Quasideterminant solutions of the generalized Heisenberg magnet model
U. Saleem 111Tel No: +92-42-99231243, Fax No: +92-42-35856892
<EMAIL_ADDRESS><EMAIL_ADDRESS>and M. Hassan
<EMAIL_ADDRESS>
Department of Physics, University of the Punjab,
Quaid-e-Azam Campus, Lahore-54590, Pakistan.

In this paper we present Darboux transformation for the generalized Heisenberg
magnet (GHM) model based on general linear Lie group $GL(n)$ and construct
multi-soliton solutions in terms of quasideterminants. Further we relate the
quasideterminant multi-soliton solutions obtained by the means of Darboux
transformation with those of obtained by dressing method. We also discuss the
model based on the Lie group $SU(n)$ and obtain explicit soliton solutions of
the model based on $SU(2)$.

PACS: 11.10.Nx, 02.30.Ik
Keywords: Integrable systems, Heisenberg model, Darboux transformation,
quasideterminants

##  1 Introduction

During the past decades, there has been an increasing interest in the study of
classical and quantum integrability of Heisenberg ferromagnet (HM) model
[1]-[15]. The Heisenberg ferromagnet (HM) model based on Hermitian symmetric
spaces has been studied in [11]-[14]. The integrability of the HM model based
on $SU(2)$ via inverse scattering method is presented in [2]-[3] and its
$SU(n)$ generalization is studied in [4]. The integrability of the GHM model
based on the general linear Lie group $GL(n)$ via Lax formalism has been
investigated in [1]. In this paper we present the Darboux transformation of
the GHM model based on general linear group $GL(n)$ with Lie algebra
$\verb"gl(n)"$ and calculate multi-soliton solutions in term of
quasideterminants. We also establish the relation between the Darboux
transformation and the well-known dressing method [16]. In the last section,
we discuss the model based $SU(n)$ and calculate an explicit expression of the
single-soliton solution of the HM model based on the Lie group $SU(2)$ using
Darboux transformation.

The Hamiltonian of the GHM model is defined by [1]

${\cal
H}=\frac{1}{2}\mbox{Tr}\left(\left(\partial_{x}U\right)^{T}\left(\partial_{x}U\right)\right),$
(1.1)

with $"T"$ is transpose and $U(x,t)$ is a matrix-valued function which takes
values in the Lie algebra $\verb"gl(n)"$ of the general linear group $GL(n)$.
The corresponding equation of motion can be expressed as

$\partial_{t}U=\\{{\cal H},\partial_{x}U\\}.$ (1.2)

The above equation (1.2) can be written as

$\partial_{t}U=\left[U,\partial^{2}_{x}U\right],$ (1.3)

where $\partial_{x}=\frac{\partial}{\partial x}$ and
$\partial_{t}=\frac{\partial}{\partial t}$. Let us assume that $U(x,t)$ is
diagonizable, i.e.,

$U=g\,T\,g^{-1},$ (1.4)

where $g\in GL(n)$ is matrix function of $(x,t)$ and $T$ is a $n\times n$
constant matrix

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccc}c_{1}&0&\cdots&0&0&0&\cdots&0&0\\\
0&c_{1}&\cdots&0&0&0&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&c_{1}&0&0&\cdots&0&0\\\ 0&0&\cdots&0&c_{2}&0&\cdots&0&0\\\
0&0&\cdots&0&0&c_{2}&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&\cdots&0&0&0&\cdots&0&c_{2}\\\ \end{array}\right),$ (1.13)

where $1\leq p\leq n$ and $c_{1},c_{2}\in\mathbb{R}$ (or $\mathbb{C}$). From
equations (1.4) and (1.13), we have

$\left[U,\left[U,\left[U,\chi\right]\right]\right]=c^{2}\left[U,\chi\right],$
(1.14)

for an arbitrary matrix function $\chi$ and $c=c_{1}-c_{2}\neq 0$. Since

$\partial_{x}U\equiv U_{x}=\left[\partial_{x}gg^{-1},U\right],$ (1.15)

implies

$\left[U,\left[U,U_{x}\right]\right]=c^{2}U_{x},$ (1.16)

The equation of motion (1.3) can also be written as the zero-curvature
condition i.e.,

$\left[\partial_{x}-\frac{1}{(1-\lambda)}U,\partial_{t}-\frac{c^{2}}{(1-\lambda)^{2}}U-\frac{1}{(1-\lambda)}\left[U,U_{x}\right]\right]=0.$
(1.17)

The above zero-curvature condition (1.17) is equivalent to the compatibility
condition of the following Lax pair

$\displaystyle\partial_{x}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{(1-\lambda)}U(x,t)\Psi(x,t;\lambda)$ (1.18)
$\displaystyle\partial_{t}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda)^{2}}U+\frac{1}{(1-\lambda)}\left[U,U_{x}\right]\right)\Psi(x,t;\lambda)$
(1.19)

where $\lambda$ is a real (or complex) parameter and $\Psi$ is an invertible
$n\times n$ matrix-valued function belonging to $GL(n)$.

In the next section, we define the Darboux transformation on matrix solutions
$\Psi$ of the Lax pair (1.18)-(1.19). To write down the explicit expressions
for matrix solutions of the GHM model, we will use the notion of
quasideterminant introduced by Gelfand and Retakh [17]-[21].

Let $X$ be an $n\times n$ matrix over a ring $R$ (noncommutative, in general).
For any $1\leq i$, $j\leq n$, let $r_{i}$ be the $i$th row and $c_{j}$ be the
$j$th column of $X$. There exist $n^{2}$ quasideterminants denoted by
$|X|_{ij}$ for $i,j=1,\ldots,n$ and are defined by

$|X|_{ij}=\left|\begin{array}[]{cc}X^{ij}&c_{j}^{\,\,i}\\\
r_{i}^{\,\,j}&\framebox(0.0,0.0)[bl]{\framebox{$x_{ij}$}}\end{array}\right|=x_{ij}-r_{i}^{\,\,j}\left(X^{ij}\right)^{-1}c_{j}^{\,\,i},$
(1.20)

where $x_{ij}$ is the $ij$th entry of $X$, $r_{i}^{\,\,j}$ represents the
$i$th row of $X$ without the $j$th entry, $c_{j}^{\,\,i}$ represents the $j$th
column of $X$ without the $i$th entry and $X^{ij}$ is the submatrix of $X$
obtained by removing from $X$ the $i$th row and the $j$th column. The
quasideterminats are also denoted by the following notation. If the ring $R$
is commutative i.e. the entries of the matrix $X$ all commute, then

$|X|_{ij}=(-1)^{i+j}\frac{\mathrm{det}X}{\mathrm{det}X^{ij}}.$ (1.21)

For a detailed account of quasideterminants and their properties see e.g.
[17]-[21]. In this paper, we will consider only quasideterminants that are
expanded about an $n\times n$ matrix over a commutative ring. Let

$\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right),$

be a block decomposition of any $K\times K$ matrix where the matrix $D$ is
$n\times n$ and $A$ is invertible. The ring $R$ in this case is the
(noncommutative) ring of $n\times n$ matrices over another commutative ring.
The quasideterminant of $K\times K$ matrix expanded about the $n\times n$
matrix $D$ is defined by

$\left|\begin{array}[]{cc}A&B\\\
C&\framebox(0.0,0.0)[bl]{\framebox{$D$}}\end{array}\right|=D-CA^{-1}B.$ (1.22)

The quasideterminants have found various applications in the theory of
integrable systems, where the multisoliton solutions of various noncommutative
integrable systems are expressed in terms of quisideterminants (see e.g.
[22]-[30]).

##  2 Darboux transformation

The Darboux transformation is one of the well-known method of obtaining multi-
soliton solutions of many integrable models [31]-[33]. We define the Darboux
transformation on the matrix solutions of the Lax pair (1.18)-(1.19), in terms
of an $n\times n$ matrix $D(x,t,\lambda)$, called the Darboux matrix. For a
general discussion on Darboux matrix approach see e.g. [34]-[39]. The Darboux
matrix relates the two matrix solutions of the Lax pair (1.18)-(1.19), in such
a way that the Lax pair is covariant under the Darboux transformation. The
one-fold Darboux transformation on the matrix solution of the Lax pair
(1.18)-(1.19) is defined by

$\Psi\left[1\right](x,t;\lambda)=D(x,t,{\lambda})\Psi(x,t;\lambda),$ (2.1)

where $D(x,t,{\lambda})$ is the Darboux matrix. For our case, we can make the
following ansatz

$D(x,t,\lambda)=\lambda I-M(x,t),\ \ $ (2.2)

where $M(x,t)$ is an $n\times n$ matrix function and $I$ is an $n\times n$
identity matrix. The new solution $\Psi\left[1\right](x,t;\lambda)$ satisfies
the following Lax pair, i.e.

$\displaystyle\partial_{x}\Psi\left[1\right](x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{1-\lambda}U\left[1\right]\Psi\left[1\right](x,t;\lambda),$
(2.3) $\displaystyle\partial_{t}\Psi\left[1\right](x,t;\lambda)$
$\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda)^{2}}U\left[1\right]+\frac{1}{1-\lambda}\left[U\left[1\right],U_{x}\left[1\right]\right]\right)\Psi\left[1\right](x,t;\lambda),$
(2.4)

where $U\left[1\right]$ satisfies the equation of motion (1.3). By operating
$\partial_{x}$ and $\partial_{t}$ on equation (2.1) and equating the
coefficients of different powers of $\lambda$, we get the following
transformation on the matrix field $U$

$\displaystyle U\left[1\right]$ $\displaystyle=$ $\displaystyle U+M_{x},$
(2.5)

and the following conditions which $M$ is required to satisfy

$\displaystyle M_{x}\left(I-M\right)$ $\displaystyle=$
$\displaystyle\left[U,M\right],$ (2.6) $\displaystyle
M_{t}\left(I-M\right)^{2}$ $\displaystyle=$
$\displaystyle\left[c^{2}U+\left[U,U_{x}\right],M\right]+M\left[U,U_{x}\right]M-\left[U,U_{x}\right]M^{2}.$
(2.7)

One can solve equations (2.6)-(2.7) to obtain an explicit expression for the
matrix function $M(x,t)$. An explicit expression for $M(x,t)$ can be found as
follows.

Let us take $n$ distinct real (or complex) constant parameters
${\lambda}_{1},\cdots,{\lambda}_{n}(\neq 1)$. Also take $n$ constant column
vectors $e_{1},e_{2},\cdots,e_{n}$ and construct an invertible non-degenerate
$n\times n$ matrix function $\Theta(x,t)$

$\Theta(x,t)=\left(\Psi({\lambda}_{1})e_{1},\cdots,\Psi({\lambda}_{n})e_{n}\right)=\left(\theta_{1},\cdots,\theta_{n}\right).$
(2.8)

Each column $\theta_{i}=\Psi({\lambda}_{i})e_{i}$ in the matrix $\Theta$ is a
column solution of the Lax pair (1.18)-(1.19) when ${\lambda}={\lambda}_{i}$
and $i=1,2,\ldots,n$ i.e.

$\displaystyle\partial_{x}\theta_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\lambda_{i}}U\theta_{i},$ (2.9)
$\displaystyle\partial_{t}\theta_{i}$ $\displaystyle=$
$\displaystyle\left(\frac{c^{2}}{(1-\lambda_{i})^{2}}U+\frac{1}{1-\lambda_{i}}\left[U,U_{x}\right]\right)\theta_{i}.$
(2.10)

Let us take an $n\times n$ invertible diagonal matrix with entries being
eigenvalues $\lambda_{i}$ corresponding to the eigenvectors $\theta_{i}$

$\Lambda=\text{diag}({\lambda}_{1},\ldots,{\lambda}_{n}).$ (2.11)

The $n\times n$ matrix generalization of the Lax pair (2.9)-(2.10) will be

$\displaystyle\partial_{x}\Theta$ $\displaystyle=$ $\displaystyle
U\Theta\left(I-\Lambda\right)^{-1},$ (2.12)
$\displaystyle\partial_{t}\Theta_{i}$ $\displaystyle=$ $\displaystyle
c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}.$
(2.13)

The $n\times n$ matrix $\Theta$ is a particular matrix solution of the Lax
pair (2.9)-(2.10) with $\Lambda$ being a matrix of particular eigenvalues. In
terms of particular matrix solution $\Theta$ of the Lax pair (2.9)-(2.10), we
make the following choice of the matrix $M(x,t)$

$M(x,t)=\Theta\Lambda\Theta^{-1}.$ (2.14)

Our next step is to check that equation (2.14) is a solution of equations
(2.6)-(2.7). In order to show this, we first operate $\partial_{x}$ on
equation (2.14) to get

$\displaystyle\partial_{x}M$ $\displaystyle=$
$\displaystyle\partial_{x}(\Theta\Lambda\Theta^{-1}),$ (2.15) $\displaystyle=$
$\displaystyle\left(\partial_{x}\Theta\right)\Lambda\Theta^{-1}+\Theta\Lambda\partial_{x}(\Theta^{-1}),$
$\displaystyle=$ $\displaystyle
U\Theta(I-\Lambda)^{-1}\Lambda\Theta^{-1}-\Theta\Lambda\Theta^{-1}U\Theta(I-\Lambda)^{-1}\Theta^{-1},$
$\displaystyle=$
$\displaystyle-U+\Theta(I-\Lambda)\Theta^{-1}j_{+}\Theta(I-\Lambda)^{-1}\Theta^{-1},$
$\displaystyle=$ $\displaystyle-U+\left(I-M\right)U\left(I-M\right)^{-1},$

which is the equation (2.6). Similarly operate $\partial_{t}$ on (2.14), we
get

$\displaystyle\partial_{t}M$ $\displaystyle=$
$\displaystyle\partial_{t}\left(\Theta\Lambda\Theta^{-1}\right)$ (2.16)
$\displaystyle=$
$\displaystyle\left(\partial_{t}\Theta\right)\Lambda\Theta^{-1}+\Theta\Theta\Lambda\partial_{t}(\Theta^{-1})$
$\displaystyle=$
$\displaystyle\left(c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}\right)\Lambda\Theta^{-1}-$
$\displaystyle\Theta\Lambda\Theta^{-1}\left(c^{2}U\Theta\left(I-\Lambda\right)^{-2}+\left[U,U_{x}\right]\Theta\left(I-\Lambda\right)^{-1}\right)\Theta^{-1},$

which is equation (2.7). This shows that the choice (2.14) of the matrix $M$
satisfies the equations (2.6)-(2.7). In other words we can say that if the
collection $\left(\Psi,U\right)$ is a solution of the Lax pair (1.18)-(1.19)
and the matrix $M$ is defined by (2.14), then $\left(\Psi[1],U[1]\right)$
defined by (2.1) and (2.5) respectively, is also a solution of the same Lax
pair. Therefore we say that

$\displaystyle\Psi[1]$ $\displaystyle=$ $\displaystyle\left(\lambda
I-\Theta\Lambda\Theta^{-1}\right)\Psi,$ $\displaystyle U[1]$ $\displaystyle=$
$\displaystyle\left(I-\Theta\Lambda\Theta^{-1}\right)U\left(I-\Theta\Lambda\Theta^{-1}\right)^{-1},$

is the required Darboux transformation on the solution $\Psi$ to the Lax pair
(1.18)-(1.19) and $U$ to the equation of motion (1.3) respectively.

##  3 Quasideterminant solutions

We have shown that the matrix $M=\Theta\Lambda\Theta^{-1}$ satisfies the
conditions (2.6)-(2.7). Therefore, the one-fold Darboux transformation (2.1)
can also be written in terms of quasideterments as

$\displaystyle\Psi[1]$ $\displaystyle\equiv$ $\displaystyle
D(x,t;\lambda)\Psi=\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi,$ (3.3) $\displaystyle=$
$\displaystyle\left|\begin{array}[]{cc}\Theta_{1}&\Psi\\\
\Theta_{1}\Lambda_{1}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda\Psi$}}\end{array}\right|.$

The above equation defines the Darboux transformation on the matrix solution
$\Psi$ of the Lax pair (1.18)-(1.19). The corresponding one-fold Darboux
transformation on the matrix field $U$ is

$\displaystyle U[1]$ $\displaystyle=$
$\displaystyle\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)U\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)^{-1},$
(3.8) $\displaystyle=$ $\displaystyle\left|\begin{array}[]{cc}\Theta_{1}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|U\left|\begin{array}[]{cc}\Theta_{1}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$

We write two-fold Darboux transformation on $\Psi$ as

$\displaystyle\Psi[2]$ $\displaystyle\equiv$ $\displaystyle
D(x,t;\lambda)\Psi[1]=\lambda\Psi[1]-\Theta_{2}[1]\Lambda_{2}\Theta^{-1}_{2}[1]\Psi[1]$
(3.12) $\displaystyle=$ $\displaystyle\lambda\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi-$
$\displaystyle\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\Lambda_{2}\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\left(\lambda
I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)\Psi,$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&\Psi\\\
\Theta_{1}\Lambda_{1}&\Theta_{2}\Lambda_{2}&\lambda\Psi\\\
\Theta_{1}\Lambda_{1}^{2}&\Theta_{2}\Lambda_{2}^{2}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda^{2}\Psi$}}\end{array}\right|.$

Similarly the expression for two-fold Darboux transformation on the matrix
field $U$ as

$\displaystyle U[2]$ $\displaystyle=$
$\displaystyle\Theta_{2}[1]\left(I-\Lambda_{2}\right)\Theta^{-1}_{2}[1]U[1]\left(\Theta_{2}[1]\left(I-\Lambda_{2}\right)\Theta^{-1}_{2}[1]\right)^{-1},$
$\displaystyle=$
$\displaystyle\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\left(I-\Lambda_{2}\right)\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\times$
$\displaystyle\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)U\left(I-\Theta_{1}\Lambda_{1}\Theta_{1}^{-1}\right)^{-1}\times$
$\displaystyle\left(\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)\left(I-\Lambda_{2}\right)\left(\Theta_{2}\Lambda_{2}-\Theta_{1}\Lambda_{1}\Theta^{-1}_{1}\Theta_{2}\right)^{-1}\right)^{-1},$
$\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|\times
U\times$ (3.20)
$\displaystyle\times\left|\begin{array}[]{ccc}\Theta_{1}&\Theta_{2}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$

The result can be generalized to obtain $N$-fold Darboux transformation on
matrix solution $\Psi$ as

$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&\Psi\\\
\Theta_{1}\Lambda_{1}&\Theta_{2}\Lambda_{2}&\cdots&\Theta_{N}\Lambda_{N}&\lambda\Psi\\\
\Theta_{1}\Lambda_{1}^{2}&\Theta_{2}\Lambda_{2}^{2}&\cdots&\Theta_{N}\Lambda_{N}^{2}&\lambda^{2}\Psi\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\Lambda_{1}^{N}&\Theta_{2}\Lambda_{2}^{N}&\cdots&\Theta_{N}\Lambda_{N}^{N}&\framebox(0.0,0.0)[bl]{\framebox{$\lambda^{N}\Psi$}}\end{array}\right|.$
(3.26)

Similarly the expression for $U[N]$ is

$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{2}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{N}&\Theta_{2}\left(I-\Lambda_{2}\right)^{N}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{N}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|\times
U\times$ (3.38)
$\displaystyle\times\left|\begin{array}[]{ccccc}\Theta_{1}&\Theta_{2}&\cdots&\Theta_{N}&I\\\
\Theta_{1}\left(I-\Lambda_{1}\right)&\Theta_{2}\left(I-\Lambda_{2}\right)&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)&0\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{2}&\Theta_{2}\left(I-\Lambda_{2}\right)^{2}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{2}&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\
\Theta_{1}\left(I-\Lambda_{1}\right)^{N}&\Theta_{2}\left(I-\Lambda_{2}\right)^{N}&\cdots&\Theta_{N}\left(I-\Lambda_{N}\right)^{N}&\framebox(0.0,0.0)[bl]{\framebox{$0$}}\end{array}\right|^{-1}.$

We now relate the quasideterminant solutions of GHM with the solutions
obtained by dressing method and the inverse scattering method. For this
purpose, we proceed as follows. From the definition of the matrix $M$, we have

$\displaystyle M\Theta$ $\displaystyle=$ $\displaystyle\Theta\Lambda.$ (3.39)

Let $\theta_{i}$ and $\theta_{j}$ be the column solutions of the Lax pair
(1.18)-(1.19) when $\lambda=\lambda_{i}$ and $\lambda=\lambda_{j}$
respectively i.e.

$\displaystyle M\theta_{i}$ $\displaystyle=$
$\displaystyle\lambda_{i}\theta_{i},\quad i=1,2,\dots,p$ $\displaystyle
M\theta_{j}$ $\displaystyle=$ $\displaystyle\lambda_{j}\theta_{j}.\quad
j=p+1,p+2,\dots,n$ (3.40)

Now we take $\lambda_{i}=\mu$ and $\lambda_{j}=\bar{\mu}$, we may write the
matrix $M$ as

$\displaystyle M$ $\displaystyle=$ $\displaystyle\mu P+\bar{\mu}P^{\perp},$
(3.41)

where $P$ is the hermitian projector i.e. $P^{\dagger}=P$. The projector $P$
satisfies $P^{2}=P$ and $P^{\perp}=1-P$. The projector $P$ is hermitian
projection on a complex space and $P^{\perp}$ as projection on orthogonal
space. Now equation (3.41) can also written as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\left(\mu-\bar{\mu}\right)P+\bar{\mu}I,$ (3.42)

where the hermitian projector can be expressed as

$\displaystyle P$ $\displaystyle=$
$\displaystyle\theta_{i}\left(\theta_{i}^{\dagger},\theta_{i}\right)^{-1}\theta_{i}^{\dagger}.$
(3.43)

The one-fold Darboux transformation (3.3) on the matrix solution $\Psi$ can
also be expressed in terms of projector $P$ as

$\displaystyle\Psi[1]$ $\displaystyle\equiv$ $\displaystyle{\cal
D}(x,t;\lambda)\Psi=\left(I-\frac{\mu-\bar{\mu}}{\lambda-\bar{\mu}}P\right)\Psi,$
(3.44)

where ${\cal D}(x,t;\lambda)$ is the rescaled Darboux-dressing function i.e.
${\cal D}(x,t;\lambda)=\left(\lambda-\mu\right)^{-1}D(x,t;\lambda)$. Similarly
the $N$-fold Darboux transformation (3.26) on the matrix solution $\Psi$ can
also be written as (take $P[1]=P$)

$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{N-1}\left(I-\frac{\mu_{N-k}-{\bar{\mu}_{N-k}}}{\lambda-{\bar{\mu}_{N-k}}}P[N-k]\right)\Psi.$
(3.45)

Now we can express the $N$-fold Darboux transformation (3.38) on the matrix
field $U$ can be written as

$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\prod_{k=0}^{N-1}\left(I-\frac{\mu_{N-k}-{\bar{\mu}_{N-k}}}{1-{\bar{\mu}_{N-k}}}P[N-k]\right)U\prod_{l=1}^{N-1}\left(I-\frac{{\bar{\mu}_{l}}-\mu_{l}}{1-{\bar{\mu}_{l}}}P[l]\right),$
(3.46)

and hermitian projector is defined as

$\displaystyle P[k]$ $\displaystyle=$
$\displaystyle\theta_{i}[k]\left(\theta_{i}^{\dagger}[k],\theta_{i}[k]\right)^{-1}\theta_{i}^{\dagger}[k].$
(3.47)

The expressions (3.45) and (3.46) can also be written as sum of $K$ terms [27]

$\displaystyle\Psi[N]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\left(I-\frac{1}{\lambda-{\bar{\mu}_{k}}}R_{k}\right)\Psi,$
(3.48)

and

$\displaystyle U[N]$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{N-1}\left(I-\frac{1}{1-{\bar{\mu}_{k}}}R_{k}\right)U\sum_{l=0}^{N-1}\left(I-\frac{1}{1-{\bar{\mu}_{l}}}R_{l}\right)^{-1},$
(3.49)

where

$\displaystyle R_{k}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{N-1}\left(\mu_{l}-\bar{\mu_{k}}\right)\theta_{i}^{(k)}\left(\theta_{i}^{(k)\dagger},\theta_{i}^{(l)}\right)^{-1}\theta_{i}^{(l)\dagger}.$
(3.50)

##  4 The explicit solutions of the GHM model

In this section we calculate explicit expression of soliton solution. First of
all we will study GHM model based on $SU(n)$. In this case the spin function
$U$ takes values in the Lie algebra $\verb"su(n)"$ so that one can decompose
the spin function into components $U=U^{a}T^{a}$, and
$T^{a},a=1,2,\dots,n^{2}$ are anti-hermitian $n\times n$ matrices with
normalization $\mbox{Tr}\left(T^{a}T^{b}\right)=\frac{1}{2}\delta^{ab}$ and
are the generators of the $SU(n)$ in the fundamental representation satisfying
the algebra

$\left[T^{a},T^{b}\right]=f^{abc}T^{c},$ (4.1)

where $f^{abc}$ are the structure constants of the Lie algebra $\verb"su(n)"$.
For any $X\in\verb"su(n)"$, we write $X=X^{a}T^{a}$ and
$U^{a}=-2\mbox{Tr}(UT^{a})$.

The matrix-field $U$ belongs to the Lie algebra $\verb"su(n)"$ of the Lie
group $SU(n)$ therefore

$\displaystyle U^{\dagger}=-U,\quad\quad\mbox{Tr}(U)=0.$ (4.2)

The equations (2.1)-(2.2) and (2.5) define a Darboux transformation for the
GHM model based on the Lie group $SU(n)$. The new solution of the equation of
motion (1.3) $U[1]$ must be $\verb"su(n)"$ valued i.e.

$\displaystyle U^{\dagger}[1]=-U[1],\quad\quad\mbox{Tr}(U[1])=0,$ (4.3)

therefore, we have the following conditions on the matrix $M$

$\displaystyle M^{\dagger}=-M,\quad\quad\mbox{Tr}(M)=0.$ (4.4)

In other words we want to make specific $M$ to satisfy the (4.4). This can be
achieved if we choose the particular solutions $\theta_{i}$ at
$\lambda=\lambda_{i}$, let us first calculate

$\displaystyle\partial_{x}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$
$\displaystyle\left(\partial_{x}\theta_{i}^{\dagger}\right)\theta_{j}+\theta_{i}^{\dagger}\left(\partial_{x}\theta_{j}\right)$
(4.5) $\displaystyle=$
$\displaystyle\left(1-\bar{\lambda_{i}}\right)^{-1}\theta_{i}^{\dagger}U^{\dagger}\theta_{j}+\left(1-\lambda_{j}\right)^{-1}\theta_{i}^{\dagger}U\theta_{j},$

using equation (4.2) the above equation (4.5) becomes

$\displaystyle\partial_{x}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$ $\displaystyle 0,$ (4.6)

when $\lambda_{i}\neq\lambda_{j}$ (i.e. $\bar{\lambda_{i}}=\lambda_{j}$).
Similarly we can check

$\displaystyle\partial_{t}\left(\theta_{i}^{\dagger}\theta_{j}\right)$
$\displaystyle=$ $\displaystyle 0.$ (4.7)

From the definition of the matrix $M$, we have

$\displaystyle\theta_{i}^{\dagger}\left(M^{\dagger}+M\right)\theta_{j}$
$\displaystyle=$
$\displaystyle\left(\bar{\lambda_{i}}+\lambda_{j}\right)\theta_{i}^{\dagger}\theta_{j},$
(4.8)

when $\lambda_{i}\neq\lambda_{j}$ then the above expression (4.8) implies

$\displaystyle\theta_{i}^{\dagger}\theta_{j}=0.$ (4.9)

The column vectors $\theta_{i}$ are linearly independent and the equation
(4.9) holds everywhere.

For the HM model based on $SU(n)$, the constant matrix (1.13) becomes

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccccc}2-\frac{2}{n}&0&\cdots&0&0&0&\cdots&0&0\\\
0&-\frac{2}{n}&\cdots&0&0&0&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&\cdots&-\frac{2}{n}&0&0&\cdots&0&0\\\
0&0&\cdots&0&-\frac{2}{n}&0&\cdots&0&0\\\
0&0&\cdots&0&0&-\frac{2}{n}&\cdots&0&0\\\
\vdots&\vdots&&\vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&\cdots&0&0&0&\cdots&0&-\frac{2}{n}\\\ \end{array}\right).$ (4.18)

Then $U^{2}$ becomes

$U^{2}=\frac{4\left(n-1\right)}{n^{2}}I+\frac{2\left(n-2\right)}{n}U.$ (4.19)

These are the constraints given in ref. [4]. For the construction of explicit
soliton solution for the $SU(n)$ HM model, we construct the matrix $M$ by
defining a Hermitian projector $P$. For this case, we take the seed solution
to be

$U_{0}\equiv U=\mbox{i}\left(\begin{array}[]{ccc}a_{1}&&\\\ &\ddots&\\\
&&a_{n}\end{array}\right),$ (4.20)

where $a_{i}$ are real constants and $\sum_{i=1}^{n}a_{i}=0$. The
corresponding solution of the Lax pair is expressed in block diagonal matrix

$\Psi(x,t;\lambda)=\left(\begin{array}[]{cc}W_{p}(\lambda)&O\\\
O&W_{n-p}(\lambda)\end{array}\right),$ (4.21)

where

$W_{p}(\lambda)=\left(\begin{array}[]{ccc}e^{\mbox{i}\omega_{1}(\lambda)}&&\\\
&\ddots&\\\ &&e^{\mbox{i}\omega_{p}(\lambda)}\end{array}\right),$ (4.22)

and

$W_{n-p}(\lambda)=\left(\begin{array}[]{ccc}e^{\mbox{i}\omega_{p+1}(\lambda)}&&\\\
&\ddots&\\\ &&e^{\mbox{i}\omega_{n}(\lambda)}\end{array}\right),$ (4.23)

are $p\times p$ and $(n-p)\times(n-p)$ matrices respectively and

$\omega_{i}(\lambda)=a_{i}\left(\frac{1}{1-\lambda}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right).$
(4.24)

Now define a particular matrix solution $\Theta$ of the Lax pair as

$\Theta=\left(\Psi(\mu)L_{1}\ ,\ \Psi(\bar{\mu})L_{2}\right),$ (4.25)

where $L_{1}$ is an $n\times p$ constant matrix of $p$ column vectors and
$L_{2}$ is the orthogonal complementary $n\times(n-p)$ matrix of $(n-p)$
column vectors. The columns of $L_{1}$ span a $p$-dimensional subspace $U$ of
$C^{n}$, and those of $L_{2}$ span the orthogonal subspace $V$. The projector
$P$ is completely characterized by the two subspaces $U=\text{Im}P$ and
$V=\text{Ker}P$ given by the condition $P^{\bot}U=0$ and $PV=0$. Let us write
$L_{1}=\left(\begin{array}[]{c}A\\\ B\end{array}\right)$ and
$L_{2}=\left(\begin{array}[]{c}C\\\ D\end{array}\right),$ where $A$, $B$, $C$
and $D$ are constant $p\times p$, $(n-p)\times n$, $p\times(n-p)$ and
$(n-p)\times(n-p)$ constant matrices respectively. Given this, the $n\times n$
matric $\Theta$ is given by

$\Theta=\left(\begin{array}[]{cc}W_{p}(\mu)A&W_{p}(\bar{\mu})C\\\
W_{n-p}(\mu)B&W_{n-p}(\bar{\mu})D\end{array}\right).$ (4.26)

We now define the projector $P$ in terms of the matrix
$\Phi=\Psi(\mu)L_{1}=(\theta_{1},\cdots\theta_{p})$ given by

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\left(\theta_{1},\cdots,\theta_{p}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}W_{p}(\mu)A\\\
W_{n-p}(\mu)B\end{array}\right).$

The projector is thus given by

$P=\left(\begin{array}[]{cc}W_{p}(\mu)A\Delta
A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})&W_{p}(\mu)A\Delta
B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})\\\ W_{n-p}(\mu)B\Delta
A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})&W_{n-p}(\mu)B\Delta
B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})\end{array}\right),$ (4.28)

where
$\Delta^{-1}=A^{{\dagger}}W_{p}^{{\dagger}}(\bar{\mu})W_{p}(\mu)A+B^{{\dagger}}W_{n-p}^{{\dagger}}(\bar{\mu})W_{n-p}(\mu)A$.
The Darboux matrix $D(\lambda)$ can now be constructed to give explicit
soliton solution of the $SU(n)$ HM model. To elaborate the result more
explicitly, we proceed with the example of $SU(2)$ HM model.

For the $SU(2)$ model, the equations (4.18) and (4.19) become

$\displaystyle T=\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\ \end{array}\right).$
(4.31)

Then $U^{2}$ becomes

$U^{2}=I.$ (4.32)

The Lax pair (1.18)-(1.19) for the $SU(2)$ model can be written as

$\displaystyle\partial_{x}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{(1-\lambda)}U(x,t)\Psi(x,t;\lambda),$ (4.33)
$\displaystyle\partial_{t}\Psi(x,t;\lambda)$ $\displaystyle=$
$\displaystyle\left(\frac{4}{(1-\lambda)^{2}}U+\frac{2}{(1-\lambda)}UU_{x}\right)\Psi(x,t;\lambda).$
(4.34)

If we take trivial solution (as seed solution), single soliton and multi-
soliton solutions can be obtained by Darboux transformation as explained
above.

We take the seed solution to be

$\displaystyle U_{0}\equiv U=\left(\begin{array}[]{cc}\mbox{i}&0\\\
0&-\mbox{i}\end{array}\right).$ (4.37)

The corresponding solution of the linear system (4.33)-(4.34) can be written
as

$\displaystyle\Psi(x,t;\lambda)=\left(\begin{array}[]{cc}e^{\mbox{i}\left(\frac{1}{\left(1-\lambda\right)}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right)}&0\\\
0&e^{-\mbox{i}\left(\frac{1}{\left(1-\lambda\right)}x+\frac{4}{\left(1-\lambda\right)^{2}}t\right)}\end{array}\right).$
(4.40)

Take $\lambda_{1}=\mu$ and $\lambda_{2}=\bar{\mu}$, the constant matrix
$\Lambda$ is given by

$\displaystyle\Lambda=\left(\begin{array}[]{cc}\mu&0\\\
0&\bar{\mu}\end{array}\right),$ (4.43)

and corresponding $2\times 2$ matrix solution $\Theta$ becomes

$\displaystyle\Theta\equiv\left(\theta_{1},\theta_{2}\right)=\left(\begin{array}[]{cc}e^{\mbox{i}\left(\frac{1}{\left(1-\mu\right)}x+\frac{4}{\left(1-\mu\right)^{2}}t\right)}&e^{\mbox{i}\left(\frac{1}{\left(1-\bar{\mu}\right)}x+\frac{4}{\left(1-\bar{\mu}\right)^{2}}t\right)}\\\
-e^{-\mbox{i}\left(\frac{1}{\left(1-\mu\right)}x+\frac{4}{\left(1-\mu\right)^{2}}t\right)}&e^{-\mbox{i}\left(\frac{1}{\left(1-\bar{\mu}\right)}x+\frac{4}{\left(1-\bar{\mu}\right)^{2}}t\right)}\end{array}\right).$
(4.46)

The matrix $M$ is given by

$\displaystyle M$ $\displaystyle=$ $\displaystyle\Theta\Lambda\Theta^{-1},$
(4.49) $\displaystyle=$
$\displaystyle\frac{1}{e^{u}+e^{-u}}\left(\begin{array}[]{ll}\mu
e^{u}+{\bar{\mu}}{e^{-u}}&\left({\bar{\mu}-\mu}\right)e^{{i}v}\\\
\left({\bar{\mu}-\mu}\right)e^{-iv}&{\bar{\mu}}{e^{u}}+{\mu}{e^{-u}}\end{array}\right),$

where the functions $u(x,t)$ and $v(x,t)$ are defined by

$\displaystyle u(x,t)$ $\displaystyle=$
$\displaystyle\mbox{i}\left(\frac{1}{\left(1-\mu\right)}-\frac{1}{\left(1-\bar{\mu}\right)}\right)x+4\mbox{i}\left(\frac{1}{\left(1-\mu\right)^{2}}-\frac{1}{\left(1-\bar{\mu}\right)^{2}}\right)t,$
$\displaystyle v(x,t)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\left(1-\mu\right)}+\frac{1}{\left(1-\bar{\mu}\right)}\right)x+4\left(\frac{1}{\left(1-\mu\right)^{2}}+\frac{1}{\left(1-\bar{\mu}\right)^{2}}\right)t.$
(4.50)

Let us take the eigenvalue to be $\mu=e^{\mbox{i}\theta}.$ The expression
(4.49) then becomes

$M=\left(\begin{array}[]{cc}\cos\theta+\mbox{i}\sin\theta\tanh
u&-\mbox{i}\left(\sin\theta\text{sech}u\right)e^{\mbox{i}v}\\\
-\mbox{i}\left(\sin\theta\text{sech}u\right)e^{-\mbox{i}v}&\cos\theta-\mbox{i}\sin\theta\tanh
u\end{array}\right),$ (4.51)

and the corresponding Darboux matrix $D\left(\lambda\right)$ in this case is

$D\left(\lambda\right)=\left(\begin{array}[]{cc}\lambda-\cos\theta-\mbox{i}\sin\theta\tanh
u&\mbox{i}\left(\sin\theta\text{sech}u\right)e^{\mbox{i}v}\\\
\mbox{i}\left(\sin\theta\text{sech}u\right)e^{-\mbox{i}v}&\lambda-\cos\theta+\mbox{i}\sin\theta\tanh
u\end{array}\right).$ (4.52)

Comparing the above equation with (3.44), we find the following expression for
the projector

$P=\left(\begin{array}[]{cc}2e^{u}\text{sech}u&-2e^{\mbox{i}v}\text{sech}u\\\
-2e^{-\mbox{i}v}\text{sech}u&2e^{-u}\text{sech}u\end{array}\right).$ (4.53)

Using (3.8) and (4.37), we get

$U[1]=\left(\begin{array}[]{cc}\mbox{i}U_{3}&U_{+}\\\ -U_{-}&-\mbox{i}U_{3}\\\
\end{array}\right),$ (4.54)

where

$\displaystyle U_{3}$ $\displaystyle=$ $\displaystyle
1-(1+\cos\theta)\mbox{sech}^{2}u,$ $\displaystyle U_{+}$ $\displaystyle\equiv$
$\displaystyle\overline{U}_{-}=-\mbox{i}e^{\mbox{i}v}\left[(1+\cos\theta)\mbox{tanh}u+\mbox{i}\sin\theta\right]\mbox{sech}u.$
(4.55)

From equation (4.55), we see that $U^{\dagger}[1]=-U[1]$ and
$\mbox{Tr}(U[1])=0$. Therefore equation (4.55) is an explicit expression of
the single-soliton solution of the HM model based on $SU(2)$ obtained by using
Darboux transformation. Similarly one can calculate explicit expression for
the multi-soliton solution of the model. The expression (4.55) is similar to
the expression of the single soliton given in [2].

##  5 Concluding remarks

In this paper, we have studied GHM model based on general linear Lie group
$GL(n)$ and expressed the multi-soliton solutions in terms of the
quasideterminant using the Darboux transformation defined on the solution of
the Lax pair. We have also established equivalence between the Darboux matrix
approach and the Zakharov-Mikhailov’s dressing method. In last section we have
reduced the GHM model into the HM model based on $SU(n)$ and calculated an
explicit expression for the single-soliton solution. It would be interesting
to study the GHM models based on Hermitian symmetric spaces. We shall address
this problem in a separate work.

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# A Review of Deep Reinforcement Learning in Serverless Computing: Function
Scheduling and Resource Auto-Scaling

Amjad Yousef Majid  Martel InnovateKeizersgracht 482Amsterdamthe Netherlands
<EMAIL_ADDRESS>and  Eduard Marin  Telefonica1 Thørväld
CircleMadridSpain<EMAIL_ADDRESS>
(2023)

###### Abstract.

In the rapidly evolving field of serverless computing, efficient function
scheduling and resource scaling are critical for optimizing performance and
cost. This paper presents a comprehensive review of the application of Deep
Reinforcement Learning (DRL) techniques in these areas. We begin by providing
an overview of serverless computing, highlighting its benefits and challenges,
with a particular focus on function scheduling and resource scaling. We then
delve into the principles of deep reinforcement learning (DRL) and its
potential for addressing these challenges. A systematic review of recent
studies applying DRL to serverless computing is presented, covering various
algorithms, models, and performances. Our analysis reveals that DRL, with its
ability to learn and adapt from an environment, shows promising results in
improving the efficiency of function scheduling and resource scaling in
serverless computing. However, several challenges remain, including the need
for more realistic simulation environments, handling of cold starts, and the
trade-off between learning time and scheduling performance. We conclude by
discussing potential future directions for this research area, emphasizing the
need for more robust DRL models, better benchmarking methods, and the
exploration of multi-agent reinforcement learning for more complex serverless
architectures. This review serves as a valuable resource for researchers and
practitioners aiming to understand and advance the application of DRL in
serverless computing.

cloud, serverless, deep reinforcement learning

††copyright: rightsretained††journalyear: 2023††doi: XXXXXXX.XXXXXXX

##  1\. Introduction

The advent of serverless computing has revolutionized the way applications are
developed and deployed in the cloud. By abstracting away the underlying
infrastructure, serverless computing allows developers to focus on writing
applications’ code, while the cloud provider manages the execution
environment, including server management, capacity planning, and scaling. This
paradigm shift, however, introduces new challenges in terms of function
scheduling and resource scaling, which are critical for optimizing performance
and cost in serverless environments.

Function scheduling in serverless computing involves deciding when and where
to execute a function, considering factors such as resource availability,
current workload, performance security requirements, or privacy regulations
(Ghobaei-Arani and Ghorbian, 2023). Resource scaling, on the other hand,
refers to the dynamic allocation and deallocation of resources based on demand
(Mampage et al., 2022). Both of these tasks are complex and dynamic problems
that require intelligent and adaptive solutions.

Traditional ML algorithms, while powerful, were designed for more static
datasets and predetermined variables. In the dynamic environment of serverless
computing, where demands and patterns can change rapidly, these algorithms
might not adapt quickly enough or may require extensive retraining.
Additionally, they typically do not handle the real-time, adaptive decision-
making that’s essential for optimizing serverless deployments, given the
importance of instantaneous feedback. For example, Smart Spread uses
supervised ML for serverless function scheduling. Consequently, functions have
to be profiled before deployment which is considered a critical drawback
(Mahmoudi et al., 2019).

Deep Reinforcement Learning (DRL), a subfield of artificial intelligence that
combines deep learning and reinforcement learning, has shown promise in
addressing these challenges (Yao et al., 2023; Qiu et al., 2022b; Mampage et
al., 2023). DRL models learn to make decisions by interacting with an
environment, receiving feedback in the form of rewards or penalties, and
adjusting their strategies to maximize the cumulative reward (Majid et al.,
2023). This ability to learn and adapt makes DRL particularly suitable for the
dynamic and uncertain environments of serverless computing.

There are many review and survey papers that target a variety of serverless
computing aspects (Cassel et al., 2022; Wen et al., 2022; Li et al., 2022;
Shafiei et al., 2022; Hassan et al., 2021; Jawaddi and Ismail, 2023). The use
of DRL for cloud orchestration—a process that coordinates and automates the
management and control of the computer systems, middleware, and services
within a cloud environment to ensure they operate in harmony. Serverless
function scheduling is intrinsically linked to cloud orchestration as it
requires efficiently coordinating where and when serverless functions run—and
resource management has also been surveyed and reviewed (Zhong et al., 2022;
Duc et al., 2019; Kar et al., 2023). However, there is no paper that is
dedicated to reviewing or surveying that work on the use of (D)RL for
optimizing both function scheduling and resource scaling in serverless
environments. This paper aims to fill this gap by providing a systematic
review of recent studies that apply DRL to serverless computing. We cover
various DRL algorithms and models, evaluate their performance metrics, and
discuss the challenges and future directions in this research area. This
review serves as a valuable resource for researchers and practitioners aiming
to understand and advance the application of DRL in serverless computing.

##  2\. Background

###  2.1. Serverless Computing

Serverless computing, also known as Function as a Service (FaaS), is a cloud
computing model where the cloud provider dynamically manages the allocation
and provisioning of servers allowing users to write and deploy code without
worrying about the underlying infrastructure (Jonas et al., 2019). Key
benefits of serverless computing include reduced operational concerns,
automatic scaling, and cost-effectiveness. However, it also presents unique
challenges, such as statelessness, cold start latency, and the need for
efficient function scheduling and resource auto-scaling in (near) real time
(Li et al., 2022).

Serverless Scheduling is about how tasks are assigned and managed within a
serverless architecture. In a serverless environment, the cloud provider
manages the server infrastructure, and tasks or functions are executed in
stateless compute containers that are event-triggered and fully managed by the
cloud provider. Serverless scheduling involves deciding when and where these
tasks or functions are run. It’s about efficiently distributing tasks among
the available resources to optimize for factors like performance, cost, and
completion time while taking security and privacy into consideration.

Resource Auto-scaling is the dynamic capability of cloud infrastructures to
automatically tailor computational resources according to the demand
(Kriushanth et al., 2013). It’s not a monolithic approach. While serverless
computing often provides a pre-defined set of computational resources for
functions, such as a fixed memory size, advanced platforms now allow diverse
computational requirements to cater to varying workloads (Mampage et al.,
2022). This complexity highlights the challenge and importance of effective
auto-scaling. Essentially, auto-scaling dynamically adjusts resources like
servers, memory, or storage based on workload requirements. Overestimating
needs lead to under-utilized resources, resulting in increased costs.
Conversely, underestimating can compromise the application’s performance or
even lead to its failure. Auto-scaling can be predictive, where resources are
allocated based on a schedule, or reactive, adjusting to real-time workload
changes (Poppe et al., 2022). The potential of Deep Reinforcement Learning
(DRL) extends to enhancing both predictive and reactive scaling strategies. To
encapsulate, while both processes focus on efficient resource utilization in a
cloud setting, resource auto-scaling modifies the volume of resources, whereas
serverless scheduling oversees the task executions within those allocated
resources.

Serverless versus Microservices Although autoscaling in serverless computing
and microservices are driven by the same objective of adjusting resources
based on workload, they differ fundamentally in their mechanisms (Castro et
al., 2019; Jawaddi et al., 2022). In serverless computing, resources are
automatically allocated for each function or task as it’s triggered, with
execution occurring in isolated environments. Scaling is virtually limitless,
and users are charged solely for the actual compute time. Contrarily,
microservices autoscaling necessitates a hands-on approach, adjusting the
number of individual microservice instances as required. This involves active
performance monitoring, orchestration tools, and an understanding of the
infrastructure, leading to charges based on reserved resources irrespective of
their full utilization (Toffetti et al., 2017).

Function scheduling in serverless platforms is largely abstracted, with events
triggering functions and the platform ensuring seamless execution. Resource
allocation in this context is handled internally, with automatic assignment of
required compute, memory, and storage (Kaffes et al., 2022). In contrast,
microservices architectures require more direct engagement: scheduling
determines which instances handle requests, often involving tools like load
balancers, and resource allocation decisions are more explicit, demanding
careful capacity planning (Toffetti et al., 2017). While serverless
autoscaling simplifies infrastructure management, microservices autoscaling
offers increased control, albeit with greater complexity.

###  2.2. Deep Reinforcement Learning

Figure 1. Deep Reinforcement Learning Overview

Deep Reinforcement Learning (DRL) is a subfield of artificial intelligence
that combines the strengths of deep learning and reinforcement learning. Deep
learning is a type of machine learning that uses neural networks with many
layers to model and understand complex patterns in data. Reinforcement
learning, on the other hand, is a type of machine learning where an agent
learns to make decisions by interacting with an environment and receiving
feedback in the form of rewards or penalties. The combination of these two
fields in DRL allows for the creation of models that can handle high-
dimensional inputs, deal with sequential data, and learn from interaction with
an environment. This interaction is often modeled as a Markov Decision Process
(MDP; Figure 1). An MDP includes: (i) Agent: The entity that makes decisions
and takes actions; (ii) Environment: The context in which the agent operates
and interacts. (iii) State: The current situation or condition of the
environment; (iv) Action: A choice made by the agent that affects the state of
the environment; (v) Reward: Feedback from the environment that indicates the
success of an action. Rewards can be positive (for successful actions) or
negative (for unsuccessful actions); (vi) Policy: The strategy that the agent
uses to decide which action to take in a given state. In DRL, the policy is
typically represented by a deep neural network.

The objective of a DRL agent is to maximize the total reward (or return) over
a series of interactions with an environment. This is facilitated by value
functions, namely the state-value function and the action-value function (or
Q-function), which provide the expected return of being in a particular state
or performing a specific action and following a specific policy thereafter
(Sutton and Barto, 2018).

Several types of RL algorithms have been developed. They are largely
categorized under three main categories:

  * •

Value-based RL algorithms optimize value functions to select the most
rewarding actions. Examples of such algorithms are SARSA (Rummery and
Niranjan, 1994), Q-learning (Sutton and Barto, 2018) and DQN (Mnih et al.,
2015) which combines Q-learning with deep neural networks (DNNs).

  * •

Policy-based RL algorithms optimize the policy directly, outputting a
probability distribution over the action space given a state. These algorithms
are particularly useful for learning stochastic policies and dealing with
high-dimensional or continuous action spaces, despite potentially being sample
inefficient. An example of this is the REINFORCE algorithm (Williams, 1992).

  * •

Actor-critic algorithms combine policy and value function learning to reduce
variance and speed up learning. These algorithms use a policy ”actor” and a
value estimator ”critic”, both parameterized by DNNs. Prominent examples
include the Advantage Actor-Critic (A2C) (Clemente et al., 2017), Asynchronous
Advantage Actor-Critic (A3C) (Mnih et al., 2016), and Deep Deterministic
Policy Gradient (DDPG) algorithms (Lillicrap et al., 2015).

DRL has been applied in a wide range of fields, including: (i) Game playing:
DRL has been used to train agents that can play complex games such as Go,
chess, and poker at a high level. For example, Google’s AlphaGo, the first
computer program to defeat a human world champion at Go, uses a form of DRL
(Silver et al., 2017); (ii) Robotics: DRL can be used to train robots to
perform tasks such as grasping objects, walking, or flying. The advantage of
DRL in this context is that it allows robots to learn from interaction with
the environment, rather than relying on pre-programmed instructions (Saaybi et
al., 2022; Haarnoja et al., 2018; Doukhi and Lee, 2022); and (iii) Resource
management in computer systems: DRL can be used to optimize resources in
computer systems. For example, it can be used to reduce energy consumption in
a data center (Ran et al., 2019), optimize the scheduling of tasks in a
computing cluster (Mao et al., 2016), or control the flow of data in a network
(Jay et al., 2019). In the context of serverless computing, DRL can be used to
optimize function scheduling, resource auto-scaling, and cold start (Qiu et
al., 2022a; Vahidinia et al., 2022), for example. In this review, we opt to
focus on the function scheduling and resource auto-scaling as these are
complex and dynamic problems that require intelligent and adaptive solutions,
which DRL is well-suited to provide.

##  3\. Deep Reinforcement Learning for Resource Auto-Scaling and Serverless
Scheduling

Recent studies have started to explore the application of DRL in serverless
computing, particularly in the areas of function scheduling and resource
scaling.

###  3.1. DRL for function scheduling in serverless computing

Current serverless FaaS platforms typically employ basic, traditional
scheduling algorithms to distribute function calls. However, they often
overlook FaaS-specific traits, like swift shifts in resource use and the
dynamic activation and idling of functions (Yu et al., 2021). Therefore, they
present FaaSRank, a function scheduler for serverless Function-as-a-Service
(FaaS) platforms. FaaSRank uses DRL to learn scheduling policies based on
monitored information from servers and functions. It is implemented in Apache
OpenWhisk, an open-source FaaS platform, and evaluated against other baseline
schedulers using real-world serverless workload traces provided by Microsoft
Azure. The results showed that FaaSRank sustained on average a lower number of
inflight invocations (i.e., the number of function instances being executed)
59.62% and 70.43% as measured on two clusters respectively.

Rather than solely concentrating on the cloud setting, Dehury et al. (2021)
explored deploying serverless functions across both fog and cloud
environments. To tackle the problem of effectively allocating functions
between these environments, they introduced DeF-DReL, a DRL-driven serverless
function scheduler. DeF-DReL determines the optimal split of user requests to
be handled by fog and cloud, considering factors like user proximity to fog
nodes, latency, user and application priorities, and resource needs. The
findings indicate that the DeF-DReL agent adopts a function distribution
approach that exhibits contrasting distribution patterns compared to the other
two traditional scheduling algorithms examined.

Yao et al. (2023) focus on optimizing serverless task offloading in an edge-
IoT environment. The authors propose an experience-sharing deep reinforcement
learning-based (ES-DRL) method for optimizing task offloading in serverless
edge computing. ES-DRL uses a distributed learning architecture and a
population-guided policy search method to enhance performance and avoid local
optima. ES-DRL’s main objective is to decide whether to execute a computation
task locally on an IoT device or to offload it to an edge server taking into
account metrics like CPU utilization and energy consumption. The method is
tested against existing DRL-based task offloading methods and shows a
reduction in average latency by up to approximately 17%.

Mampage et al. (2023) present a DRL technique based on DQN (Mnih et al., 2015)
for function scheduling in multi-tenant serverless computing environments. The
proposed method addresses challenges such as resource contention and the
ephemeral nature of serverless functions. The DRL model optimizes the trade-
off between application response time and resource usage cost. The technique,
evaluated using the Kubeless serverless framework, shows significant
improvements in both response time and resource usage cost compared to
baseline techniques, demonstrating the effectiveness of DRL in managing
resource allocation in multi-tenant serverless environments.

###  3.2. DRL for Resource Auto-Scaling

Schuler et al. (2021) investigate the use of RL to optimize auto-scaling in
serverless computing environments. The authors argue that serverless
computing, while offering significant advantages such as scalability and cost-
effectiveness, presents challenges in resource management due to fluctuating
demand. They propose an RL model based on Q-learning to dynamically determine
the optimal level of concurrency for individual workloads in a serverless
environment. The authors implemented this model in Knative, an open-source
serverless platform, and evaluated its performance. The results showed that
the proposed model was able to learn an effective scaling policy within a
limited number of iterations, improving performance compared to the default
auto-scaling configuration. While the results obtained are specific to a
single application, limiting their general applicability, the authors
nevertheless conclude reinforcement learning offers a promising approach for
optimizing auto-scaling in serverless computing environments. (Yuvaraj et al.,
2021) improved the performance of Q-learning by combining it with Grey Wolf
Optimization (GWO) algorithm: a nature-inspired metaheuristic method that
iteratively evaluates candidate solutions to approach an optimal one
(Mirjalili et al., 2014).

Zafeiropoulos et al. (2022) studied the effectiveness of three (D)Rl agents
(namely, Q-learning, DynaQ+, and DQL) to optimize auto-scaling in serverless
computing. The agents are trained in real and simulated environments using the
Kubeless platform. The results validate the effectiveness of these agents in
balancing application performance and resource usage, demonstrating the
potential of RL in optimizing serverless computing.

Bensalem et al. (2023) present a DRL and RL approach to auto-scaling
serverless functions in edge networks. The authors argue that traditional
cloud-based auto-scaling mechanisms are not directly applicable to edge
networks due to their distributed nature, the complexity of optimal resource
allocation, and the delay sensitivity of workloads. They propose a (D)RL-based
solution to efficiently scale and allocate resources for serverless functions
in edge networks. The paper compares these RL and DRL algorithms with
empirical, monitoring-based heuristics, particularly for delay-sensitive
applications. The simulation results show that the RL algorithm outperforms
standard, monitoring-based algorithms in terms of the total delay of function
requests, achieving an improvement in delay performance by up to 50%. Results
showed that RL is as good as DRL, which allows us to use the basic RL as a
fast and efficient solution.

Qiu et al. (2022a, b) discuss the application of DRL for resource management
in multi-tenant serverless platforms. The authors present a customized multi-
agent DRL algorithm based on Proximal Policy Optimization (MA-PPO). They argue
that the state-of-the-art single-agent RL algorithm (S-RL) suffers from high
function tail latency degradation on multi-tenant serverless FaaS platforms
and struggles to converge during training. In contrast, MA-PPO allows each
agent to be trained until convergence and provides online performance
comparable to S-RL in single-tenant cases with less than 10% degradation.
Furthermore, MA-PPO offers a 4.4x improvement in S-RL performance in multi-
tenant cases. The authors evaluate their approach using both real-world and
synthetic function invocation patterns, and they use benchmarks from widely
used open-source FaaS benchmark suites.

Wang et al. (2019) propose SIREN, a distributed machine learning framework
based on serverless architecture. SIREN utilizes DRL to dynamically control
the number and memory size of the serverless functions used in each training
epoch. It is able to achieve higher parallelism and elasticity while reducing
system configuration overhead. The prototype implementation on AWS Lambda
showed a reduction in model training time by up to 44% compared to traditional
machine learning training benchmarks on AWS EC2 at the same cost. The paper
argues that serverless architecture is a more cost-effective and manageable
solution for machine learning practitioners and data scientists.

##  4\. Methodology

In this study, we conduct a systematic review of the application of RL and DRL
in serverless computing, focusing on function scheduling and resource scaling.
We perform a comprehensive literature search in several databases, including
IEEE Xplore, ACM Digital Library, Springer, and Google Scholar. The search
terms include combinations of ”serverless computing”, ”function scheduling”,
”resource scaling”, ”deep reinforcement learning”, and ”DRL”. We also manually
search the reference lists of the included studies to identify additional
relevant papers. We include studies that apply RL and/or DRL to serverless
computing, particularly in the areas of function scheduling and resource
scaling. We exclude studies that do not use DRL, do not focus on serverless
computing, or do not provide sufficient detail about the DRL models and their
performance.

##  5\. Findings and Discussion

Our systematic review reveals a significant interest in the application of DRL
for function scheduling and resource scaling in serverless computing. The
reviewed studies demonstrate the potential of DRL to address the dynamic and
complex challenges of serverless computing. It is evident that DRL can learn
from the environment and adapt to changing conditions, which are key
characteristics of serverless computing environments.

DRL-based serverless schedulers such as FaaSRank (Yu et al., 2021), DeF-DReL
(Dehury et al., 2021), ES-DRL (Yao et al., 2023), and DQN-based model (Mampage
et al., 2023) have shown DRL algorithms are superior to classical heuristics
in optimizing serverless function scheduling. In terms of resource auto-
scaling, Q-learning based models (Schuler et al., 2021), Grey Wolf
Optimization combined with Q-learning (Yuvaraj et al., 2021), DRL and RL based
models (Bensalem et al., 2023; Zafeiropoulos et al., 2022), MA-PPO (Qiu et
al., 2022a, b), and SIREN (Wang et al., 2019) demonstrated that DRL can more
optimally manage dynamic resource demands in serverless computing environments
compared to the current state of the art algorithms.

Despite the promising results demonstrated by the reviewed studies, there are
several challenges and open questions that need to be addressed.

First, the problem of efficient function scheduling and resource auto-scaling
in serverless computing is a complex, multi-dimensional optimization problem.
It requires managing trade-offs among competing objectives such as cost,
performance, and resource utilization. The existing DRL models mainly focus on
one or two objectives, leaving the multi-objective optimization an open
problem. Second, serverless computing operates in a highly dynamic and
uncertain environment, with fluctuating workloads and resource availability.
The current DRL models need to deal with this high level of uncertainty and
variability. Incorporating robustness and uncertainty quantification in the
learning process is therefore crucial for the successful application of DRL in
serverless computing. Third, the evaluation and comparison of different DRL
models is challenging due to the lack of standardized benchmarks and
performance metrics and datasets. Each study uses different evaluation
scenarios, metrics, and baselines, making it difficult to compare the results
and identify the best-performing models. Finally, there are practical
challenges in deploying DRL in serverless computing, such as the high
computational cost of DRL, the difficulty in debugging and understanding the
learned policies, and the sensitivity of DRL to hyperparameter settings.

##  6\. Future Research Directions

Based on the findings and discussion, we suggest several directions for future
research in the application of DRL for serverless computing:

  * •

Multi-objective optimization: Future research should investigate DRL models
that can handle multiple objectives simultaneously and balance trade-offs
among cost, performance, and resource utilization.

  * •

Robustness and uncertainty quantification: Developing DRL models that can
handle uncertainty and variability in serverless computing environments is
crucial. Techniques such as adversarial training, Bayesian neural networks,
ensemble methods, and meta-learning should be explored to develop robust DRL-
based serverless solutions.

  * •

Benchmarks and performance metrics: The development of standardized benchmarks
and performance metrics for evaluating and comparing DRL models in serverless
computing would be beneficial.

  * •

Practical deployment: Research is needed to address the practical challenges
in deploying DRL in serverless computing, such as reducing the computational
cost of DRL, improving the interpretability of learned policies, and making
the DRL models more robust to different hyperparameter settings. Solutions
could include developing more efficient training algorithms, creating tools
for visualizing and interpreting DRL policies, and designing hyperparameter
tuning strategies specifically for serverless computing environments.

  * •

Real-time adaptation: As serverless computing environments are highly dynamic,
DRL models need to adapt to changes in real-time. Future work could focus on
online learning strategies and methods for continual learning, enabling DRL
models to adjust their policies as the environment changes.

  * •

Collaborative learning: In the context of collaborative learning in serverless
computing, federated learning emerges as a promising approach. Federated
learning enables multiple (potentially untrusted) stakeholders to
collaboratively train a model without directly sharing their data. Instead,
they share model updates or gradients, ensuring data privacy. This
decentralized training approach is particularly suitable for serverless
computing environments, where data might be distributed across various nodes
or locations. By leveraging federated learning, DRL models can benefit from
diverse data sources without compromising on data security or privacy, leading
to more robust and generalized policies.

  * •

Privacy and security: As DRL models interact with the environment and collect
data for learning, it’s essential to consider privacy and security issues.
Future research could explore methods for privacy-preserving learning and
mechanisms to ensure the security of DRL models in serverless computing
environments.

##  7\. Conclusion

In conclusion, the application of DRL in serverless computing presents
promising opportunities and significant challenges. While existing studies
have demonstrated the potential of DRL for function scheduling and resource
scaling, much work remains to be done. Future research should focus on
addressing the challenges and exploring the suggested research directions to
advance the field and realize the full potential of DRL in serverless
computing.

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