trajectories of the poles $\lambda_{k}$ are given
by

$\displaystyle\lambda_{k,z}=\frac{-2\lambda_{k}^{2}}{\lambda_{k}^{2}+\rho^{2}},$
(12)
$\displaystyle\lambda_{k,\rho}=\frac{2\lambda_{k}\rho}{\lambda_{k}^{2}+\rho^{2}};$

and the solution of these equations is given by the solutions of the quadratic
equation

$\lambda_{k}^{2}+2z\lambda_{k}-\rho^{2}=2w_{k}\lambda_{k},$ (13)

where the $w_{k}$ are complex arbitrary constants. Poles are than given by

$\lambda_{k}=(w_{k}-z)\pm\sqrt{(w_{k}-z)^{2}+\rho^{2}},$ (14)

as one can see the root is always positive for real poles ($w_{k}$ real) and
therefore the solitonic solution will not experience discontinuity and it will
be present in all space-time in contrast to the case of not stationary
gravisolitons.

The condition $det(g)=-\rho^{2}$, necessary to have a physical solution, be
satisfied. To obtain the physical solution, $g^{(f)}$, we need to compute the
determinant of the solution $g$; the calculation leads to

$det(g)=(-1)^{N}\rho^{2N}\bigg{(}\prod_{k=1}^{N}\frac{1}{\lambda_{k}^{2}}\bigg{)}det(g_{b}).$
(15)

The 15 shows that if the background solution is such that
$det(g_{b})=-\rho^{2}$ then the solution must necessarily be an even number of
solitons, $N=2n$, otherwise the sign of the determinant of $g$ would change
and this would lead to a non-physical metric. Hence, the simplest solution is
the one with $N=2$ solitons and can be shown Manzoni:2021dij  that tuning in a
suitable way parameters this solution returns Kerr black hole and more
generally a Taub-NUT space-time Belinski:2001ph .

##  3 Asymptotic behavior and asymptotic symmetries

Let us consider the axisymmetric gravisolitons metric 7, explicitly

$\displaystyle
g_{22}^{(ph)}=16C(g_{b})_{22}\rho^{-\frac{N^{2}}{2}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}^{N+1}\bigg{(}\prod_{k>l=1}^{N}\frac{1}{(\lambda_{k}-\lambda_{l})^{2}}\bigg{)}\det(\Gamma);$
(16) $\displaystyle
g_{00}^{(ph)}=\pm\frac{(g)_{00}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}=\pm\frac{\big{(}(g_{b})_{00}-\sum_{k,l=1}^{N}D_{kl}\lambda_{k}^{-1}\lambda_{l}^{-1}L_{0}^{(k)}L_{0}^{(l)}\big{)}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)};$
$\displaystyle
g_{11}^{(ph)}=\pm\frac{(g)_{11}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}=\pm\frac{\big{(}(g_{b})_{11}-\sum_{k,l=1}^{N}D_{kl}\lambda_{k}^{-1}\lambda_{l}^{-1}L_{1}^{(k)}L_{1}^{(l)}\big{)}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)};$
$\displaystyle
g_{10}^{(ph)}=g_{01}^{(ph)}=\pm\frac{(g)_{10}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)}=\pm\frac{\big{(}(g_{b})_{10}-\sum_{k,l=1}^{N}D_{kl}\lambda_{k}^{-1}\lambda_{l}^{-1}L_{1}^{(k)}L_{0}^{(l)}\big{)}}{\rho^{N}}\bigg{(}\prod_{k=1}^{N}\lambda_{k}\bigg{)},$

Our task is to develop an asymptotic expansion of this metric in order to
study the subgroup of diffeomorphisms which take the asymptotic metric
invariant. In cylindrical coordinates the asymptotic expansion is performed
taking the limits

$\rho\rightarrow\infty,\ |z|\rightarrow\infty\ \ \mathrm{s.t.}\ \
\frac{\rho}{z}\sim O(1);$ (17)

however, for some reasons it is simpler taking limits in spherical
coordinates222The change of coordinates is given by
$(\phi,\rho,z)\rightarrow(\phi=\phi,r=\sqrt{\rho^{2}+z^{2}},\theta=acrtang\big{(}\frac{\rho}{z}\big{)})$,
or in other terms $\rho=rsin(\theta),\phi=\phi,z=rcos(\theta)$.

$r\rightarrow\infty.$ (18)

It is therefore necessary to study the asymptotic behavior of the various
factors appearing in 16.

We will not fix the background metric in order to work in full generality

$ds^{2}=(g_{b})_{22}(d\rho^{2}+dz^{2})+(g_{b})_{ij}dx^{i}dx^{j}\ \ \ i,j=0,1;$
(19)

we only adopt the necessary requests for consistency on $(g_{b})_{ij}$: a
symmetric matrix with determinant given by $det(g_{b})=-\rho^{2}$

$(g_{b})_{ij}=\begin{bmatrix}(g_{b})_{00}:=A&(g_{b})_{01}:=B\\\
(g_{b})_{10}:=B&(g_{b})_{11}:=C\end{bmatrix}\ \ \mathrm{s.t.}\ \
det((g_{b})_{ij})=AC-B^{2}=-\rho^{2}.$ (20)

Since we are interested in asymptotic behaviours, let us give an asymptotic
expansion for elements $A,B,C$. We have assumed that background metric has a
time-like Killing vector hence the metric is stationary; this implies we can
define Komar-like quantities of the background metric. Despite these
quantities could not have physical reasoning they are, however, conserved
quantities associate to the two Killing vectors of the background metric;
therefore, under the physical requirement that conserved quantities do not
diverge we can restrict the possible choice of elements $A,B,C$.

Let us now switch to spherical coordinates $(t,\phi,r,\theta)\equiv(0,1,2,3)$;
the background metric reads

$ds^{2}=(g_{b})_{22}(dr^{2}+r^{2}d\theta^{2})+(g_{b})_{ij}dx^{i}dx^{j}\ \ \
i,j=0,1;$ (21)

and metric elements are now function of $r$ and $\theta$.

Komar-like quantities are proportional to the surface integral of Killing
forms, hence

$\displaystyle K_{t}\propto\int_{\partial S}*dk,\ \ \
K_{\phi}\propto\int_{\partial S}*dm;$ (22)

as shown in Appendix A some combination of the background metric elements have
to be a specific asymptotic behavior in order to make finite Komar-like
quantities:

$\displaystyle\frac{C}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
A}{\partial r}\sim r^{n_{1}}f_{1}(\theta);$ (23)
$\displaystyle\frac{B}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
C}{\partial r}\sim r^{n_{2}}f_{2}(\theta);$
$\displaystyle\frac{C}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
B}{\partial r}\sim r^{n_{3}}f_{3}(\theta);$

with

$\displaystyle n_{1},n_{2},n_{3}\leq 1$ (24)

and $f_{1}(\theta),f_{2}(\theta),f_{3}(\theta)$ arbitrary function of
azimuthal angle.

We now assume a power law behavior for the background metric elements when
$r\rightarrow\infty$

$\displaystyle A\sim r^{\alpha}\mathcal{A}(\theta),\ \ B\sim
r^{\beta}\mathcal{B}(\theta),\ \ C\sim r^{\gamma}\mathcal{C}(\theta),\ \
(g_{b})_{rr}\sim r^{\delta}\mathcal{D}(\theta);$ (25)

the condition on the determinant can be made explicit in two ways as
summarized in Table 1

1 | $\alpha+\gamma=2,\ \ \ \beta=1,\ \ \ \mathcal{AC}(\theta)-\mathcal{B}^{2}(\theta)=-sin^{2}(\theta)$
---|---
2 | $\alpha+\gamma=2,\ \ \ \mathcal{AC}(\theta)=-sin^{2}(\theta),\ \ \ \mathcal{B}^{2}(\theta)=0$
Table 1: Possible explicit choices for the determinant condition.

The possible choices of exponents, coherent with the condition on the
determinant, are summarized in the following Table 2.

a | $\alpha\neq 0,\beta\neq 0,\gamma\neq 0$
---|---
b | $\alpha\neq 0,\beta\neq 0,\gamma=0$
c | $\alpha=0,\beta\neq 0,\gamma\neq 0$
e | $\alpha=0,\beta=0,\gamma\neq 0$
f | $\alpha\neq 0,\beta=0,\gamma=0$
g | $\alpha\neq 0,\beta=0,\gamma\neq 0$
Table 2: Possible coherent choices of exponents due to the determinant
condition.

These cases have to be combined with one of the two ways to made explicit the
determinant condition, so, for example the c2 case means case c combined to
the conditions 2 of the determinant condition. In the following we study in
details the case a1 then proceeding with the other cases in a similar way.

#### Case a1

Plugging the asymptotic expression into 23 we get

$\displaystyle\alpha r^{\gamma+\frac{\gamma}{2}-\frac{\delta}{2}+\alpha-1}\sim
r^{n_{1}}\Rightarrow\frac{3\gamma}{2}-\frac{\delta}{2}+\alpha-1=n_{1};$ (26)
$\displaystyle\gamma r^{\beta+\frac{\gamma}{2}-\frac{\delta}{2}+\gamma-1}\sim
r^{n_{2}}\Rightarrow\beta+\frac{3\gamma}{2}-\frac{\delta}{2}-1=n_{2};$
$\displaystyle\beta r^{\gamma+\frac{\gamma}{2}-\frac{\delta}{2}+\beta-1}\sim
r^{n_{3}}\Rightarrow\frac{3\gamma}{2}-\frac{\delta}{2}+\beta-1=n_{3}.$

We get the following complete matrix for our system

$M|b=\begin{bmatrix}1&0&1&0&\vline&2\\\ 0&2&0&0&\vline&2\\\
1&0&\frac{3}{2}&-\frac{1}{2}&\vline&n_{1}-1\\\
0&1&\frac{3}{2}&-\frac{1}{2}&\vline&n_{2}-1\\\
0&1&\frac{3}{2}&-\frac{1}{2}&\vline&n_{3}-1\\\ \end{bmatrix}$ (27)

From Rouché-Capelli theorem, a linear system admits solutions if only if
$Rk(M)=Rk(M|b)$ where $M$ and $M|b$ are the complete and incomplete matrices
associated to the linear system. In our case, both matrices have rank 4 if and
only if $n_{2}=n_{3}$; moreover the solution space is a subspace of dimension
$4$, hence the solution is unique. Therefore given the asymptotic data
$\\{n_{1},n_{2}\\}$ we get the unique radial asymptotic expansion of the
metric components fixed by the unique set of solution

$\\{\alpha=n_{1}-n_{2}+1,\beta=1,\gamma=-n_{1}+n_{2}+1,\delta=-3n_{1}+n_{2}+7\\}.$
(28)

This solution can be expressed in terms of two parameters $\alpha$ and
$\delta$ as

$\\{\alpha=n_{1}-n_{2}+1,\beta=1,\gamma=-\alpha+2,\delta=-3n_{1}+n_{2}+7\\};$
(29)

we can note that, since $n_{1},n_{2}\leq 1$ we must have

$\begin{cases}\alpha\leq 2&\ \ \ if\ n_{2}\geq 0;\\\ \alpha>2&\ \ \ if\
n_{2}<0.\\\ \end{cases}$ (30)

Moreover, we get also relation between the angular functions

$f_{2}(\theta)=f_{3}(\theta)=\frac{\mathcal{BC(\theta)}}{sin^{2}(\theta)}\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}},\
\ \
f_{1}(\theta)=\frac{\mathcal{AC(\theta)}}{sin^{2}(\theta)}\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$
(31)

#### Case a2

Also in the a2 case we must have $n_{2}=n_{3}$ to hava a solution;

$\\{\alpha=-2n_{1}+8-\delta,\beta=-n_{1}+n_{2}+\alpha,\gamma=2n_{1}-6+\delta\\}$
(32)

where $\delta$ is undetermined by the system. For the angular function we get

$f_{2}(\theta)=f_{3}(\theta)=0,\ \ \
f_{1}(\theta)=-\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$ (33)

#### Case b1

We have

$\\{\alpha=2,\beta=1,\gamma=0,\delta=-2n_{3}=2(1-n_{1})\\};$ (34)

while for the angular

$f_{2}(\theta)=\frac{\mathcal{BC(\theta)}}{sin^{2}(\theta)}\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}},\
\ \
f_{1}(\theta)=\frac{\mathcal{AC(\theta)}}{sin^{2}(\theta)}\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$
(35)

#### Case b2

For the case b2 we have

$\\{\alpha=2,\beta=n_{3}-n_{1}+2,\gamma=0,\delta=2(1-n_{1})\\};$ (36)

the angular relations are

$f_{2}(\theta)=0,\ \ \
f_{1}(\theta)=-\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$ (37)

#### Case c1

In this case we get

$\\{\alpha=0,\beta=1,\gamma=2,\delta=2(3-n_{2})=2(3-n_{3})\\};$ (38)

while the angular relations read

$f_{2}(\theta)=f_{3}(\theta)=\frac{\mathcal{BC}(\theta)}{sin^{2}(\theta)}\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$
(39)

#### Case c2

We have

$\\{\alpha=0,\beta=\delta/2+n_{2}-2=\delta/2+n_{3}-2,\gamma=2,\delta\\},$ (40)

where $\delta$ is undetermined and the relations

$f_{2}(\theta)=f_{3}(\theta)=0.$ (41)

#### Case e2

For this case we have

$\\{\alpha=0,\beta=0,\gamma=2,\delta=2(2-n_{2})\\},$ (42)

moreover

$f_{2}(\theta)=0.$ (43)

#### Case f2

We get

$\\{\alpha=2,\beta=0,\gamma=0,\delta=2(1-n_{1})\\},$ (44)

and

$f_{1}(\theta)=-\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$ (45)

#### Case g2

For the final case we get

$\\{\alpha=-2n_{1}+4-\delta=-n_{2}/5-\delta/10,\beta=0,\gamma=-\alpha+2,\delta\\},$
(46)

where $\delta$ is undetermined and we have

$f_{1}(\theta)=-\sqrt{\frac{\mathcal{C}(\theta)}{\mathcal{D}(\theta)}}.$ (47)

###  3.1 Asymptotic behavior of the $N$-solitons metric

In order to find the asymptotic expansion of the metric 16 we need to compute
the asymptotic behaviour of

$\displaystyle(D)_{kl}:=(\Gamma^{-1})_{kl};$ (48)
$\displaystyle(\Gamma)_{kl}:=m^{(k)}_{i}(g_{b})_{ij}m_{j}^{(l)}(\rho^{2}+\lambda_{k}\lambda_{l})^{-1};$
$\displaystyle L_{i}^{(k)}:=m_{j}^{(k)}(g_{b})_{ji};$ $\displaystyle
m_{i}^{(k)}:=m_{2j}^{(k)}(\psi_{b}^{-1}(\lambda_{k},\rho,z))_{ji}.$

Let us start with $\psi_{b}^{-1}(\lambda_{k},\rho,z))_{ji}$, which is
determined by system

$\displaystyle\bigg{(}\partial_{z}-\frac{2\lambda^{2}}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\bigg{)}\psi_{b}=\frac{\rho
V_{b}-\lambda U_{b}}{\lambda^{2}+\rho^{2}}\psi_{b},$ (49)
$\displaystyle\bigg{(}\partial_{\rho}+\frac{2\lambda\rho}{\lambda^{2}+\rho^{2}}\partial_{\lambda}\bigg{)}\psi_{b}=\frac{\rho
V_{b}+\lambda U_{b}}{\lambda^{2}+\rho^{2}}\psi_{b};$

we can rewrite it in spherical coordinates333Derivatives become
$\frac{\partial}{\partial\rho}=\frac{\partial}{\partial
r}\frac{\partial\rho}{\partial
r}+\frac{\partial}{\partial\theta}\frac{\partial\rho}{\partial\theta}=sin(\theta)\frac{\partial}{\partial
r}+cos(\theta)\frac{\partial}{\partial\theta}$ and $\frac{\partial}{\partial
z}=\frac{\partial}{\partial r}\frac{\partial z}{\partial
r}+\frac{\partial}{\partial\theta}\frac{\partial
z}{\partial\theta}=cos(\theta)\frac{\partial}{\partial
r}-sin(\theta)\frac{\partial}{\partial\theta}$

$\displaystyle\bigg{(}cos(\theta)\frac{\partial}{\partial
r}-sin(\theta)\frac{\partial}{\partial\theta}-\frac{2\lambda^{2}}{\lambda^{2}+r^{2}sin^{2}(\theta)}\partial_{\lambda}\bigg{)}\psi_{b}=\frac{rsin(\theta)V_{b}-\lambda
U_{b}}{\lambda^{2}+r^{2}sin^{2}(\theta)}\psi_{b},$ (50)
$\displaystyle\bigg{(}sin(\theta)\frac{\partial}{\partial
r}+cos(\theta)\frac{\partial}{\partial\theta}+\frac{2\lambda\rho}{\lambda^{2}+r^{2}sin^{2}(\theta)}\partial_{\lambda}\bigg{)}\psi_{b}=\frac{rsin(\theta)V_{b}+\lambda
U_{b}}{\lambda^{2}+r^{2}sin^{2}(\theta)}\psi_{b};$

and expand them asymptotically, as done in Appendix B, to find a solution for
the generating matrix. However the result in not handle and we prefer to
follow another path. Indeed from the general theory about gravisolitons we
know relations 10 and 11, which written in spherical coordinates read

$g_{b}(r,\theta)={\psi}_{b}(0,r,\theta);$ (51)

therefore

${\psi}_{b}(\lambda,r,\theta)=g_{b}(r,\theta)+\sum_{n=1}^{\infty}\lambda^{n}F^{(n)}_{b}(r,\theta)=\sum_{n=0}^{\infty}\lambda^{n}F_{b}^{(n)}(r,\theta),$
(52)

where $F_{b}^{(0)}(r,\theta)=g_{b}(r,\theta)$. We assume a power law expansion
in the radial coordinate

$F_{b}^{(n)}=\sum_{k\in\mathbb{Z}}r^{k}f^{(k)}(\theta)\Rightarrow{\psi}_{b}(\lambda,r,\theta)=\sum_{n=0}^{\infty}\sum_{k\in\mathbb{Z}}\lambda^{n}r^{k}f^{(k)}(\theta).$
(53)

Now we note that due to a version of Moore-Osgood theorem we have

$\lim_{r\rightarrow\infty}\bigg{(}\lim_{\lambda\rightarrow
0}{\psi}_{b}(\lambda,r,\theta)\bigg{)}=\lim_{r\rightarrow\infty}g_{b}(r,\theta)=\lim_{\lambda\rightarrow
0}\bigg{(}\lim_{r\rightarrow\infty}{\psi}_{b}(\lambda,r,\theta)\bigg{)}$ (54)

since the generating matrix is well-behaved, i.e. the limit for
$\lambda\rightarrow 0$ converges uniformly to $g_{b}(r,\theta)$ and from 51 we
have
$\lim_{r\rightarrow\infty}g_{b}(r,\theta)=\lim_{r\rightarrow\infty}{\psi}_{b}(0,r,\theta)$.
From relation 54 and 53 we have

$\displaystyle\lim_{r\rightarrow\infty}g_{b}(r,\theta)$
$\displaystyle=\lim_{\lambda\rightarrow
0}\bigg{(}\lim_{r\rightarrow\infty}{\psi}_{b}(\lambda,r,\theta)\bigg{)}=\lim_{\lambda\rightarrow
0}\bigg{(}\lim_{r\rightarrow\infty}\sum_{n=0}^{\infty}\sum_{k\in\mathbb{Z}}\lambda^{n}r^{k}f^{(k)}(\theta)\bigg{)}=$
(55) $\displaystyle=\lim_{\lambda\rightarrow
0}\bigg{(}\lim_{r\rightarrow\infty}\sum_{n=0}^{\infty}\sum_{k\in\mathbb{Z}}r^{n}(\pm
1-cos(\theta))^{n}r^{k}f^{(k)}(\theta)\bigg{)}$

therefore the $\lambda\rightarrow 0$ limit has to suppress all orders in $r$
which are not the leading order of the background metric; for example if the
leading order for the background metric is $\mathcal{O}(r^{p})$ we must have

$(\pm 1-cos(\theta))^{n}f^{(k)}(\theta)=0\ \ \ if\ n\neq p-k.$ (56)

Now, if the leading order in $r$ of ${\psi}_{b}(\lambda,r,\theta)$ was
$\lambda$-dependent, in the $\lambda\rightarrow 0$ limit the leading order
would be vanishing contradicting the non-vanishing leading order, solution of
system 26, of $g_{b}(r,\theta)$. Then, 56 must hold also for general
$\lambda$-dependence of ${\psi}_{b}(\lambda,r,\theta)$ and its leading order
in $r$ coincide with those of $g_{b}(r,\theta)$. However, is a fact of life
that in same cases, for example for Minkowski space-time Manzoni:2021dij ,
when we compute $\psi_{b}(\lambda_{k},r,\theta)$ the leading order of some
elements of the generating matrix are different from those of the background
matrix. This is a very peculiar structure due to the appearing in the the
generating matrix elements of a specific combination, that is

$r^{2}sin^{2}(\theta)-2rcos(\theta)\lambda-\lambda^{2};$ (57)

indeed for $\lambda=\lambda_{k}$ this is the LHS of equation 13 and reduces to
$-2w_{k}\lambda_{k}$. This particular feature is not captured by the general
reasoning above and in order to compute the $\theta$-dependent coefficients
and the subleading orders in $r$ we need to solve order by order the set of
equations 98 but this is not a simple task in a fully general setting.Case by
case is easier since one knows the leading order of the background metric and
can solve equations 98 to get the leading order of the background generating
matrix. We therefore introduce new factors to parametrize the leading order
behavior of the background generating matrix on the poles

$(\psi_{b})_{00}(\lambda_{k},r,\theta)\sim
r^{\epsilon_{1}}\mathcal{E}_{1}(\theta),\ \ \
(\psi_{b})_{10}(\lambda_{k},r,\theta)\sim
r^{\epsilon_{2}}\mathcal{E}_{2}(\theta),\ \ \
(\psi_{b})_{11}(\lambda_{k},r,\theta)\sim
r^{\epsilon_{3}}\mathcal{E}_{3}(\theta).$ (58)

We can now expand the interesting quantities to compute the asymptotic
$N$-solitons metric. We have

$\displaystyle
m_{0}^{(k)}\sim\frac{m_{21}^{(k)}r^{\epsilon_{2}}\mathcal{E}_{2}(\theta)-m_{20}^{(k)}r^{\epsilon_{3}}\mathcal{E}_{3}(\theta)}{r^{\epsilon_{1}+\epsilon_{3}}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}}\mathcal{E}_{2}^{2}(\theta)};$
(59) $\displaystyle
m_{1}^{(k)}\sim\frac{m_{21}^{(k)}r^{\epsilon_{1}}\mathcal{E}_{1}(\theta)-m_{20}^{(k)}r^{\epsilon_{2}}\mathcal{E}_{2}(\theta)}{r^{\epsilon_{1}+\epsilon_{3}}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}}\mathcal{E}_{2}^{2}(\theta)},$

from which

$\displaystyle
L_{0}^{(k)}\sim\frac{\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)}{r^{\epsilon_{1}+\epsilon_{3}}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}}\mathcal{E}_{2}^{2}(\theta)};$
(60) $\displaystyle
L_{1}^{(k)}\sim\frac{\mathcal{G}^{(k)\beta,\gamma,\gamma,\beta}_{1(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)}{r^{\epsilon_{1}+\epsilon_{3}}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}}\mathcal{E}_{2}^{2}(\theta)};$

where

$\displaystyle\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)=m_{20}^{(k)}[r^{\epsilon_{3}+\alpha}\mathcal{A}\mathcal{E}_{3}(\theta)-r^{\epsilon_{2}+\beta}\mathcal{B}\mathcal{E}_{2}(\theta)]+m_{21}^{(k)}[r^{\epsilon_{1}+\beta}\mathcal{B}\mathcal{E}_{1}(\theta)-r^{\epsilon_{2}+\alpha}\mathcal{A}\mathcal{E}_{2}(\theta)];$
(61a)
$\displaystyle\mathcal{G}^{(k)\beta,\gamma,\gamma,\beta}_{1(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)=m_{20}^{(k)}[r^{\epsilon_{3}+\beta}\mathcal{B}\mathcal{E}_{3}(\theta)-r^{\epsilon_{2}+\gamma}\mathcal{C}\mathcal{E}_{2}(\theta)]+m_{21}^{(k)}[r^{\epsilon_{1}+\gamma}\mathcal{C}\mathcal{E}_{1}(\theta)-r^{\epsilon_{2}+\beta}\mathcal{B}\mathcal{E}_{2}(\theta)]$
(61b)

Using these expansions we get

$\displaystyle\Gamma_{kl}\sim$
$\displaystyle\frac{{A_{kl}}+{B_{kl}}+{C_{kl}}+{D_{kl}}}{2(1\mp
cos(\theta))(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))^{2}}.$
(62)

where

$\displaystyle{A_{kl}}=r^{2\epsilon_{3}+\alpha}\mathcal{A}(\theta)[m_{20}^{(k)}m_{20}^{(l)}\mathcal{E}^{2}_{3}(\theta)+m_{21}^{(k)}m_{21}^{(l)}\mathcal{E}^{2}_{2}(\theta)]-r^{\epsilon_{2}+\epsilon_{3}+\alpha}\mathcal{E}_{2}\mathcal{E}_{3}\mathcal{A}(\theta)[m_{20}^{(k)}m_{21}^{(l)}+m_{21}^{(k)}m_{20}^{(l)}];$
(63a)
$\displaystyle{B_{kl}}=-2r^{\epsilon_{2}+\epsilon_{3}+\beta}\mathcal{E}_{2}\mathcal{E}_{3}\mathcal{B}(\theta)m_{20}^{(k)}m_{20}^{(l)}-2r^{\epsilon_{1}+\epsilon_{2}+\beta}\mathcal{E}_{1}\mathcal{E}_{2}\mathcal{B}(\theta)m_{21}^{(k)}m_{21}^{(l)};$
(63b)
$\displaystyle{C_{kl}}=r^{\epsilon_{1}+\epsilon_{3}+\beta}\mathcal{E}_{1}\mathcal{E}_{3}\mathcal{B}(\theta)[m_{20}^{(k)}m_{21}^{(l)}+m_{21}^{(k)}m_{20}^{(l)}]+r^{2\epsilon_{2}+\beta}\mathcal{E}^{2}_{2}\mathcal{B}(\theta)[m_{20}^{(k)}m_{21}^{(l)}+m_{21}^{(k)}m_{20}^{(l)}];$
(63c)
$\displaystyle{D_{kl}}=r^{2\epsilon_{2}+\gamma}\mathcal{C}(\theta)[m_{20}^{(k)}m_{20}^{(l)}\mathcal{E}^{2}_{2}(\theta)+m_{21}^{(k)}m_{21}^{(l)}\mathcal{E}^{2}_{1}(\theta)]-r^{\epsilon_{1}+\epsilon_{2}+\gamma}\mathcal{E}_{1}\mathcal{E}_{2}\mathcal{C}(\theta)[m_{20}^{(k)}m_{21}^{(l)}+m_{21}^{(k)}m_{20}^{(l)}].$
(63d)

To find the asymptotic behaviour of $(D)_{kl}$ we write it as

$(D)_{kl}=(\Gamma^{-1})_{kl}=\frac{(Adj(\Gamma))_{kl}}{\det(\Gamma)}$ (64)

and we have to taken into account that for an $N$-solitons solution
$(\Gamma)_{kl}$ is an $N\times N$ matrix. The determinant is given by

${\displaystyle\det(\Gamma)=\sum_{\sigma\in
S_{N}}\left(\operatorname{sgn}(\sigma)\prod_{s=0}^{N}(\Gamma)_{s,\sigma_{s}}\right)};$
(65)

asymptotically, the generic addend is

$\displaystyle\operatorname{sgn}(\sigma)\prod_{s=0}^{N}(\Gamma)_{s,\sigma_{s}}$
$\displaystyle\sim\operatorname{sgn}(\sigma)\bigg{(}\frac{1}{2(1\mp
cos(\theta))(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))^{2}}\bigg{)}^{N}\times$
(66)
$\displaystyle\times\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}].$

Using the expression above for the determinant we can also compute the
asymptotic behaviour of the cofactor which is computed by a determinant of an
$(N-1)\times(N-1)$ matrix; therefore we compute that

$\displaystyle(D)_{kl}$ $\displaystyle\sim[{2(1\mp
cos(\theta))(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))^{2}}]\times$
(67) $\displaystyle\times\frac{\sum_{\tilde{\sigma}\in
S_{N-1}}\operatorname{sgn}(\tilde{\sigma})\prod_{s=0}^{N-1}[{A_{s\tilde{\sigma}}}+{B_{s\tilde{\sigma}}}+{C_{s\tilde{\sigma}}}+{D_{s\tilde{\sigma}}}]}{\sum_{\sigma\in
S_{N}}\operatorname{sgn}({\sigma})\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]};$

where $s\neq k$ and $\tilde{\sigma}\neq l$. One may wonder and questioning
about the effective usefulness and easy handling of expansions 59, 60, 62 and
67; the point is that they are completely general but, case by case, one knows
exponents $\\{\alpha,\beta,\gamma,\epsilon_{1},\epsilon_{2},\epsilon_{3}\\}$
and angular functions
$\\{\mathcal{A}(\theta),\mathcal{B}(\theta),\mathcal{C}(\theta),\mathcal{E}_{1}(\theta),\mathcal{E}_{2}(\theta),\mathcal{E}_{3}(\theta)\\}$
and can extract the leading orders in a simple way.

We can finally expand the $N$-solitons metric; let us start with the
combination appearing in the $g_{ij}$ elements, for example in $g_{00}$

$\displaystyle\sum_{k,l=1}^{N}(D)_{kl}\lambda_{k}^{-1}\lambda_{l}^{-1}L_{0}^{(k)}L_{0}^{(l)}\sim$
(68a)
$\displaystyle\sim\sum_{k,l=1}^{N}K\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\mathcal{G}^{(l)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\frac{\sum_{\tilde{\sigma}\in
S_{N-1}}\operatorname{sgn}(\tilde{\sigma})\prod_{s=0}^{N-1}[{A_{s\tilde{\sigma}}}+{B_{s\tilde{\sigma}}}+{C_{s\tilde{\sigma}}}+{D_{s\tilde{\sigma}}}]}{\sum_{\sigma\in
S_{N}}\operatorname{sgn}({\sigma})\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]},$

where $K={2\frac{(1\mp cos(\theta))}{(\pm 1-cos(\theta))^{2}}}=\frac{2}{(1\mp
cos(\theta))}$ since $(\pm 1-cos(\theta))^{2}=(1\mp cos(\theta))^{2}$. Similar
expansions hold for $g_{01}=g_{10}$ and $g_{11}$. Putting it all together, the
metric component on the $N$-gravisoliton have the following asymptotic
expansion

$\displaystyle
g_{22}^{(ph)}\sim\mathcal{M}_{N}(\theta)\bigg{(}\frac{1}{(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))^{2}}\bigg{)}^{N}r^{\delta-\frac{N^{2}}{2}+2N}\times$
(69) $\displaystyle\ \ \ \ \ \ \ \ \ \times\sum_{\sigma\in
S_{N}}\big{(}\operatorname{sgn}(\sigma)\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]\big{)};$
$\displaystyle
g_{00}^{(ph)}\sim\pm\mathcal{N}_{N}(\theta)\bigg{[}r^{\alpha}\mathcal{A}(\theta)-\sum_{k,l=1}^{N}K\mathcal{K}\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\mathcal{G}^{(l)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\bigg{]};$
$\displaystyle
g_{11}^{(ph)}\sim\pm\mathcal{N}_{N}(\theta)\bigg{[}r^{\gamma}\mathcal{C}(\theta)-\sum_{k,l=1}^{N}K\mathcal{K}\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{1(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\mathcal{G}^{(l)\beta,\gamma,\gamma,\beta}_{1(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\bigg{]};$
$\displaystyle
g_{10}^{(ph)}=g_{01}^{(ph)}\sim\pm\mathcal{N}_{N}(\theta)\bigg{[}r^{\beta}\mathcal{B}(\theta)-\sum_{k,l=1}^{N}K\mathcal{K}\mathcal{G}^{(k)\alpha,\beta,\beta,\alpha}_{0(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\mathcal{G}^{(l)\beta,\gamma,\gamma,\beta}_{1(\epsilon_{3},\epsilon_{2},\epsilon_{1}\epsilon_{2})}(r,\theta)\bigg{]};$

where

$\displaystyle\mathcal{M}_{N}(\theta)=4^{N(N-1)+2}C\mathcal{D}(\theta)[(\pm
1-cos(\theta))]^{N^{2}+N}sin(\theta)^{-\frac{N^{2}}{2}}(2\mp
2cos(\theta))^{-N};$ (70)
$\displaystyle\mathcal{K}=\frac{\sum_{\tilde{\sigma}\in
S_{N-1}}\operatorname{sgn}(\tilde{\sigma})\prod_{s=0}^{N-1}[{A_{s\tilde{\sigma}}}+{B_{s\tilde{\sigma}}}+{C_{s\tilde{\sigma}}}+{D_{s\tilde{\sigma}}}]}{\sum_{\sigma\in
S_{N}}\operatorname{sgn}({\sigma})\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]};$
$\displaystyle\mathcal{N}_{N}=(\theta)\bigg{[}\frac{(\pm
1-cos(\theta))}{sin(\theta)}\bigg{]}^{N}.$

We note that for the $g_{22}^{(ph)}$ component we have more choices for the
asymptotic expansion according to the relative signs in front of the square
roots in the definition of the poles due to the presence of the factors
$\frac{1}{(\lambda_{k}-\lambda_{l})^{2}}$ which give $\frac{1}{4r^{2}}$ if the
signs disagree while give $\frac{1}{w_{k}-w_{l}}$ if signs agree. Therefore
the most general expansion for $g_{22}^{(ph)}$ is

$\displaystyle
g_{22}^{(ph)}\sim\mathcal{M}_{N}(\theta)\bigg{(}\frac{1}{(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))^{2}}\bigg{)}^{N}r^{\delta-\frac{N^{2}}{2}+N(N+1)-N(N-1)+2\\#}\times$
(71) $\displaystyle\ \ \ \ \ \ \ \ \
\times\bigg{(}\prod_{j=1}^{\\#}\frac{1}{(w_{k}-w_{l})_{j}^{2}}\bigg{)}\sum_{\sigma\in
S_{N}}\big{(}\operatorname{sgn}(\sigma)\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]\big{)};$

where

$\displaystyle\mathcal{M}_{N}(\theta)=4^{N(N-1)-2\\#+2}C\mathcal{D}(\theta)[(\pm
1-cos(\theta))]^{N^{2}+N}sin(\theta)^{-\frac{N^{2}}{2}}(2\mp
2cos(\theta))^{-N},$ (72)

and $\\#$ is the number of poles couple with concordant root signs.

One can check that choosing Minkowski metric as background and computing the
asymptotic behaviour of the background generating matrix, the asymptotic
behaviour of
$m_{0}^{(k)},m_{1}^{(k)},L_{0}^{(k)},l_{1}^{(k)},\Gamma_{kl},D_{kl}$ and of
the metric component is exactly what was expected Belinski:2001ph
,Manzoni:2021dij . We note that the physical information of how many solitons
make up the solution is reached by all the metric components meaning again
that non-diagonal solution can be generated even if the background solution
was diagonal; this is the case of Kerr black hole $2$-solitons solution.
Moreover, there are no reasons why the asymptotic behavior of the chosen
background solution must be preserved by the $N$-soliton solution and in
general this is no the case. Indeed, let us consider the following example.
Consider a background metric with $\delta=1$ and with
$\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=0$. This is no restrictive since the
determination of the asymptotic behavior of the background generating matrix
does not depend on $\delta$. In this case the $N$-solitons solution give us a
$g_{22}^{ph}$ metric component with asymptotic behavior
$r^{1-\frac{N^{2}}{2}}r^{h(\alpha,\beta,\gamma)}$ where
$h(\alpha,\beta,\gamma)\in\frac{1}{2}\mathbb{Z}$ has to derived case by case
depending on $\alpha,\beta$ and $\gamma$ but then it is fixed. However, surely
there exist a $N$ for which $1-\frac{N^{2}}{2}+h(\alpha,\beta,\gamma)\neq 0$.
This show that the $N$-solitons solution has no to preserve the asymptotic
behavior of the background metric in general, at least in the assumption of
Komar-like quantities are finite. Therefore it is a well posed question ask
for asymptotic symmetries with expectation that these may depend on the number
of solitons that make up the solution.

###  3.2 Leading order of the asymptotic symmetries and the UCS group

In order to discuss asymptotic symmetries let us find the asymptotic Killing
vectors solutions of the asymptotic Killing equation, i.e. Killing equation
for the asymptotic $N$-solitons metric. As known, a vector field $\xi$ is a
Killing field if the Lie derivative with respect to $\xi$ of the metric
vanishes

${\mathcal{L}}_{{\xi}}g=0,$ (73)

in terms of the Levi-Civita connection, we can write it as

${\displaystyle
g\left(\nabla_{Y}\xi,Z\right)+g\left(Y,\nabla_{Z}\xi\right)=0,}$ (74)

for all vectors $Y$ and $Z$. Moreover in local coordinates expressed by a
coordinate fields, Killing equation assume its standard form

${\displaystyle\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}=\xi_{(\mu;\nu)}=\partial_{\mu}\xi_{\nu}+\partial_{\nu}\xi_{\mu}-2\Gamma^{\rho}_{\mu\nu}\xi_{\rho}=0};$
(75)

moreover, we assume a homogeneous expansion for the Killing vectors of the
form

$\xi_{\mu}=\sum_{l\in\mathbb{Z}}\frac{\bar{\xi}_{\mu}^{(l)}(\theta)}{r^{l}}\ \
\ \ \ \mu=0,1,2,3.$ (76)

Our goal is to determine the first possible $l$ of the expansion above; this
information can be obtained looking at the Killing equation and requiring that
it is satisfied only asymptotically, i.e. it is not really necessary for it to
be exactly zero as long as it is satisfied within the limit of large radial
distances. We focus on the Killing equation with $(\mu,\nu)=(3,3)$ this
reduces, using results in C, to

$2\partial_{3}\xi_{3}-2\Gamma_{33}^{3}\xi_{3}\sim\mathcal{O}(r^{0}).$ (77)

For $N=0$ we have, inserting the Killing vector expansion

$2\sum_{l\in\mathbb{Z}}\frac{\bar{\xi}_{3}^{{}^{\prime}(l)}(\theta)}{r^{l}}-\frac{\mathcal{D}^{\prime}(\theta)}{\mathcal{D}(\theta)}\sum_{l\in\mathbb{Z}}\frac{\bar{\xi}_{3}^{(l)}(\theta)}{r^{l}}\sim\mathcal{O}(r^{0}),$
(78)

which means that the leading order is $l=0$ since for $l>0$ we have subleading
terms while for $l<0$ we have overleading terms; moreover coefficients must
satisfy the equation

$2\bar{\xi}_{3}^{{}^{\prime}(l)}(\theta)-\frac{\mathcal{D}^{\prime}(\theta)}{\mathcal{D}(\theta)}\bar{\xi}_{3}^{(l)}(\theta)=0\
\ \ \ \ \forall l\in[0,+\infty),$ (79)

Let us continue with the case of $N=2$; in this case using the Crhistoffel
symbols derived in 105 the equation is

$2\sum_{l\in\mathbb{Z}}\frac{\bar{\xi}_{3}^{{}^{\prime}(l)}(\theta)}{r^{l}}-2\big{(}\Gamma^{3}_{33}\big{)}_{N=2}\sum_{l\in\mathbb{Z}}\frac{\bar{\xi}_{3}^{(l)}(\theta)}{r^{l}}\sim\mathcal{O}(r^{0}),$
(80)

since $\big{(}\Gamma^{3}_{33}\big{)}_{N=2}$ turn out to has a leading order
independent of $r$ then, again, the leading order of the Killing vector is
$l=0$; moreover coefficients satisfy

$2\bar{\xi}_{3}^{{}^{\prime}(l)}(\theta)-2\big{(}\Gamma^{3}_{33}\big{)}_{N=2}\bar{\xi}_{3}^{(l)}(\theta)=0\
\ \ \ \ \forall l\in[0,+\infty).$ (81)

The same kind of analysis can be done in the general $N$ case but we note that
despite the leading order is the same for all $N$ the angular dependence can
be different; this can be seen since the Christoffel symbols have different
angular dependence depending on $N$ since the $N$-solitons metric has.
Therefore, in the case of Killing vectors whose components fall-off with the
same behaviour, the leading order is $r^{0}$ while the specific case
coefficients have to be derived case by case but their angular dependence
depend from the number of solitons.
The point now is to compute the algebra by investigating the vector field
bracket in order to make a link with the corner proposal. The corner proposal
ciambelli2023asymptotic  is based on the fact that gauge symmetries are
classical redundancies of the system that are not expected to survive in
quantum gravity. This applies in particular to diffeomorphisms that are pure
gauge; however, there are diffeomorphisms that are asymptotic symmetries which
are not redundancy, they physically act on the field space. The corner
proposal focuses on these symmetries, and posits that they survive in quantum
gravity. The central point is that a gravitational theory is described by a
set of charges and their algebra at corners.

we can expand a vector field in the coordinate basis on the corner obtained in
the radial limit at fixed time $\partial_{i}$ with $i=1,3$ and the "normal"
coordinates $\partial_{a}$ with $a=0,2$. Therefore

$\xi^{\mu}\partial_{\mu}=\xi^{i}\partial_{i}+\xi^{a}\partial_{a}.$ (82)

It is ease to see that these Killing vectors generate, as expected, a
subalgebra of the ucs algebra and therefore its exponential generate a
subgroup of the UCS group. This is because the Lie bracket gives us, where
expressed in coordinates

$[\xi^{i}\partial_{i}+\xi^{a}\partial_{a},\eta^{i}\partial_{i}+\eta^{a}\partial_{a}]=[\xi^{i}\partial_{i},\eta^{i}\partial_{i}]^{j}\partial_{j}+(\xi^{i}\partial_{i}\eta^{a}-\eta^{i}\partial_{i}\xi^{a})\partial_{a};$
(83)

the first term is $\textfrak{Diff}(S)$ while the second one is the action of
$\textfrak{Diff}(S)$ on $\mathbb{R}^{2}$ where $S$ is the infinite distance
corner whose coordinates are the angles. Therefore the axialgravisoliton
corner symmetry algebra (agcsa) generated by the leading order Killing vectors
is

$\textfrak{agcsa}=\textfrak{Diff}(S)\opluslhrim\mathbb{R}^{2}$ (84)

##  4 Conclusion

Gravitational solitons or gravisolitons are particularly exact solution of the
Einstein gravity that can be extended to Einstein-Maxwell theory
1978ZhETF..75.1953B ; 1980JETPL..32..277A . The theory of gravitational
solitons is essentially linked to the Inverse Scattering Method (ISM) and many
of the most useful metrics used, like those of black holes, are nothing but
gravisolitons Belinski:2001ph ; Manzoni:2021dij . Moreover gravitational
solitons are a good candidate of dark matters since they have the right
properties and the right characteristics Asselmeyer_Maluga_2020 ;
nevertheless, they are fundamental for the completeness hypothesis thanks to
the charges they carry mcnamara2021gravitational .
In this work we investigate the asymptotic behaviour of a particular class of
gravisolitons: axialgravisolitons. First of all we study the background metric
compatible with the requests on the determinant of the metric which is an
essential condition to have physical metrics. We find the expansion for any
value of the soliton numbers giving minimal assumption on the background
metric. The most important requirement we make is the finitude of the Komar-
like quantities associated to the background metric. According to the
parameters the final $N$-soliton metric can develop different asymptotic
behaviour with respect the background solution used to generate the solitons.
Since the $N$-soliton metric has no to preserve the asymptotically behaviour
of the background metric, it is a well posed question ask for asymptotic
symmetries with the expectation that these may depend on the number of
solitons that make up the solution. This is in fact the case since the leading
order coefficients satisfy different differential equations according to the
number of solitons, however the radial leading order turn out to be the same
in every case. The algebra generated by these Killing vectors turn out to be a
subalgebra of the universal corner symmetry algebra given by
$\textfrak{agcsa}=\textfrak{Diff}(S)\opluslhrim\mathbb{R}^{2}$.
It would be interesting to extent what we done to the case of higher
derivative and modified $f(R)$ gravity as well as Einstein-spinors and SUGRA
theories; moreover one could study the case of non-axialsymmetric
gravisolitons.

### Acknowledgment

I want to to thanks Paolo Maria Santini who introduced me, at the time of my
master’s degree, to non-linear waves and gravitational solitons. I thank my
other half, Maria Luisa Limongi for having, once again, supported my work by
also helping me when some computations were struggling to move forward. I
thank Luca Ciambelli for the interesting seminar with which he introduced me
to the corner proposal; I also thank him for interesting discussions on the
matter. Finally, I thank my PhD advisor Dario Francia.

##  Appendix A Evaluation of Komar-like quantities

In an explicit way metric and its inverse read

$(g_{b})_{\mu\nu}\begin{bmatrix}A&B&0&0\\\ B&C&0&0\\\ 0&0&(g_{b})_{rr}&0\\\
0&0&0&r^{2}(g_{b})_{rr}\end{bmatrix},\ \ \ \ \
(g_{b})^{\mu\nu}\begin{bmatrix}-\frac{C}{r^{2}sin^{2}(\theta)}&\frac{B}{r^{2}sin^{2}(\theta)}&0&0\\\
\frac{B}{r^{2}sin^{2}(\theta)}&-\frac{A}{r^{2}sin^{2}(\theta)}&0&0\\\
0&0&\frac{1}{(g_{b})_{rr}}&0\\\
0&0&0&\frac{1}{r^{2}(g_{b})_{rr}}.\end{bmatrix}$ (85)

Let us start with the first Komar-like integral, we need $*dk$.
$k^{\mu}=(1,0,0,0)$ and so $k_{\mu}=(g_{b})_{\mu\nu}k^{\nu}=(g_{b})_{\mu t}$;
the form associated is $k=(g_{b})_{\mu t}dx^{\mu}=Adt+Bd\phi$ and its exterior
derivative reads

$dk=\frac{\partial A}{\partial r}dr\wedge dt+\frac{\partial
A}{\partial\theta}d\theta\wedge dt+\frac{\partial B}{\partial r}dr\wedge
d\phi+\frac{\partial B}{\partial\theta}d\theta\wedge d\phi.$ (86)

Taking the Hodge dual we get

$\displaystyle(*dk)_{\mu\nu}=\frac{\epsilon_{\mu\nu\rho\sigma}}{2}(g_{b})^{\rho\rho^{\prime}}(g_{b})^{\sigma\sigma^{\prime}}(dk)_{\rho^{\prime}\sigma^{\prime}}=$
(87a) $\displaystyle=\frac{\epsilon_{\mu\nu\rho\sigma}}{2}[(g_{b})^{\rho
r}(g_{b})^{\sigma t}(dk)_{rt}+(g_{b})^{\rho\theta}(g_{b})^{\sigma
t}(dk)_{\theta t}+(g_{b})^{\rho
r}(g_{b})^{\sigma\phi}(dk)_{r\phi}+(g_{b})^{\rho\theta}(g_{b})^{\sigma\phi}(dk)_{\theta\phi}].$

A similar procedure can be done for computing $*dm$: $m=Bdt+Cd\phi$ and the
final result is

$\displaystyle(*dm)_{\mu\nu}=\frac{\epsilon_{\mu\nu\rho\sigma}}{2}(g_{b})^{\rho\rho^{\prime}}(g_{b})^{\sigma\sigma^{\prime}}(dm)_{\rho^{\prime}\sigma^{\prime}}=$
(88a) $\displaystyle=\frac{\epsilon_{\mu\nu\rho\sigma}}{2}[(g_{b})^{\rho
r}(g_{b})^{\sigma t}(dm)_{rt}+(g_{b})^{\rho\theta}(g_{b})^{\sigma
t}(dm)_{\theta t}+(g_{b})^{\rho
r}(g_{b})^{\sigma\phi}(dm)_{r\phi}+(g_{b})^{\rho\theta}(g_{b})^{\sigma\phi}(dm)_{\theta\phi}],$

with

$dm=\frac{\partial B}{\partial r}dr\wedge dt+\frac{\partial
B}{\partial\theta}d\theta\wedge dt+\frac{\partial C}{\partial r}dr\wedge
d\phi+\frac{\partial C}{\partial\theta}d\theta\wedge d\phi.$ (89)

To calculate the Komar quantities $K_{t}$ and $K_{\phi}$ one needs to choose
an appropriate boundary surface. As underlined in 1984JMP….25..992C  and
Modak_2011  a good choice is the boundary of a spatial three volume
characterised by a constant $r$ and $dt=-\frac{A}{B}d\phi$; In this way,
infinitesimally close points are simultaneous events. Since $r$ is constant
the dual Killing forms reduced to

$\displaystyle(*dk)_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}[(g_{b})^{\rho
r}(g_{b})^{\sigma t}(dk)_{rt}+(g_{b})^{\rho
r}(g_{b})^{\sigma\phi}(dk)_{r\phi}]=$ (90)
$\displaystyle=\frac{(g_{b})^{rr}}{2}\\{(dk)_{rt}[\epsilon_{\mu\nu
rt}(g_{b})^{tt}+\epsilon_{\mu\nu r\phi}(g_{b})^{\phi
t}]+(dk)_{r\phi}[\epsilon_{\mu\nu rt}(g_{b})^{t\phi}+\epsilon_{\mu\nu
r\phi}(g_{b})^{\phi\phi}]\\};$
$\displaystyle(*dm)_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}[(g_{b})^{\rho
r}(g_{b})^{\sigma t}(dm)_{rt}+(g_{b})^{\rho
r}(g_{b})^{\sigma\phi}(dm)_{r\phi}]=$
$\displaystyle=\frac{(g_{b})^{rr}}{2}\\{(dm)_{rt}[\epsilon_{\mu\nu
rt}(g_{b})^{tt}+\epsilon_{\mu\nu r\phi}(g_{b})^{\phi
t}]+(dm)_{r\phi}[\epsilon_{\mu\nu rt}(g_{b})^{t\phi}+\epsilon_{\mu\nu
r\phi}(g_{b})^{\phi\phi}]\\}$

and the only non-vanishing elements are

$\displaystyle(*dk)_{\theta\phi}=\frac{(g_{b})^{rr}}{2}\epsilon_{\theta\phi
rt}[(g_{b})^{tt}(dk)_{rt}+(g_{b})^{t\phi}(dk)_{r\phi}];$ (91)
$\displaystyle(*dk)_{\theta t}=\frac{(g_{b})^{rr}}{2}\epsilon_{\theta
tr\phi}[(g_{b})^{\phi t}(dk)_{rt}+(g_{b})^{\phi\phi}(dk)_{r\phi}],$

for $*dk$ and

$\displaystyle(*dm)_{\theta\phi}=\frac{(g_{b})^{rr}}{2}\epsilon_{\theta\phi
rt}[(g_{b})^{tt}(dm)_{rt}+(g_{b})^{t\phi}(dm)_{r\phi}];$ (92)
$\displaystyle(*dm)_{\theta t}=\frac{(g_{b})^{rr}}{2}\epsilon_{\theta
tr\phi}[(g_{b})^{\phi t}(dm)_{rt}+(g_{b})^{\phi\phi}(dm)_{r\phi}],$

for $*dm$.

Since we are performing an integral over a over a surface of simultaneous
events terms $(*dk)_{\theta t}$ and $(*dm)_{\theta t}$ do not have to be
included since then their integrals should be subtracted because doing cycle
along the compact direction $\phi$ would bring a shift in time
1984JMP….25..992C ,Modak_2011 ,PhysRevLett.39.1641 .
In order to avoid divergences in the Komar-like quantities, $(*dk)_{\theta t}$
and $(*dm)_{\theta t}$ have to be a good radial falloff behavior, therefore

$\displaystyle(g_{b})^{rr}(g_{b})^{tt}(dk)_{rt}=-\frac{C}{r^{2}sin^{2}(\theta)(g_{b})_{rr}}\frac{\partial
A}{\partial
r}\Rightarrow\frac{C}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
A}{\partial r}\sim r^{n_{1}}f_{1}(\theta)$ (93)
$\displaystyle(g_{b})^{rr}(g_{b})^{t\phi}(dk)_{r\phi}=\frac{B}{r^{2}sin^{2}(\theta)(g_{b})_{rr}}\frac{\partial
C}{\partial
r}\Rightarrow\frac{B}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
C}{\partial r}\sim r^{n_{2}}f_{2}(\theta)$
$\displaystyle(g_{b})^{rr}(g_{b})^{tt}(dm)_{rt}=-\frac{C}{r^{2}sin^{2}(\theta)(g_{b})_{rr}}\frac{\partial
B}{\partial
r}\Rightarrow\frac{C}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
B}{\partial r}\sim r^{n_{3}}f_{3}(\theta)$
$\displaystyle(g_{b})^{rr}(g_{b})^{t\phi}(dm)_{r\phi}=\frac{B}{r^{2}sin^{2}(\theta)(g_{b})_{rr}}\frac{\partial
C}{\partial
r}\Rightarrow\frac{B}{sin^{2}(\theta)}\sqrt{\frac{C}{(g_{b})_{rr}}}\frac{\partial
C}{\partial r}\sim r^{n_{4}}f_{4}(\theta)$

with

$\displaystyle n_{1},n_{2},n_{3},n_{4}\leq 1$ (94)

and $f_{1}(\theta),f_{2}(\theta),f_{3}(\theta),f_{3}(\theta)$ arbitrary
function of azimuthal angle; we have also taken into account the square root
of the induced metric determinant that enter in the computation of the Komar-
like integrals $\sqrt{|\gamma|}=r\sqrt{C(g_{b})_{rr}}$.

##  Appendix B Asymptotic generating matrix equations

Let us start with the asymptotic expansion for poles $\lambda_{k}$

$\displaystyle\lambda_{k}$
$\displaystyle=(w_{k}-z)\pm\sqrt{(w_{k}-z)^{2}+\rho^{2}}=$ (95)
$\displaystyle=(w_{k}-rcos(\theta))\pm\sqrt{(w_{k}-rcos(\theta))^{2}+r^{2}sin^{2}(\theta)}=$
$\displaystyle=(w_{k}-rcos(\theta))\pm\sqrt{(w^{2}_{k}+r^{2}+2w_{k}cos(\theta))}\sim
r(-cos(\theta)\pm 1);$

Assuming the asymptotic behaviour of the spectral parameter is the same of the
poles444This is suggested from the fact that the spectral parameter meets the
poles $\lambda_{k}$ along its trajectory; if the poles are considered on the
asymptotic boundary also the whole trajectory has to be consider
asymptotically and it has to meet poles on the asymptotic boundary. we have to
expand the following coefficients:

$\displaystyle\frac{2\lambda^{2}}{\lambda^{2}+\rho^{2}}\sim\frac{2(-rcos(\theta)\pm
r)^{2}}{(-rcos(\theta)\pm
r)^{2}+r^{2}sin^{2}(\theta)}=\frac{2[r^{2}cos^{2}(\theta)+r^{2}\mp
2r^{2}cos(\theta)]}{2r^{2}\mp 2r^{2}cos(\theta)}=$ (96)
$\displaystyle=1\mp\frac{cos(\theta)}{1\mp
cos(\theta)}+\frac{cos^{2}(\theta)}{1\mp cos(\theta)};$
$\displaystyle\frac{2\lambda\rho}{\lambda^{2}+\rho^{2}}\sim\frac{2(-rcos(\theta)\pm
r)rsin(\theta)}{2r^{2}\mp
2r^{2}cos(\theta)}=-\frac{cos(\theta)sin(\theta)}{1\mp
cos(\theta)}\pm\frac{sin(\theta)}{1\mp cos(\theta)};$
$\displaystyle\frac{\rho}{\lambda^{2}+\rho^{2}}\sim\frac{rsin(\theta)}{2r^{2}\mp
2r^{2}cos(\theta)}=\frac{1}{2r}\frac{sin(\theta)}{1\mp cos(\theta)};$
$\displaystyle\frac{\lambda}{\lambda^{2}+\rho^{2}}\sim\frac{-rcos(\theta)\pm
r}{2r^{2}\mp 2r^{2}cos(\theta)}=\frac{1}{2r}\frac{-cos(\theta)\pm 1}{1\mp
cos(\theta)}.$

We can sum and subtract equations in system 49 to get

$\displaystyle\bigg{(}[c(\theta)+s(\theta)]\frac{\partial}{\partial
r}+[c(\theta)-s(\theta)]\frac{\partial}{\partial\theta}+\frac{2\lambda\rho-2\lambda^{2}}{\lambda^{2}+r^{2}s^{2}(\theta)}\partial_{\lambda}-\frac{2rs(\theta)V_{b}}{\lambda^{2}+r^{2}s^{2}(\theta)}\bigg{)}\psi_{b}=0,$
(97) $\displaystyle\bigg{(}[c(\theta)-s(\theta)]\frac{\partial}{\partial
r}-[s(\theta)+c(\theta)]\frac{\partial}{\partial\theta}-\frac{2\lambda^{2}+2\lambda\rho}{\lambda^{2}+r^{2}s^{2}(\theta)}\partial_{\lambda}+\frac{2\lambda
U_{b}}{\lambda^{2}+r^{2}s^{2}(\theta)}\bigg{)}\psi_{b}=0;$

where $c(\theta)=cos(\theta)$ and $s(\theta)=sin(\theta)$. Inserting the
asymptotic expansions 96

$\displaystyle\bigg{(}[c(\theta)+s(\theta)]\frac{\partial}{\partial
r}+[c(\theta)-s(\theta)]\frac{\partial}{\partial\theta}+\mathcal{F}^{\uparrow\downarrow}_{1}(\theta)\partial_{\lambda}\bigg{)}\psi_{b}=\frac{1}{r}\frac{s(\theta)}{1\mp
c(\theta)}V_{b}\psi_{b},$ (98)
$\displaystyle\bigg{(}[c(\theta)-s(\theta)]\frac{\partial}{\partial
r}-[s(\theta)+c(\theta)]\frac{\partial}{\partial\theta}+\mathcal{F}^{\uparrow\downarrow}_{2}(\theta)\partial_{\lambda}\bigg{)}\psi_{b}=-\frac{1}{r}\frac{-c(\theta)\pm
1}{1\mp c(\theta)}U_{b}\psi_{b};$

where

$\displaystyle\mathcal{F}^{\uparrow\downarrow}_{1}(\theta)=\frac{1\mp
2c(\theta)+c^{2}(\theta)+c(\theta)s(\theta)\mp s(\theta)}{1\mp c(\theta)}=\mp
s(\theta)\mp c(\theta)+1;$ (99)
$\displaystyle\mathcal{F}^{\uparrow\downarrow}_{2}(\theta)=\frac{1\mp
2c(\theta)+c^{2}(\theta)-c(\theta)s(\theta)\pm s(\theta)}{1\mp c(\theta)}=\pm
s(\theta)\mp c(\theta)+1;$

where the apexes ${\uparrow\downarrow}$ indicate respectively the up or down
choice for the signs; matrices $V_{b}$ and $U_{b}$ could be expanded to get

$\displaystyle V_{b}=\rho(g_{b})_{,z}(g_{b})^{-1}\sim$ (100)
$\displaystyle\sim\frac{1}{rsin(\theta)}\begin{bmatrix}r^{\gamma}\mathcal{C}sin(\theta)r^{\alpha}\mathcal{A}^{\prime}-r^{\beta}\mathcal{B}r^{\beta}sin(\theta)\mathcal{B}^{\prime}&\
\ \
-r^{\beta}\mathcal{B}sin(\theta)r^{\alpha}\mathcal{A}^{\prime}+r^{\alpha}\mathcal{A}r^{\beta}sin(\theta)\mathcal{B}^{\prime}\\\
r^{\beta}\mathcal{B}^{\prime}sin(\theta)r^{\gamma}\mathcal{C}-r^{\beta}\mathcal{B}r^{\gamma}sin(\theta)\mathcal{C}^{\prime}&\
\ \
-r^{\beta}\mathcal{B}sin(\theta)r^{\beta}\mathcal{B}^{\prime}+r^{\alpha}\mathcal{A}r^{\gamma}sin(\theta)\mathcal{C}^{\prime}\end{bmatrix};$
$\displaystyle U_{b}=\rho(g_{b})_{,\rho}(g_{b})^{-1}\sim$ (101)
$\displaystyle\sim\frac{1}{rsin(\theta)}\begin{bmatrix}-r^{\gamma}\mathcal{C}cos(\theta)r^{\alpha}\mathcal{A}^{\prime}+r^{\beta}\mathcal{B}r^{\beta}cos(\theta)\mathcal{B}^{\prime}&\
\ \
r^{\beta}\mathcal{B}cos(\theta)r^{\alpha}\mathcal{A}^{\prime}-r^{\alpha}\mathcal{A}r^{\beta}cos(\theta)\mathcal{B}^{\prime}\\\
-r^{\beta}\mathcal{B}^{\prime}cos(\theta)r^{\gamma}\mathcal{C}+r^{\beta}\mathcal{B}r^{\gamma}cos(\theta)\mathcal{C}^{\prime}&\
\ \
r^{\beta}\mathcal{B}cos(\theta)r^{\beta}\mathcal{B}^{\prime}-r^{\alpha}\mathcal{A}r^{\gamma}cos(\theta)\mathcal{C}^{\prime}\end{bmatrix}.$

Note that asymptotically we have $V_{b}\sim cotang(\theta)U_{b}$.

##  Appendix C Christoffel symbols for the $N$ solitons solutions

Let us derive here the non-vanishing Christoffel symbols for a metric like

$ds^{2}=g_{22}(dr^{2}+r^{2}d\theta^{2})+g_{ij}dx^{i}dx^{j}\ \ \ i,j=0,1;$
(102)

which is of the same kind of the solitons metric. The non-vanishing symbols
are

$\displaystyle\Gamma^{2}_{00}=-\frac{1}{2}g^{22}\partial_{2}g_{00};\ \
\Gamma^{3}_{00}=-\frac{1}{2r^{2}}g^{22}\partial_{3}g_{00};\ \
\Gamma^{2}_{01}=-\frac{1}{2}g^{22}\partial_{2}g_{01};\ \
\Gamma^{3}_{01}=-\frac{1}{2r^{2}}g^{22}\partial_{3}g_{01};$ (103)
$\displaystyle\Gamma^{0}_{20}=\frac{1}{2}g^{00}\partial_{2}g_{00}+\frac{1}{2}g^{10}\partial_{2}g_{01};\
\
\Gamma^{1}_{20}=\frac{1}{2}g^{01}\partial_{2}g_{00}+\frac{1}{2}g^{11}\partial_{2}g_{01};$
$\displaystyle\Gamma^{0}_{30}=\frac{1}{2}g^{00}\partial_{3}g_{00}+\frac{1}{2}g^{10}\partial_{3}g_{01};\
\
\Gamma^{1}_{30}=\frac{1}{2}g^{01}\partial_{3}g_{00}+\frac{1}{2}g^{11}\partial_{3}g_{01};$
$\displaystyle\Gamma^{2}_{11}=-\frac{1}{2}g^{22}\partial_{2}g_{11};\ \
\Gamma^{3}_{11}=-\frac{1}{2r^{2}}g^{22}\partial_{3}g_{11};\ \
\Gamma^{0}_{22}=-\frac{1}{2}g^{00}\partial_{2}g_{22};\ \
\Gamma^{2}_{22}=-\frac{1}{2}g^{22}\partial_{2}g_{22};$
$\displaystyle\Gamma^{0}_{12}=\frac{1}{2}g^{00}\partial_{2}g_{01}+\frac{1}{2}g^{10}\partial_{2}g_{11};\
\
\Gamma^{1}_{12}=\frac{1}{2}g^{01}\partial_{2}g_{11}+\frac{1}{2}g^{11}\partial_{2}g_{11};$
$\displaystyle\Gamma^{0}_{13}=\frac{1}{2}g^{00}\partial_{3}g_{10}+\frac{1}{2}g^{10}\partial_{3}g_{11};\
\
\Gamma^{1}_{12}=\frac{1}{2}g^{01}\partial_{3}g_{10}+\frac{1}{2}g^{11}\partial_{3}g_{11};$
$\displaystyle\Gamma^{3}_{22}=-\frac{1}{2r^{2}}g^{22}\partial_{3}g_{22};\ \
\Gamma^{2}_{23}=\frac{1}{2}g^{22}\partial_{3}g_{22};\ \
\Gamma^{3}_{23}=\frac{1}{2r^{2}}g^{22}\partial_{2}g_{33};\ \
\Gamma^{2}_{31}=-\frac{1}{2}g^{22}\partial_{2}g_{33};$
$\displaystyle\Gamma^{3}_{33}=\frac{1}{2r^{2}}g^{22}\partial_{3}g_{33}.$

The one we are more interested to deduce which is the leading order of the
killing vectors is the last one and for $N=0$ we simply have

$\big{(}\Gamma^{3}_{33}\big{)}_{N=0}=\frac{\mathcal{D}^{\prime}(\theta)}{2\mathcal{D}(\theta)}$
(104)

while for the case $N=2$ we get

$\displaystyle\big{(}\Gamma^{3}_{33}\big{)}_{N=2}$
$\displaystyle=\mathcal{M}_{2}(\theta)\bigg{[}{(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))}\bigg{]}^{-4}r^{\delta+2+2\\#}\bigg{(}\prod_{j=1}^{\\#}\frac{1}{(w_{k}-w_{l})_{j}^{2}}\bigg{)}\Xi_{2}\
\times$ (105)
$\displaystyle\times\bigg{[}\frac{\mathcal{M}^{\prime}_{2}(\theta)}{\mathcal{M}_{2}(\theta)}-4\frac{r^{\epsilon_{1}+\epsilon_{3}+1}(\mathcal{E}_{1}\mathcal{E}_{3}(\theta))^{\prime}-r^{2\epsilon_{2}+1}(\mathcal{E}_{2}^{2}(\theta))^{\prime}}{r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}_{2}^{2}(\theta)}+\frac{\Xi^{\prime}_{2}}{\Xi_{2}}\bigg{]}\frac{1}{2g^{22}}=$
$\displaystyle=\frac{1}{2}\bigg{[}\frac{\mathcal{M}^{\prime}_{2}(\theta)}{\mathcal{M}_{2}(\theta)}-4\frac{r^{\epsilon_{1}+\epsilon_{3}+1}(\mathcal{E}_{1}\mathcal{E}_{3}(\theta))^{\prime}-r^{2\epsilon_{2}+1}(\mathcal{E}_{2}^{2}(\theta))^{\prime}}{r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}_{2}^{2}(\theta)}+\frac{\Xi^{\prime}_{2}}{\Xi_{2}}\bigg{]};$

where $\Xi_{2}:=\sum_{\sigma\in
S_{2}}\big{(}\operatorname{sgn}(\sigma)\prod_{s=0}^{2}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]\big{)}$.
The first term has no power of the radial cooordinate while the second and
third terms are of the form

$\frac{r^{a}f^{\prime}(\theta)+r^{b}g^{\prime}(\theta)}{r^{a}f(\theta)+r^{b}g(\theta)},$
(106)

where $a$ and $b$ are certain numbers; if $r^{a}$ is dominant with respect to
$r^{b}$ we will get $\frac{f^{\prime}(\theta)}{f(\theta)}$ and if $r^{b}$ is
dominant with respect to $r^{a}$ we will get
$\frac{g^{\prime}(\theta)}{g(\theta)}$. Therefore, in any case, the leading
order is independent of $r$ but strongly depends on the initial data. For the
general $N$ case the story is no different; we get

$\displaystyle\big{(}\Gamma^{3}_{33}\big{)}_{N}$
$\displaystyle=\mathcal{M}_{N}(\theta)\bigg{[}{(r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}^{2}_{2}(\theta))}\bigg{]}^{-2N}r^{\bar{\delta}}\bigg{(}\prod_{j=1}^{\\#}\frac{1}{(w_{k}-w_{l})_{j}^{2}}\bigg{)}\Xi_{N}\
\times$ (107)
$\displaystyle\times\bigg{[}\frac{\mathcal{M}^{\prime}_{N}(\theta)}{\mathcal{M}_{N}(\theta)}-2N\frac{r^{\epsilon_{1}+\epsilon_{3}+1}(\mathcal{E}_{1}\mathcal{E}_{3}(\theta))^{\prime}-r^{2\epsilon_{2}+1}(\mathcal{E}_{2}^{2}(\theta))^{\prime}}{r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}_{2}^{2}(\theta)}+\frac{\Xi^{\prime}_{N}}{\Xi_{N}}\bigg{]}\frac{1}{2g^{22}}=$
$\displaystyle=\frac{1}{2}\bigg{[}\frac{\mathcal{M}^{\prime}_{N}(\theta)}{\mathcal{M}_{N}(\theta)}-2N\frac{r^{\epsilon_{1}+\epsilon_{3}+1}(\mathcal{E}_{1}\mathcal{E}_{3}(\theta))^{\prime}-r^{2\epsilon_{2}+1}(\mathcal{E}_{2}^{2}(\theta))^{\prime}}{r^{\epsilon_{1}+\epsilon_{3}+1}\mathcal{E}_{1}\mathcal{E}_{3}(\theta)-r^{2\epsilon_{2}+1}\mathcal{E}_{2}^{2}(\theta)}+\frac{\Xi^{\prime}_{N}}{\Xi_{N}}\bigg{]};$

where now $\Xi_{N}:=\sum_{\sigma\in
S_{N}}\big{(}\operatorname{sgn}(\sigma)\prod_{s=0}^{N}[{A_{s\sigma_{s}}}+{B_{s\sigma_{s}}}+{C_{s\sigma_{s}}}+{D_{s\sigma_{s}}}]\big{)}$,
$\bar{\delta}:=\delta-\frac{N^{2}}{2}+N(N+1)-N(N-1)+2\\#$ and the same
consideration above for the case $N=2$ can be applied.

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# A Few-Shot Approach to Dysarthric Speech Intelligibility Level
Classification Using Transformers

Paleti Nikhil Chowdary Vadlapudi Sai Aravind Gorantla V N S L Vishnu Vardhan
Center for Computational Engineering and Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>Center for Computational Engineering and Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>Center for Computational Engineering and
Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>Menta Sai Akshay Menta Sai Aashish
Jyothish Lal. G  Center for Computational Engineering and Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>Center for Computational Engineering and Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>Center for Computational Engineering and Networking
Amrita Vishwa Vidyapeetham
Coimbatore, India
<EMAIL_ADDRESS>
###### Abstract

Dysarthria is a speech disorder that hinders communication due to difficulties
in articulating words. Detection of dysarthria is important for several
reasons as it can be used to develop a treatment plan and help improve a
person’s quality of life and ability to communicate effectively. Much of the
literature focused on improving ASR systems for dysarthric speech. The
objective of the current work is to develop models that can accurately
classify the presence of dysarthria and also give information about the
intelligibility level using limited data by employing a few-shot approach
using a transformer model. This work also aims to tackle the data leakage that
is present in previous studies. Our whisper-large-v2 transformer model trained
on a subset of the UASpeech dataset containing medium intelligibility level
patients achieved an accuracy of 85%, precision of 0.92, recall of 0.8
F1-score of 0.85, and specificity of 0.91. Experimental results also
demonstrate that the model trained using the ’words’ dataset performed better
compared to the model trained on the ’letters’ and ’digits’ dataset. Moreover,
the multiclass model achieved an accuracy of 67%.

###### Index Terms:

Dysarthria, UA-Speech, Whisper-large-v2, Few Shot Learning, PEFT, LORA,
Transfer Learning, Voice Pathology

##  I Introduction

Dysarthria, a neuro-motor impairment affecting speech articulation and
coordination, significantly impacts an individual’s ability to produce
coherent and intelligible verbal communication. In [1] F.Rudzicz et al.
claimed that dysarthria arises from congenital conditions or traumatic events
that impact the neuromotor system involved in speech production. The
congenital causes of dysarthria encompass conditions like brain asphyxiation
during birth, which result in long-term speech impairments. On the other hand,
traumatic causes of dysarthria include events such as stroke, cerebral palsy,
multiple sclerosis, Parkinson’s disease, myasthenia gravis, and amyotrophic
lateral sclerosis (ALS). Individuals with dysarthria encounter difficulties
related to articulation, speech rate, breath control, resonance, and overall
communication [2, 3, 4]. These challenges can result in diminished
comprehensibility, limited expressive abilities, and obstacles in social
interactions.

The field of dysarthria research has seen advancements in automatic speech
recognition (ASR) systems [5] for aiding individuals with dysarthria in
communication. However, the automatic classification of dysarthria and its
severity levels remain limited. Using the Frenchay Dysarthria Assessment [6],
doctors undertake perceptual evaluations of speech to determine the kind and
severity of the disease. Subjective assessments by clinicians are costly,
time-consuming, and prone to biases, raising concerns about their reliability.
This motivates the development of an impartial objective technique for
evaluating dysarthric speech.

More and more researchers are employing deep learning and machine learning
algorithms to develop automatic dysarthria identification in order to
objectively and reliably identify individuals with the condition. Many
researchers extract characteristics from voice signals using various feature
extraction techniques [7]. For example, Stephanie et al. [8] used Teager
Energy Operator (TEO) and the glottal waveform features. Chitralekha et al.
[9] utilized audio descriptors or features that are often used to determine
the timbre of musical instruments. Dong et al. [10] and Amlu et al. [11] used
MFCC-based features. N.P. Narendra et al. [12] used Two sets of glottal
features and acoustic features. Then, deep learning and machine learning
techniques, including convolutional neural networks (CNNs), artificial neural
networks (ANNs), CNN-LSTM (long short-term memory), CNN-GRU (Gated Recurrent
Unit), SVM, and other models, are used to detect dysarthria.

This research aims to develop an automatic tool that leverages vocal acoustics
to detect the presence of dysarthria and accurately determine its severity
level. Additionally, we investigate the efficacy of different speech tasks,
such as words, letters, and digits, in training the detection model.
Furthermore, we explore the feasibility of employing transformer models in
pathology detection, specifically dysarthria, utilizing few-shot transfer
learning techniques [13]. The training process utilizes a portion of the
UASpeech Dataset [14], while the remaining dataset is reserved for testing
purposes. Log Mel spectrogram features are extracted from the audio files and
are employed for training the Whisper Model [15] which is a large language
model, trained on 680,000 hours of multilingual audio data procured from the
internet. The whisper model family comprises of five different models with
varying model sizes. The large variant was considered in this research which
has 1550 million parameters. Considering the computational complexity involved
in training models of enormous size, various efficient training approaches
were considered and LORA [16] was used to make the training process efficient
and cost-effective.

The rest of the paper is organized as follows. Section 2 describes related
works while Section 3 gives a detailed description of the methodology used.
Section 4 presents the results and discussion and we conclude in Section 5.

##  II Related Works

TABLE I: Literature Review Reference No |  Title |  Author and Year |  Dataset |  Methodology |  Results
---|---|---|---|---|---
[8] |  Cross-Database Models for the Classification of Dysarthria Presence |  Stephanie Gillespie, Yash-Yee Logan, Elliot Moore, Jacqueline Laures-Gore, Scott Russell, Rupal Patel (2017) |  UA SPEECH, AMSDC |  Prosodic, spectral, Teager Energy Operators (TEO), and glottal features of various types were recovered. To further minimize the size of the feature subsets, 10-fold cross-validation sequential forward feature selection (SFFS) was utilized. SVM is employed for categorization. |  Using prosodic characteristics, the UA-Speech dataset maximizes word and participant-level accuracy at 75.3% and 92.9%, respectively.
[9] |  Automatic Assessment of Dysarthria Severity Level Using Audio Descriptors |  Chitralekha Bhat, Bhavik Vachhani, Sunil Kumar Kopparapu (2017) |  UA Speech, TORGO Database |  10 different sets of features called audio descriptors including STFT and Harmonic based features can be extracted using Multi-taper spectral estimation. An Artificial Neural Network (ANN) was used as the classifier for dysarthria severity classification |  Accuracy of 96.44% for UA speech corpus and 98.7% for TORGO database.
[10] |  Dysarthria Speech Detection Using Convolutional Neural Networks with Gated Recurrent Unit |  Dong-Her Shih, Ching-Hsien Liao, Ting-Wei Wu, Xiao-Yin Xu, Ming-Hung Shih (2022) |  UA SPEECH |  The model combines Convolutional Neural Networks (CNNs) and Gated Recurrent Units (GRUs). These methods allow for faster detection of dysarthria patients |  The findings demonstrate that the suggested CNN-GRU model outperforms the CNN, LSTM, CNN-LSTM, and models developed by other researchers, with the greatest accuracy of 98.38%.
[11] |  Automated Dysarthria Severity Classification:A Study on Acoustic Features and Deep Learning Techniques |  Amlu Anna Joshy, Rajeev Rajan (2022) |  UA SPEECH, TORGO Database |  The MFCC’s are computed framewise during the feature extraction stage. In the classification phase, four deep learning techniques DNN, CNN, LSTM and GRU are used. |  DNN gave 93.97% accuracy for speaker-dependent scenario and 49.22% for speaker-independent scenario
[12] |  Dysarthric speech classification from coded telephone speech using glottal features |  N P Narendra, Paavo Alku (2019) |  UA SPEECH, TORGO Database |  Support vector machine classifiers are trained using the suggested technique using both acoustic and glottal characteristics that were collected from coded speech utterances and their corresponding healthy/dysarthric labels. Individual and combined glottal and auditory characteristics are used to train different classifiers. |  An Accuracy of 95.4% and 96.38% is obtained for both datasets in different instances with different feature sets

There have been numerous techniques and models developed to predict the
presence of dysarthria. Some of the approaches are discussed in this section
and Table I presents the overview of the literature review.

In [8], Stephanie et al. employed a cross-database training strategy in their
study to distinguish speech samples with and without dysarthria. Specifically,
they trained their model on the UA-Speech database and evaluated its
performance on the AMSDC database. To mitigate the issue of repeated speech
samples from the same individual, one channel per participant was randomly
selected for analysis. The current analysis contains elements based on the
Teager Energy Operator (TEO) and the glottal waveform in addition to
conventional spectral and prosodic aspects. Baseline findings employing
prosodic features on the UA-Speech dataset to optimize word and participant-
level accuracy at 75.3% and 92.9%. However, the UA-Speech cross-training
evaluated on the AMSDC maximizes word- and participant-level accuracy at 71.3%
and 90%, respectively, based on TEO features.

In [9], Chitralekha et al. adopted audio descriptors or features commonly
employed to characterize the timbre of musical instruments and adapted them
for the purpose of their study. They utilized a dataset consisting of
dysarthric utterances, including utterances associated with 10 digits and 19
computer commands, collected from all patients. Features based on multi-
tapered spectral estimates were calculated and employed for classification.
With the use of the TORGO database and the Universal Access dysarthric speech
corpus, an Artificial Neural Network (ANN) was trained to categorize speech
into different severity levels. For the UA speech corpus and the TORGO
database, average classification accuracy was 96.44% and 98.7%, respectively.

In [10], Dong et al. used features based on MFCC Coefficients They utilized a
dataset consisting of dysarthric utterances, including utterances associated
with numbers 1 to 10, the 26 letters, collected from all patients. , and they
used Convolutional Neural Networks (CNNs) and Gated Recurrent Units (GRUs) to
enable faster dysarthria detection. Their experimental results demonstrate
that the CNN-GRU model achieves an accuracy of 98.38%, surpassing the
performance of other models like CNN, LSTM, and CNN-LSTM.

In [11] Amlu et al. employ the deep neural network (DNN), the convolutional
neural network (CNN), and the gated recurrent units(GRU) Long short term
memory (LSTM) to classify the severity of dysarthric speech. Mel frequency
cepstral coefficients (MFCCs) and their derivatives are the characteristics
used in this investigation. For the UA-Speech database, they used 4,500 test
files and 6,975 training files. Using the UA-Speech corpus and the TORGO
database, The findings show that DNN gave 93.97% accuracy for speaker-
dependent scenarios and 49.22% for speaker-independent scenarios.

In [12] N.P. Narendra et al. suggested a unique technique for classifying
dysarthric speech from coded telephone voice using glottal characteristics.
Each speaker’s spoken utterances were utilized. calculated using a glottal
inverse filtering technique based on deep neural networks. The openSMILE
toolbox is used to integrate glottal information (time- and frequency-domain
parameters and PCA-based parameters) with acoustic characteristics. Glottal
and auditory characteristics are used to train both separate and mixed support
vector machine classifiers. Studies using the TORGO and UA-Speech databases
show that the glottal factors produced a classification accuracy range of
63–77%.In [17] Amlu Anna Joshy et al. also classified dysarthria using multi-
head attention.

It was clear from the above literature that many methods had data leakage as
audio files from the same patient were split across train and test sets. And
there has not been much research conducted on few-shot learning techniques for
pathology classification, which is important because the amount of audio data
for pathology tasks is limited. The novelty of this work lies in exploring the
effectiveness of the few-shot learning approach using transformer models like
whisper-large-v2 for dysarthria detection and comparing which dataset task
(Words or letters and digits) performs the best.

##  III Methodology

###  III-A Dataset

The goal of the UA-Speech database [14] is to encourage the creation of user
interfaces for talkers who have spastic dysarthria and severe neuromotor
diseases. It consists of isolated-word recordings made using a 7-channel
microphone array mounted on top of a computer display from 15 dysarthric
speakers and 13 control speakers. Age, Gender, and Speech intelligibility of
speakers list the dataset’s varied dysarthric speakers’ levels of
intelligibility. This is represented in the table II.

TABLE II: Speaker-wise severity distribution for UASPEECH Database Speaker | Gender | Age | Speech Intelligibility (%)
---|---|---|---
M09 | Male |  18 | High (86%)
M14 | Male |  44 | High (90%)
M10 | Male |  21 | High (93%)
M08 | Male |  28 | High (95%)
F05 | Female |  22 | High (95%)
M05 | Male |  21 | Mid (58%)
M11 | Male |  48 | Mid (62%)
F04 | Female |  18 | Mid (62%)
M07 | Male |  58 | Low (28%)
F02 | Female |  30 | Low (29%)
M16 | Male |  40 | Low (43%)
M04 | Male |  $>18$ | Very Low (2%)
F03 | Female |  51 | Very Low (6%)
M12 | Male |  19 | Very Low (7%)
M01 | Male |  $>18$ | Very Low (17%)

Each patient has a total of 765 files which is comprised of 10 Digits of 3
repetitions, 26 Letters of 3 repetitions, 19 Computer Commands of 3
repetitions, 100 Common Words of 3 repetitions, and 300 Uncommon Words of 1
repetition.

For various experiments conducted in this study, various subsets of the
dataset are considered. First, a dataset is prepared for the purpose of
building binary classification models. This dataset is constructed by
exclusively using only the common words and uncommon words of the speakers. A
single repetition of common words (100 words) and all uncommon words (300
words) are combined together. In order to avoid data leakage, files from two
control patients and files from two pathology patients are used for training,
and files from all other patients are used for testing. The training set
contained a total of 1600 audio files (800 control and 800 pathology) and the
test set contained a total of 9,600 files (4400 control and 5200 pathology).
various experiments are conducted by considering pathology patients with
various intelligibility levels. A detailed description of this data is
presented in table III.

TABLE III: Patient distribution of Binary Class Dataset Experiment | Train | Test
---|---|---
Name |  Control Patients |  Pathology Patients |  Control |  Pathology
High Model |  800 |  800 |  4400 |  5200
(CM01 |  (F05 |  |
,CF02) |  ,M14) |  |
Medium Model |  800 |  800 |  4400 |  5200
(CM01 |  (F04 |  |
,CF02) |  ,M11) |  |
Low Model |  800 |  800 |  4400 |  5200
(CM01 |  (F02 |  |
,CF02) |  ,M16) |  |
Very Low Model |  800 |  800 |  4400 |  5200
(CM01 |  (F03 |  |
,CF02) |  ,M12) |  |

For the purpose of determining which dataset task gives better accuracy for
multiclass models. A new dataset is created using the letters and numbers
audio files. Each patient had 36 files ( 26 letters + 10 numbers) and again to
avoid data leakage only two patients were considered in the training set of
each class and all other patients were considered in the test set. The
training set contained a total of 360 audio files (72 control and 288
pathology) and the test set contained a total of 648 audio files (396 control
and 252 pathology). The detailed description of the multiclass dataset is
presented in table V.

TABLE IV: Patient distribution of Multiclass Dataset using Common and Uncommon words Data Split | Class | Patients | Patient Files
---|---|---|---
Train | Control | 2(CM01,CF02) | 800
High | 2(M14,F05) | 800
Medium | 2(M11,F04) | 800
Low | 2(M16,F02) | 800
Very-Low | 2(M12,F03) | 800
Test | Control | 11 | 396
High | 3 | 1200
Medium | 1 | 400
Low | 1 | 400
Very-Low | 2 | 800
TABLE V: Patient distribution of Multiclass Dataset using Digits and Letters Data Split | Class | Patients | Patient Files
---|---|---|---
Train | Control | 2(CM01,CF02) | 72
High | 2(M14,F05) | 72
Medium | 2(M11,F04) | 72
Low | 2(M16,F02) | 72
Very-Low | 2(M12,F03) | 72
Test | Control | 11 | 4400
High | 3 | 108
Medium | 1 | 36
Low | 1 | 36
Very-Low | 2 | 72

All of the input audio samples are resampled to 16,000 Hz for data
preprocessing, and a representation of an 80-channel log magnitude Mel
spectrogram is produced on 25-millisecond windows with a stride of 10
milliseconds. The whisper models are trained using this representation of the
preprocessed data.

###  III-B Whisper Model

Whisper [15] is an Automatic Speech Recognition (ASR) system developed by
OpenAI. It was trained using 680,000 hours of supervised, multilingual, and
multitasking web data. The details about various architectural parameters of
the whisper family models is presented in table VI. Since whisper was trained
with the intention of achieving high-quality results in a zero-shot setting,
this makes it very powerful and able to handle a wide range of tasks,
including speech recognition. Whisper is an encoder-decoder-based architecture
but since the task at hand requires only the encoder part of the model, we
extracted it and added a classification head as seen in Fig. 1. Using the
classification head, Given the log mel spectrogram of a speech uttered by a
subject, the model will predict the probability that the subject has
dysarthria and also the level of it in case of multiclass classification.

Figure 1: Modified Whisper Architecture TABLE VI: Whisper Model Family Model | Layers | Width | Heads | Parameters
---|---|---|---|---
Large | 910 | 1280 | 64 | 1550M
Modified Large(only encoder) | 361 | 1280 | 32 | 642M

###  III-C PEFT and LoRA

Training large language models typically requires huge clusters of GPUs and
vast amounts of data. In order to make training accessible for everyone,
various techniques are explored and presented. parameter-efficient fine-tuning
(PEFT) [18] selectively updates a subset of the model’s parameters,
specifically targeting the most influential ones for the new task. This
approach significantly reduces the computational resources required for fine-
tuning, resulting in improved efficiency without compromising performance. By
focusing on updating only the essential parameters, we ensured effective
training while minimizing unnecessary computations.

Among various methods included in PEFT, LoRA (LOW-RANK ADAPTATION OF LARGE
LANGUAGE MODELS) [16], developed by Microsoft, is by far the most popular
method. It is a technique used to fine-tune large language models (LLMs) by
freezing most of the parameters and updating only a small subset specific to
the task. It achieves parameter reduction by employing singular value
decomposition (SVD), which decomposes a matrix into three matrices. By
retaining a reduced number of singular values and their corresponding vectors,
the LLM can be efficiently fine-tuned while maintaining performance.

We utilized INT8 tuning along with PEFT, LoRA and bitsandbytes [19]. This
approach optimized memory usage and improved training efficiency, allowing us
to overcome the memory limitations and successfully train our model.

###  III-D Training

We opted to train the model using a cloud-rented machine provided by
Lambdalabs. The system had 30vCPUs, 200GiB RAM, and 1.4 TiB SSD, and it was
equipped with an Nvidia A10 GPU with compute capability of 8.6 and cost about
0.6$ per hour at the time of writing.

After preprocessing the data, the model was loaded into memory in 8-bit
precision and then optimized using Lora with a projection rank of 32. Then the
optimized model was put to training for 10 epochs using a batch size of 8 and
a learning rate of ${10^{-3}}$.

##  IV Results and Discussion

We used standard evaluation metrics such as accuracy, precision, recall, and
specificity. Accuracy is the percentage of the images that were correctly
classified. Precision is the accuracy of the positive predictions. The recall
is the fraction of the positives that were correctly predicted. F1-Score is
the harmonic mean of precision and recall. Specificity is the fraction of the
negatives that were correctly predicted.

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$

$Recall=\frac{TP}{TP+FN}$

$Precision=\frac{TP}{TP+FP}$

$F1-Score=\frac{2(Precision*Recall)}{Precision+Recall}$

$Specificity=\frac{TN}{TN+FP}$

The results obtained from the binary classification experiment are summarized
in table VII. Among the four experiments conducted, the model trained using
pathology patients with medium intelligibility levels performed the best,
giving an accuracy of 85%, precision of 0.92, recall of 0.8, F1-score of 0.85,
and specificity of 0.91. This indicates that the model finds it easy to
predict dysarthria if trained on data containing patients with medium
intelligibility levels compared to models trained with other intelligibility
levels such as very low, low, and high. Our model achieved around 10%
improvement in accuracy in comparison with the work presented in [12], where
they reported a word-level accuracy of 75.3%.

Table VIII and Table IX show accuracy, precision, recall, F1-Score, and
specificity of models trained on words dataset and digits and letters dataset,
respectively for multiclass classification which includes the classes:
Control, High, Medium, Low and Very Low. Both models are trained with two
patients belonging to each class. The accuracy of multiclass classification
using the words dataset is 67% while the accuracy for multiclass
classification using the ‘digits’ and ‘letters’ dataset is 58%. The multiclass
model trained on the words dataset achieved 9% better accuracy than its
counterpart. Analyzing the results from the table, we can see that both models
are performing the best in the control class compared with other classes, and
both models have a hard time predicting patients from the Low class. Both
models have good precision for the high class, but the class has a very low
score for other evaluation metrics.

TABLE VII: Binary Classification results Experiment name |  Accuracy (%) |  Precision |  Recall |  F1-Score |  Specificity
---|---|---|---|---|---
High |  81 |  0.85 |  0.79 |  0.82 |  0.85
Medium |  85 |  0.92 |  0.80 |  0.85 |  0.91
Low |  76 |  0.78 |  0.77 |  0.77 |  0.77
Very-Low |  69 |  0.99 |  0.42 |  0.59 |  0.99
TABLE VIII: Multiclass Classification Results using words dataset Class |  Precision |  Recall |  F1-Score |  Specificity
---|---|---|---|---
Control |  0.93 |  0.91 |  0.92 |  0.90
High |  0.73 |  0.02 |  0.04 |  0.02
Medium |  0.22 |  0.62 |  0.32 |  0.61
Low |  0.07 |  0.16 |  0.10 |  0.15
Very Low |  0.62 |  0.41 |  0.50 |  0.41
Accuracy = 67%
TABLE IX: Multiclass Classification Using Digits and Letters dataset Class |  Precision |  Recall |  F1-Score |  Specificity
---|---|---|---|---
Control |  0.83 |  0.81 |  0.82 |  0.81
High |  1.00 |  0.01 |  0.03 |  0.018
Medium |  0.23 |  0.44 |  0.30 |  0.44
Low |  0.03 |  0.11 |  0.05 |  0.11
Very Low |  0.37 |  0.23 |  0.29 |  0.234
Accuracy = 58%

##  V Conclusion and Future Works

This work explores a few-shot learning approach for dysarthria detection using
the encoder of the whisper-large-v2 model. The main contributions of the
proposed study are:

  * •

Compared with the previous methods, pathology detection has improved
considerably using transformer models and we are able to demonstrate the
potential use of few-shot learning for pathology detection.

  * •

From our study we have determined that to detect dysarthria, a model trained
using patients having medium-level intelligibility performs better.

  * •

We also determined that the dataset built using audio recordings of words will
result in better model performance.

Potential future works include determining the minimum number of patients to
accurately classify dysarthria using few-shot learning and comparative
analysis can be done using a wide spectrum of deep learning models to
determine which architecture performs the best.

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