# Ion-acoustic rogue waves in multi-ion plasmas

M. Hassan∗,1, M. H. Rahman1, N. A. Chowdhury∗∗,1, A. Mannan1,3, and A. A.
Mamun1,2 1Department of Physics, Jahangirnagar University, Savar, Dhaka-1342,
Bangladesh
2Wazed Miah Science Research Center, Jahangirnagar University, Savar,
Dhaka-1342, Bangladesh
3Institut für Mathematik, Martin Luther Universität Halle-Wittenberg, Halle,
Germany
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The basic properties of nonlinear ion-acoustic (IA) waves (IAWs), particularly
finite amplitude IA rogue waves (IARWs) in a plasma medium (containing pair
ions, iso-thermal positrons and non-thermal electrons) are theoretically
investigated by deriving the nonlinear Schrödinger equation (NLSE). The
criteria for the modulational instability of IAWs, and the basic features of
finite amplitude IARWs are identified. The modulationally stable and unstable
regions are determined by the sign of the ratio of the dispersive coefficient
to the nonlinear coefficient of NLSE. The latter is analyzed to obtain the
region for the existence of the IARWs, which corresponds to the unstable
region. The shape of the profile of the rogue waves depends on the non-thermal
parameter $\alpha$ and the ratio of electron temperature to positron
temperature. It is found that the increase in the value of the non-thermal
parameter enhances both the amplitude and width of IARWs, and that the
enhancement of positron (electron) temperature reduces (enhances) the
amplitude and width of IARWs. It is worth to mention that our present
investigation may be useful for understanding the salient features of IARWs in
space (viz., upper region of Titan’s atmosphere, cometary comae, and Earth’s
ionosphere, etc.) and laboratory (viz., plasma processing reactor and neutral
beam sources, etc.) plasmas.

###### keywords:

NLSE , modulational instability , rogue waves.

††journal: “High Energy Density Physics”

##  1 Introduction

The signature of pair-ion (PI) plasma (PIP) has been observed by the Cassini
spacecraft [1] in spaces, specially, in upper region of Titan’s atmosphere,
and is also generated by Oohara et al. [2, 3, 4] in laboratory experiments. PI
has also identified in cometary comae [5, 6, 7], Earth’s ionosphere [8, 9, 10,
11, 12], plasma processing reactor [15], neutral beam sources [16], and
successively has encouraged many researchers to understand the intrinsic
contribution of the PI in recognizing the collective properties of the PIP by
investigating ion-plasma waves [3], intermediate frequency waves [3], ion-
acoustic (IA) solitary waves (IASWs) [5], IA shock waves (IASHWs) [6], IA
Gardner solitons (IAGSs) [7], IA double layers (IADLs) [8], and IA rogue waves
(IARWs) [10, 11, 12, 13, 14], etc.

Non-thermal particles are appeared in a complex system due to the presence of
the external force field, and are governed by the Cairns distribution/non-
thermal distribution [17, 18, 19], and have also been identified by the Freja
[20] and Viking [21] satellites in the magnetosphere and upper ionosphere of
the auroral zone. Paul and Bandyopadhyay [22] investigated dust-ion-acoustic
solitary waves in a four components plasma medium having non-thermal plasma
species, and observed that the height of the positive super-solitons increases
with increasing the value of non-thermality of the plasma species. Selim [23]
studied IASWs in a three components PIP medium (PIPM) by considering non-
thermal electrons, and found that the height and thickness of the compressive
solitons increase with increasing the value of electron’s non-thermality.
Singh and Lakhina [24] considered a three components plasma model having non-
thermal electrons, and studied IA super-solitons, and found that the width of
the super-solitons increases with an increase in the value of electrons non-
thermality.

Electrostatic rogue waves (RWs) are considered to have appeared due to the
nonlinear property of the plasma medium as well as the modulational
instability (MI) of the carrier waves, and are governed by the nonlinear
Schrödinger equation (NLSE), and have also been identified in the plasma
physics [10, 11, 12, 25]. El-Tantawy et al. [10] considered a two components
plasma model having inertial ions and non-thermal electrons, and studied the
mechanism of the formation of IARWs, and also found that behind the critical
value of the non-thermal parameter, the nonlinearity and height of the IARWs
increase with the non-thermal parameter. Abdelwahed et al. [11] considered a
three components PIPM and observed the IARWs in presence of super-thermal
electrons, and found that the height of the IARWs decreases with the positive
ion number density. Elwakil et al. [12] analyzed the MI of the IAWs in a PIPM
having non-thermal electrons. Rahman et al. [25] studied dust-acoustic RWs in
a non-thermal dusty plasma medium, and found that the height of the RWs
increases with the non-thermality of the plasma species.

Recently, Sabry et al. [5] studied IASWs in a three components PIPM having
non-thermal electrons, and found that the presence of negative ion enhances
the amplitude but decreases the width of the solitary pulse. El-Labany et al.
[8] reported IADLs in a three components PIPM in presence of non-thermal
electrons. Jannat et al. [6] examined IASHWs in a four components PIPM having
inertial PI as well as inertialess electrons and positrons. It is inevitable
to investigate the MI of the IAWs and the formation of the IARWs in a four
components PIPM by considering inertial double ions and inertialess iso-
thermal positrons as well as non-thermal Cairns distributed electrons.

The outline of the paper is as follows: The governing equations describing our
plasma model are presented in Section 2. The standard NLSE is derived in
Section 3. The MI of the IAWs is examined in Section 4. The first-order and
second-order RWs are discussed in Section 5. Finally, a brief conclusion is
provided in Section 6.

##  2 Governing Equations

We consider a four components plasma model consisting of negative ions,
positive ions, electrons, and positrons. Overall, the charge neutrality
condition can be written as $Z_{+}n_{+0}+n_{p0}=Z_{-}n_{-0}+n_{e0}$; where
$n_{+0}$, $n_{p0}$, $n_{-0}$, and $n_{e0}$ are, respectively, the equilibrium
number densities of positive ions, iso-thermal positrons, negative ions, and
non-thermal Cairn’s distributed electrons, and also $Z_{+}$ and $Z_{-}$ are,
respectively, the charge state of the positive ions and negative ions. Now,
the basic set of normalized equations can be written as

$\displaystyle\frac{\partial n_{+}}{\partial t}+\frac{\partial}{\partial
x}(n_{+}u_{+})=0,$ (1) $\displaystyle\frac{\partial u_{+}}{\partial
t}+u_{+}\frac{\partial u_{+}}{\partial x}=-\frac{\partial\phi}{\partial x},$
(2) $\displaystyle\frac{\partial n_{-}}{\partial t}+\frac{\partial}{\partial
x}(n_{-}u_{-})=0,$ (3) $\displaystyle\frac{\partial u_{-}}{\partial
t}+u_{-}\frac{\partial u_{-}}{\partial
x}=\delta_{1}\frac{\partial\phi}{\partial x},$ (4)
$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=\delta_{2}n_{e}-\delta_{3}n_{p}+(1-\delta_{2}+\delta_{3})n_{-}-n_{+},$
(5)

where $n_{+}$ and $n_{-}$ are the positive and negative ion number density
normalized by their equilibrium value $n_{+0}$ and $n_{-0}$, respectively;
$u_{-}$ and $u_{+}$ are the negative and positive ion fluid speed normalized
by wave speed $C_{+}=(Z_{+}k_{B}T_{e}/m_{+})^{1/2}$ (with $T_{e}$ being the
temperature non-thermal electron, $m_{+}$ being the positive ion mass, and
$k_{B}$ being the Boltzmann constant); $\phi$ is the electrostatic wave
potential normalized by $k_{B}T_{e}/e$ (with $e$ being the magnitude of single
electron charge); the time and space variables are normalized by
$\omega_{p+}^{-1}=(m_{+}/4\pi Z_{+}^{2}e^{2}n_{+0})^{1/2}$ and
$\lambda_{D+}=(k_{B}T_{e}/4\pi e^{2}Z_{+}n_{+0})^{1/2}$, respectively. Other
plasma parameters are defined as $\delta_{1}=Z_{-}m_{+}/Z_{+}m_{-}$,
$\delta_{2}=n_{e0}/Z_{+}n_{+0}$, and $\delta_{3}=n_{p0}/Z_{+}n_{+0}$. Now, the
expression for electron number density which is obeying non-thermal Cairn’s
distribution [17] is given by

$\displaystyle n_{e}=(1-\beta\phi+\beta\phi^{2})\exp(\phi),$ (6)

where $\beta=4\alpha/(1+3\alpha)$, with $\alpha$ being the parameter
determining the fast particles present in our plasma model. The positron
number density which is iso-thermally distributed is given by

$\displaystyle n_{p}=\exp(-\delta_{4}\phi),$ (7)

where $\delta_{4}=T_{e}/T_{p}$ and $T_{e}>T_{p}$. Now, by substituting Eq. (6)
and (7) into Eq. (5), and expanding up to third order of $\phi$, we get

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}+n_{+}=(\delta_{2}-\delta_{3})+(1-\delta_{2}+\delta_{3})n_{-}$
$\displaystyle\hskip
8.5359pt+H_{1}\phi+H_{2}\phi^{2}+H_{3}\phi^{3}+\cdot\cdot\cdot,$ (8)

where

$\displaystyle H_{1}=\delta_{2}(1-\beta)+\delta_{3}\delta_{4},$ $\displaystyle
H_{2}=[\delta_{2}-\delta_{3}\delta_{4}^{2}]/2,$ $\displaystyle
H_{3}=[\delta_{2}(1-\beta+6\beta)+\delta_{3}\delta_{4}^{3}]/6.\ $

We note that the term on the right hand side of the Eq. (8) is the
contribution of electron and positron species.

##  3 Derivation of the NLSE

To study the MI of IAWs, we want to derive the NLSE by employing the reductive
perturbation method (RPM) and for that case, we can write the stretched
coordinates in the form [26, 27]

$\displaystyle\xi={\epsilon}(x-v_{g}t),$ (9)
$\displaystyle\tau={\epsilon}^{2}t,$ (10)

where $v_{g}$ is the group velocity and $\epsilon$ ($\epsilon\ll 1$) is a
small parameter. Then, we can write the dependent variables as [26, 27]

$\displaystyle
n_{+}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{+l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (11) $\displaystyle
u_{+}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{+l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (12) $\displaystyle
n_{-}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{-l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (13) $\displaystyle
u_{-}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{-l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (14)
$\displaystyle\phi=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}\phi_{l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],\ $ (15)

where $k$ ($\omega$) is real variables representing the carrier wave number
(frequency). The derivative operators in above equations are treated as
follows:

$\displaystyle\frac{\partial}{\partial t}\rightarrow\frac{\partial}{\partial
t}-\epsilon
v_{g}\frac{\partial}{\partial\xi}+\epsilon^{2}\frac{\partial}{\partial\tau},$
(16) $\displaystyle\frac{\partial}{\partial
x}\rightarrow\frac{\partial}{\partial
x}+\epsilon\frac{\partial}{\partial\xi}.$ (17)

Now, by substituting Eq. (9)-(17) into Eq. (1)-(4) and Eq. (8), and collecting
the terms containing $\epsilon$, the first order ($m=1$ with $l=1$) reduced
equations can be written as

$\displaystyle n_{+1}^{(1)}=\frac{k^{2}}{\omega^{2}}\phi_{1}^{(1)},$ (18)
$\displaystyle u_{+1}^{(1)}=\frac{k}{\omega}\phi_{1}^{(1)},$ (19)
$\displaystyle
n_{-1}^{(1)}=-\frac{k^{2}\delta_{1}}{\omega^{2}}\phi_{1}^{(1)},$ (20)
$\displaystyle u_{-1}^{(1)}=-\frac{k\delta_{1}}{\omega}\phi_{1}^{(1)}.\ $ (21)

These relation provides the dispersion relation for IAWs

$\displaystyle\omega^{2}=\frac{k^{2}[1+\delta_{1}(1-\delta_{2}+\delta_{3})]}{k^{2}+H_{1}}.$
(22)

Figure 1: Plot of $P/Q$ vs $k$ for different values of $\delta_{1}$ when other
parameters are $\alpha=0.7$, $\delta_{2}=0.4$, $\delta_{3}=0.4$, and
$\delta_{4}=2.0$. Figure 2: Plot of $P/Q$ vs $k$ for different values of
$\delta_{3}$ when other parameters are $\alpha=0.7$, $\delta_{1}=0.8$,
$\delta_{2}=0.4$, and $\delta_{4}=2.0$. Figure 3: Plot of $|\Phi_{1}|$ vs
$\xi$ for different values of $\alpha$ when other parameters are $k=0.7$,
$\tau=0$, $\delta_{1}=0.8$, $\delta_{2}=0.4$, $\delta_{3}=0.4$, and
$\delta_{4}=2.0$. Figure 4: Plot of $|\Phi_{1}|$ vs $\xi$ for different values
of $\delta_{2}$ when other parameters are $k=0.7$, $\tau=0$, $\alpha=0.7$,
$\delta_{1}=0.8$, $\delta_{3}=0.4$, and $\delta_{4}=2.0$. Figure 5: Plot of
$|\Phi_{1}|$ vs $\xi$ for different values of $\delta_{4}$ when other
parameters are $k=0.7$, $\tau=0$, $\alpha=0.7$, $\delta_{1}=0.8$,
$\delta_{2}=0.4$, and $\delta_{3}=0.4$.

The second order ($m=2$ with $l=1$) equations are given by

$\displaystyle
n_{+1}^{(1)}=\frac{k^{2}}{\omega^{2}}\phi_{1}^{(2)}+\frac{2ik(v_{g}k-\omega)}{\omega^{3}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(23) $\displaystyle
u_{+1}^{(2)}=\frac{k}{\omega}\phi_{1}^{(2)}+\frac{i(v_{g}k-\omega)}{\omega^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(24) $\displaystyle
n_{-1}^{(2)}=-\frac{\delta_{1}k^{2}}{\omega^{2}}\phi_{1}^{(2)}-\frac{2ik\delta_{1}(v_{g}k-\omega)}{\omega^{3}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(25) $\displaystyle
u_{-1}^{(2)}=-\frac{\delta_{1}k}{\omega}\phi_{1}^{(2)}-\frac{i\delta_{1}(v_{g}k-\omega)}{\omega^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},\
$ (26)

with the compatibility condition

$\displaystyle
v_{g}=\frac{\omega[1+\delta_{1}(1-\delta_{2}+\delta_{3})-\omega^{2}]}{k[1+\delta_{1}(1-\delta_{2}+\delta_{3})]}.$
(27)

The coefficients of $\epsilon$ for $m=2$ and $l=2$ provide the second order
harmonic amplitudes which are found to be proportional to
$|\phi_{1}^{(1)}|^{2}$

$\displaystyle n_{+2}^{(2)}=H_{4}|\phi_{1}^{(1)}|^{2},$ (28) $\displaystyle
u_{+2}^{(2)}=H_{5}|\phi_{1}^{(1)}|^{2},$ (29) $\displaystyle
n_{-2}^{(2)}=H_{6}|\phi_{1}^{(1)}|^{2},$ (30) $\displaystyle
u_{-2}^{(2)}=H_{7}|\phi_{1}^{(1)}|^{2},$ (31)
$\displaystyle\phi_{2}^{(2)}=H_{8}|\phi_{1}^{(1)}|^{2},\ $ (32)

where

$\displaystyle H_{4}=\frac{3k^{4}+2H_{8}\omega^{2}k^{2}}{2\omega^{4}},$
$\displaystyle H_{5}=\frac{k^{3}+2\omega^{2}kH_{8}}{2\omega^{3}},$
$\displaystyle
H_{6}=\frac{3\delta_{1}^{2}k^{4}-2\delta_{1}H_{8}\omega^{2}k^{2}}{2\omega^{4}},$
$\displaystyle
H_{7}=\frac{\delta_{1}^{2}k^{3}-2\omega^{2}k\delta_{1}H_{8}}{2\omega^{3}},$
$\displaystyle
H_{8}=\frac{2H_{2}\omega^{4}+3\delta_{1}k^{4}(1-\delta_{2}+\delta_{3})-3k^{4}}{2\omega^{2}k^{2}-2\omega^{4}(4k^{2}+H_{1})+2\delta_{1}\omega^{2}k^{2}(1-\delta_{2}+\delta_{3})}.\
$

Now, we consider the expression for ($m=3$ with $l=0$) and ($m=2$ with $l=0$),
which leads zeroth harmonic modes. Thus, we obtain

$\displaystyle n_{+0}^{(2)}=H_{9}|\phi_{1}^{(1)}|^{2},$ (33) $\displaystyle
u_{+0}^{(2)}=H_{10}|\phi_{1}^{(1)}|^{2},$ (34) $\displaystyle
n_{-0}^{(2)}=H_{11}|\phi_{1}^{(1)}|^{2},$ (35) $\displaystyle
u_{-0}^{(2)}=H_{12}|\phi_{1}^{(1)}|^{2},$ (36)
$\displaystyle\phi_{0}^{(2)}=H_{13}|\phi_{1}^{(1)}|^{2},\ $ (37)

where

$\displaystyle H_{9}=\frac{2v_{g}k^{3}+\omega
k^{2}+\omega^{3}H_{13}}{v_{g}^{2}\omega^{3}},$ $\displaystyle
H_{10}=\frac{k^{2}+\omega^{2}H_{13}}{v_{g}\omega^{2}},$ $\displaystyle
H_{11}=\frac{2v_{g}\delta_{1}^{2}k^{3}+\omega\delta_{1}^{2}k^{2}-\delta_{1}\omega^{3}H_{13}}{v_{g}^{2}\omega^{3}},$
$\displaystyle
H_{12}=\frac{\delta_{1}^{2}k^{2}-\delta_{1}\omega^{2}H_{13}}{v_{g}\omega^{2}},$
$\displaystyle
H_{13}=\frac{2H_{2}v_{g}^{2}\omega^{3}+2v_{g}\delta_{1}^{2}k^{3}(1-\delta_{2}+\delta_{3})+F1}{\omega^{3}[1+\delta_{1}(1-\delta_{2}+\delta_{3})-H_{1}v_{g}^{2}]},\
$

where $F1=\omega
k^{2}\delta_{1}^{2}(1-\delta_{2}+\delta_{3})-2v_{g}k^{3}-\omega k^{2}$.
Finally, the third harmonic modes ($m=3$) and ($l=1$) and with the help of Eq.
(18)-(40), give a set of equations, which can be reduced to the following
NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q\mid\Phi\mid^{2}\Phi=0,$
(38)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. In Eq. (38), $P$ is the dispersion
coefficient which can be written as

$\displaystyle P=\frac{3v_{g}(v_{g}k-\omega)}{2\omega k},\ $

and also $Q$ is the nonlinear coefficient which can be written as

$\displaystyle
Q=\frac{\omega^{3}}{2k^{2}[1+\delta_{1}(1-\delta_{2}+\delta_{2})]}\times\Bigg{[}-\frac{2k^{3}(H_{5}+H_{10})}{\omega^{3}}$
$\displaystyle+2H_{2}(H_{8}+H_{13})+3H_{3}-\frac{k^{2}(H_{4}+H_{9})}{\omega^{2}}$
$\displaystyle-\frac{\delta_{1}k^{2}(1-\delta_{2}+\delta_{2})(H_{6}+H_{11})}{\omega^{2}}$
$\displaystyle-\frac{2\delta_{1}k^{3}(1-\delta_{2}+\delta_{2})(H_{7}+H_{12})}{\omega^{3}}\Bigg{]}.\
$

It may be noted here that both $P$ and $Q$ are function of various plasma
parameters such as $\alpha$, $\delta_{1}$, $\delta_{2}$, $\delta_{3}$,
$\delta_{4}$ and $k$. So, all the plasma parameters are used to maintain the
nonlinearity and the dispersion properties of the PIPM.

##  4 Modulational instability

The stable and unstable regions of the IAWs are organized by the sign of the
dispersion ($P$) and nonlinear ($Q$) coefficients of the standard NLSE (30).
The stability of IAWs in four components PIPM is governed by the sign of $P$
and $Q$ [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. When $P$ and $Q$ have
same sign (i.e., $P/Q>0$), the evolution of the IAWs amplitude is
modulationally unstable in presence of the external perturbations. On the
other hand, when $P$ and $Q$ have opposite sign (i.e., $P/Q<0$), the IAWs are
modulationally stable. The plot of $P/Q$ against $k$ yields stable and
unstable regions for the IAWs. The point, at which transition of $P/Q$ curve
intersects with $k$-axis, is known as threshold or critical wave number $k$
($=k_{c}$) [26, 27, 28, 32, 33, 34, 35, 36].

We have numerically analyzed the behaviour of $P/Q$ with respect to $k$ for
different values of $\delta_{1}$ and $\delta_{3}$ in Figs. 1 and 2,
respectively. It can be seen from Fig. 1 that (a) the modulationally stable
region corresponding to the criteria (i.e., $P/Q<0$) as well as modulationally
unstable region corresponding to the criteria (i.e., $P/Q>0$) can be found
from this figure; (b) the $k_{c}$ decreases with an increase in the value of
$\delta_{1}$; (c) the mass of the negative ions causes to increase the value
of $k_{c}$ while the mass of the positive ions causes to decrease the value of
$k_{c}$ for a constant value of negative and positive ions charge state (via
$\delta_{1}$). Figure 2 demonstrates the presence of the Maxwellian
inertialess positrons reduces the $k_{c}$ as well as allows to generate IARWs
associated with IAWs within the condition $P/Q>0$ in a PIPM for a small value
of $k$ when the number density as well as the charge state of the positive ion
remain constant (via $\delta_{3}$). On the other hand, modulationally stable
region (i.e., $P/Q<0$) as well as $k_{c}$ enhances with increasing the value
of number density and charge state of the positive ion for a constant value of
positron population.

##  5 Rogue waves

The NLSE (38) has a variety of solutions, among them there is a hierarchy of
rational solution that are localized in both the $\xi$ and $\tau$ variables.
Each solution of the hierarchy represents a unique event in space and time, as
it increases its amplitude quickly along each variable, reaching its maximum
value and finally decays, just as quickly as it appeared [37]. Thus, these
waves were nicknamed “waves that appear from nowhere and disappear without a
trace [38]”. The first-order rational solution of NLSE (38) is given as [32,
33, 34, 35, 36, 37, 38, 39, 40, 41]

$\displaystyle\Phi_{1}(\xi,\tau)=\sqrt{\frac{2P}{Q}}\Big{[}\frac{4+16i\tau
P}{1+4\xi^{2}+16\tau^{2}P^{2}}-1\Big{]}\mbox{exp}(2i\tau P).$ (39)

The nonlinear superposition of the two or more first-order RWs gives rise
higher-order RWs and form a higher amplitude more complicated nonlinear
structure. The second-order rational solution is expressed as [32, 33, 34, 35,
36, 37, 38, 39, 40, 41, 39, 38]

$\displaystyle\Phi_{2}(\xi,\tau)=\sqrt{\frac{P}{Q}}\Big{[}1+\frac{G_{2}(\xi,\tau)+iM_{2}(\xi,\tau)}{D_{2}(\xi,\tau)}\Big{]}\mbox{exp}(i\tau
P),$ (40)

where

$\displaystyle
G_{2}(\xi,\tau)=\frac{3}{8}-6(P\xi\tau)^{2}-10(P\tau)^{4}-\frac{3\xi^{2}}{2}-9(P\tau)^{2}-\frac{\xi^{4}}{2},$
$\displaystyle
M_{2}(\xi,\tau)=-P\tau\Big{[}\xi^{4}+4(P\xi\tau)^{2}+4(P\tau)^{4}-3\xi^{2}+2(P\tau)^{2}-\frac{15}{4}\Big{]},$
$\displaystyle
D_{2}(\xi,\tau)=\frac{\xi^{6}}{12}+\frac{\xi^{4}(P\tau)^{2}}{2}+\xi^{2}(P\tau)^{4}+\frac{2(P\tau)^{6}}{3}+\frac{\xi^{4}}{8}$
$\displaystyle+\frac{9(P\tau)^{4}}{2}-\frac{3(P\xi\tau)^{2}}{2}+\frac{9\xi^{2}}{16}+\frac{33(P\tau)^{2}}{8}+\frac{3}{32}.\
$

The solutions (39) and (40) represent the profile of the first-order and
second-order RWs within the modulationally unstable region, which concentrate
a significant amount of energy into a relatively small area.

(a)

(b)

Figure 6: Profile of the (a) first-order rational solution and (b) second-
order rational solution. Figure 7: Plot of first-order (dashed green curve)
and second-order (solid blue curve) rational solutions of NLSE (38) at
$\tau=0$.

The properties of the first-order rational solution [by using Eq. (39)] of
NLSE (38) in a PIPM can be observed in Figs. 3-5 corresponding different
values of $\alpha$, $\delta_{2}$, and $\delta_{4}$. Figure 3 shows the
structure of the first-order IARWs according to the value of electrons non-
thermality ($\alpha$). The increase in the value of $\alpha$ does not only
cause to increase the height of the first-order IARWs associated IAWs in the
modulationally unstable region (i.e., $P/Q>0$) but also causes to increase the
thickness of the first-order IARWs associated IAWs in the modulationally
unstable region (i.e., $P/Q>0$).

The effects of the ratio of the number density of electron to positive ion as
well as the charge state of the positive ion (via $\delta_{2}$) can be seen
from Fig. 4, and it is obvious from this figure that (a) the increase in the
value of $\delta_{4}$ causes to decrease the height of the first-order IARWs
in the modulationally unstable region (i.e., $P/Q>0$) as well as causes to
decrease the thickness of the first-order IARWs in the modulationally unstable
region (i.e., $P/Q>0$); (b) physically, the presence of the excess number of
electrons reduces the nonlinearity of the plasma medium as well as the height
and thickness of the first-order IARWs while the presence of the excess number
of positive ions increases the nonlinearity of the PIPM as well as the height
and thickness of the first-order IARWs when the charge state of the positive
ion remains constant (via $\delta_{2}$).

The nature of the first-order IARWs with the variation of $\delta_{4}$ can be
observed from Fig. 5 which clearly indicates that (a) the height and thickness
of the first-order IARWs associated with IAWs in the modulationally unstable
region (i.e., $P/Q>0$) decreases with increasing the value of $\delta_{4}$;
(b) the temperature of the positron in the modulationally unstable region of
IAWs enhances the nonlinearity of the plasma medium as well as the height and
thickness of the first-order IARWs while the temperature of the electron in
the modulationally unstable region of IAWs reduces the nonlinearity of the
plasma medium as well as the height and thickness of the first-order IARWs.

The space and time evolution of the first-order and second-order rational
solutions of the NLSE (38) can be observed from Figs. 6(a) and 6(b),
respectively. Figure 7 indicates the first-order and second-order solution at
$\tau=0$, and it is clear form this figure that (a) the second-order rational
solution has double structures compared with first-order rational solution;
(b) the height of the second-order rational solution is always greater than
the height of the first-order rational solution; (c) the potential profile of
the second-order rational solution becomes more spiky (i.e., the taller height
and narrower width) than the first-order rational solution; (d) the second
(first) order rational solution has four (two) zeros symmetrically located on
the $\xi$-axis; (e) the second (first) order rational solution has three (one)
local maxima.

The existence of highly energetic rogue waves has already been confirmed
experimentally [42, 43, 44, 45] and theoretically [46]. The second-order
rational solution was experimentally observed by Chabchoub et al. [43] in a
“Water Wave Tank”, and the experimental result regarding the amplification is
a nice agreement with the theoretical result. Bailung et al. [44] demonstrated
an experiment in a multi-component plasma medium to observe RWs, and found a
slowly amplitude modulated perturbation undergoes self modulation and gives
rise to a high amplitude localized pulse. Rogue waves also observed in fiber
optics [45].

##  6 Conclusion

The amplitude modulation of IAWs has been theoretically investigated in an
unmagnetized four components PIPM consisting of inertial positively and
negatively charged ions as well as inertialess non-thermal electrons and iso-
thermal positrons. A NLSE, which governs the MI of IAWs and the formation of
electrostatic first-order and second-order RWs in PIPM, is derived by using
the RPM. To conclude, we hope that our results may be useful in understanding
the nonlinear phenomena (viz., MI of IAWs and IARWs, etc.) in space PIPM
(viz., cometary comae [5, 6, 7] and Earth’s ionosphere [8, 9, 10, 11, 12]) and
laboratory PIPM (viz., plasma processing reactor [15] and neutral beam sources
[16], etc.).

## Acknowledgements

M Hassan is thankful to the Bangladesh Ministry of Science and Technology for
awarding the National Science and Technology (NST) Fellowship. A Mannan thanks
the Alexander von Humboldt Foundation for a Postdoctoral Fellowship.

## References

  * [1] A. J. Coates, F. J. Crary, G. R. Lewis, D. T. Young, J. H. Waite, and E. C. Sittler, Geophys. Res. Lett. 34 (2007) 22103.
  * [2] W. Oohara and R. Hatakeyama, Phys. Rev. Lett. 91 (2003) 205005.
  * [3] W. Oohara, D. Date, and R. Hatakeyama, Phys. Rev. Lett. 95 (2005) 175003.
  * [4] W. Oohara, Y. Kuwabara, and R. Hatakeyama, Phys. Rev. E 75 (2007) 056403.
  * [5] R. Sabry, W. M. Moslem, and P. K. Shukla, Phys. Plasmas 16 (2009) 032302.
  * [6] N. Jannat, M. Ferdousi, and A. A. Mamun, Commun. Theor. Phys. 64 (2015) 479.
  * [7] N. Jannat, M. Ferdousi, and A. A. Mamun, Plasma Phys. Rep. 42 (2016) 678.
  * [8] S. K. El-Labany, R. Sabry, W. F. El-Taibany, and E. A. Elghmaz Astrophys. Space Sci. 340 (2012) 77.
  * [9] H. Massey, Negative Ions, 3rd ed., Cambridge University Press, Cambridge (1976).
  * [10] S. A. El-Tantawy, N. A. El-Bedwehy, and S. K. El-Labany, Phys. Plasmas 20 (2013) 072101.
  * [11] H. G. Abdelwahed, E. K. El-Shewy, M. A. Zahran, and S. A. Elwakil, Phys. Plasmas 23 (2016) 022102.
  * [12] S. A. Elwakil, E. K. El-Shewy, and H. G. Abdelwahed, Phys. Plasmas 17 (2010) 052301.
  * [13] N. A. Chowdhury, A. Mannan, M. M. Hasan, and A. A. Mamun, Chaos 27 (2017) 093105.
  * [14] N. Ahmed, A. Mannan, N. A. Chowdhury, and A. A. Mamun, Chaos 28 (2018) 123107.
  * [15] R. A. Gottscho, C. E. Gaebe, IEEE Trans. Plasma Sci. 14 (1986) 92.
  * [16] M. Bacal, G. W. Hamilton, Phys. Rev. Lett. 42 (1979) 1538.
  * [17] R. A. Cairns, A. A. Mamun, R. Bingham, R. Boström, R. O. Dendy, C. M. C. Nairn, and P. K. Shukla, Geophys. Res. Lett. 22 (1995) 2709.
  * [18] M. H. Rahman, A. Mannan, N. A. Chowdhury, and A. A. Mamun, Phys. Plasmas 25 (2018) 102118.
  * [19] N. A. Chowdhury, A. Mannan, M. R. Hossen, and A. A. Mamun, Contrib. Plasma Phys. 58 (2018) 870.
  * [20] P. O. Dovner, A. I. Eriksson, R. Boström, and B. Holback, Geophys. Res. Lett. 21 (1994) 1827.
  * [21] R. Boström, G. Gustafsson, B. Holback, G. Holmgren, H. Koskinen, and P. Kintner, Phys. Rev. Lett. 61 (1988) 82.
  * [22] A. Paul and A. Bandyopadhyay, Astrophys. Space Sci. 361 (2016) 172.
  * [23] M. M. Selim, Eur. Phys. J. Plus 131 (2016) 93.
  * [24] S. V. Singh and G. S. Lakhina, Commun. Nonlinear. Sci. Numer. Simulat. 23 (2015) 274.
  * [25] M. H. Rahman, N. A. Chowdhury, A. Mannan, M. Rahman, and A. A. Mamun, Chin. J. Phys. 56 (2018) 2061.
  * [26] I. Kourakis and P. K. Shukla, Phys. Plasmas 10 (2003) 3459.
  * [27] I. Kourakis and P. K. Shukla, Nonlinear Proc. Geophys. 12 (2003) 407.
  * [28] R. Fedele, Phys. Scr. 65 (2002) 502.
  * [29] N. A. Chowdhury, A. Mannan, and A. A. Mamun, Phys. plasmas 24 (2017) 113701.
  * [30] S. Jahan, N. A. Chowdhury, A. Mannan, and A. A. Mamun, Commun. Theor. Phys. 71 (2019) 327.
  * [31] N. A. Chowdhury, A. Mannan, M. M. Hasan, and A. A. Mamun, Vacuum 147 (2018) 31.
  * [32] S. Guo, L. Mei, A. Sun, Ann. Phys. 332 (2013) 38.
  * [33] S. Guo, L. Mei, and W. Shi, Phys. Lett. A 377 (2013) 2118.
  * [34] S. Guo, L. Mei, Phys. Plasmas 21 (2014) 112303.
  * [35] S. K. El-Labany, W. F. El-Taibany, and N. A. Zedan, Phys. Plasmas 22 (2015) 073702.
  * [36] S. A. El-Tantawy, N. A. El-Bedwehy, and S. K. El-Labany, Phys. Plasmas 20 (2019) 072102.
  * [37] A. Ankiewicz, P. A. Clarkson, and N. Akhmediev, J. Phys. A 43 (2010) 12002.
  * [38] A. Ankiewicz, N. Devine, and N. Akhmediev, Phys. Lett. A 373 (2009) 3997.
  * [39] N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601.
  * [40] N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A 373 (2009) 675.
  * [41] Z. Yan, Phys. Lett. A 374 (2010) 672.
  * [42] A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Phys. Rev. Lett. 106 (2011) 204502.
  * [43] A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev, Phys. Rev. X 2(2012) 011015 .
  * [44] H. Bailung, S. K. Sharma, and Y. Nakamura, Phys. Rev. Lett. 107 (2011) 255005.
  * [45] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, Nat. Phys. 6 (2010) 790.
  * [46] Shalini, N. S. Saini, and A. P. Misra, Phys. Plasmas 22 (2015) 092124.


# Tunneling of Hawking radiation for BTZ black hole revisited

Bijan Bagchi
e-mail<EMAIL_ADDRESS>
Sauvik Sen
e-mail<EMAIL_ADDRESS>Department of Physics, School of Natural
Sciences, Shiv Nadar University,
Gautam Buddha Nagar, Uttar Pradesh 201314, India

(November 2021)

We re-examine Hawking radiation for a nonrotating (2+1)-dimensional BTZ black
hole and evaluate the transmission probability of tunneling through the
barrier of the event horizon employing the standard method of WKB
approximation. Our results are presented for both uncharged and charged cases.
We also explore the associated thermodynamics in terms of Hawking temperature
and provide estimates of black hole parameters like the surface gravity and
entropy.

Keywords: BTZ black hole, Hawking radiation, WKB approximation, tunneling,
thermodynamics.

##  I Introduction

In the theory of general relativity (GTR) the black hole solutions are
generally described by four types of metrics. The simplest one is of course
the Schwarzschild metric with no electic charge and where there is spherical
symmetry. To deal with the electric charge, Reissner–Nordström metric is the
relevant one that addresses the geometry of empty space surrounding a charged,
nonrotating black hole. Kerr metric gives a generalization of the
Schwarzschild metric wherein spin effects are taken into account. It was shown
much earlier by Newman et al. [1, 2] that the Kerr solution is derivable from
the Schwarzschild solution for the vacuum Einstein case and further Kerr-
Newman solution follows from the Reissner–Nordström metric for the Einstein-
Maxwell theory through a set of complex-coordinate transformations in terms of
a null tetrad of basis vectors. In a wider perspective, Kerr-Newman metric
deals with both charge and spin of the black hole (see, for a recent review,
[3]).

To consolidate quantum gravity with other forces of nature one has to take
into account extremely strong gravitational effects. A black hole is a natural
place to look for them. Interest in (2+1)-dimensional gravity, which serves as
a toy model for understanding how gravity operates in higher dimensions, owes
to the foundational work of Bañados et al. (BTZ) who showed that such a
simplified formulation had an embedded black hole solution [4]. In contrast to
Schwarzschild and Kerr types of black holes, the BTZ is asymptotically anti-de
Sitter (AdS) and not being asymptotically flat. The metric shows no curvature
singularity at the origin. The static BTZ geometry is a solution of the
Einstein equations with a fixed value of the negative cosmological constant
$-1/l^{2}$ [5, 6]. In [4] was included the cosmological constant possessing a
negative sign in the formalism of three-dimensional vacuum Einstein theory.

For detailed analyses of general types of BTZ black holes having charge and
spin see [9, 7, 10, 8]. Another interesting aspect of the BTZ black hole
solution is that it reveals thermodynamical properties akin to the
Schwardschild’s black hole of (3+1)-dimensions. See [11] for an early review.
BTZ black hole has been studied in the context of quantum gravity and Hawking
temperature allowing for the presence of nonzero spacetime noncommutativity
[12]. For further related works see [13, 14].

In recent times, the construction of AdS charged BTZ black hole was carried
out and a class of plausible solutions were explored in [15]. The dynamical
and thermodynamical behaviour were investigated and Hawking radiation [16] was
estimated by Parikh and Wilczek [17] in the background of conventional quantum
mechanical tunneling (see also [18, 19, 20]). It needs to be emphasized that
the study of Hawking radiation in the context of BTZ black hole and null-
geodesic formalism within the scenario of [17] was also taken up by Wu and
Jiang [21] for the rotational uncharged BTZ black hole. An idea to measure
Hawking-like features and the Unruh effect in the laboratory was proposed in
[22]. Furthermore, the effect of generalized uncertainty principle was
inquired to assess Hawking radiation in (2+1)-dimensions [23]. In addition,
the effect of quantum gravity on the stability aspects of black holes was
discussed in [24]. The BTZ black hole was also analyzed for massive gravity in
a more general context [25] to ascertain Hawking radiation and provide an
estimate of tunneling probability. The particular work of Kim [26] is
especially interesting in that he studied viable black hole solutions in the
context of its global structure [27]. In a similar spirit he attempted a
derivation as in [1, 2] but concentrated only for the three-dimensional
spacetime. His analysis rested on the interpretation that the three-
dimensional geometry is just a $\theta=\frac{\pi}{2}$ cut of the four-
dimensional spacetime with $\theta$ representing the polar angle. In
particular, his nonrotating result correctly resembled the BTZ solution.

One of the purposes of this work is to use the BTZ metric to reassess the
transmission probability of radiation for nonrotating black holes in
(2+1)-dimensions corresponding to the uncharged and charged cases. We remark
that apart from [17], study of tunneling was also taken up for the dilatonic
black hole in [28]. However, as will be clear from the steps provided in the
following section, the approach and analysis of the present work is quite
different. Noting that the picture for the (2+1)-dimension is markedly
different from the (3+1)-case in that as per our study the temperature is
found to be proportional to the square root of mass rather than inverse of it
for both uncharged and charged black holes [11], we demonstrate that the
tunneling probability for the charged black holes emerges smaller than the
uncharged one. Interestingly, our finding is in tune with what is obtained in
[12] by including noncommutative effects when rotational effect is ignored.

In Section II, we set up the metric by transforming the Painlevé time to
Schwarzschild time with the help of a suitable shift involving an arbitrary
function of the radial coordinate. This facilitates arriving at the
generalized structure of the metric for the (2+1)-dimensional black hole. It
is then exploited to address the issue of the tunneling probability of Hawking
radiation for an uncharged black hole employing the WKB approximation and
comparing the result with the existing estimates made from different
perspectives. In section III, we consider in a similar way the case of the
charged black hole. In Section IV, we compute the essential thermodynamic
quantities [29] like the surface gravity and entropy of a black hole. Finally,
in section V, we present a short summary.

##  II Tunneling for uncharged nonrotating BTZ black hole

We begin by representing the BTZ metric for a nonrotating uncharged black hole
of mass M in spherical polar coordinates by

$ds^{2}=Z(r)dt^{2}-Z^{-1}(r)dr^{2}-r^{2}d\phi^{2}$ (2.1)

$Z(r)=-M+\displaystyle\frac{r^{2}}{l^{2}}$ (2.2)

Throughout this work we will employ natural units. In (2.1) $t$ is the
Painlevé time and we have followed Kim’s [26] treatment by picking the slice
$\theta=\pi/2$. As noted by Pantoja et al [30], the metric (2.1) is a solution
of the vacuum Einstein field equations taking the cosmological constant value
to be $\Lambda=-\frac{1}{l^{2}}$. As elaborated there, corresponding to a
strictly positive mass value, the horizon of the black hole is at the point
$r=\sqrt{M}l$, for a negative mass value, the black hole is non-existent while
the interval $-1<M<0$ is linked to the metric generated by a point source at
the origin. The vacuum state corresponds to $M=0$. In the present problem,
because of $M>0$, the positive values of the outer $(r_{out})$ and inner
horizon $(r_{in})$ are automatically obeyed.

Without loss of generality, let us transform to the Schwarzschild time
($t_{Sch}$) by making the shift

$t=t_{Sch}-\xi(r)$ (2.3)

where $\xi(r)$ is an arbitrary function of r and the prime denotes the
derivative with respect to r. Accordingly, $dt^{2}$ becomes

$dt^{2}=dt_{Sch}^{2}+\xi^{\prime 2}(r)dr^{2}-2\xi^{\prime}(r)drdt_{Sch}$ (2.4)

Substituting (2.4) in (2.1) we derive the following form for $ds^{2}$

$\begin{split}ds^{2}=Z(r)dt^{2}_{Sch}+[Z(r)\xi^{\prime
2}(r)-Z^{-1}(r)]dr^{2}-2Z(r)\xi^{\prime}(r)drdt_{Sch}-r^{2}d\phi^{2}\end{split}$
(2.5)

Since $\xi(r)$ is arbitrary we choose it to render the coefficient of $dr^{2}$
unity. Thus we are led to the following pair of relations

$\begin{split}Z(r)\xi^{\prime}(r)^{2}-Z^{-1}(r)=1\hskip
5.69054pt\longrightarrow\hskip
5.69054pt\xi^{\prime}(r)=\pm\sqrt{Z^{-1}(r)+Z^{-2}(r)}\end{split}$ (2.6)

We can therefore recast (2.5) in the form

$\begin{split}ds^{2}=Z(r)dt^{2}_{Sch}+dr^{2}-r^{2}d\phi^{2}-2Z(r)\sqrt{Z^{-1}(r)+Z^{-2}(r)}drdt_{Sch}\end{split}$
(2.7)

The radial null geodesic will obey

$\begin{split}Z(r)dt_{Sch}^{2}+dr^{2}-2Z(r)\sqrt{Z^{-1}(r)+Z^{-2}(r)}drdt_{Sch}=0\end{split}$
(2.8)

and hence we have

$\begin{split}dt_{Sch}^{2}\Big{(}Z(r)+\big{(}\displaystyle\frac{dr}{dt_{Sch}}\big{)}^{2}-2Z(r)\sqrt{Z^{-1}(r)+Z^{-2}(r)}\big{(}\displaystyle\frac{dr}{dt_{Sch}}\big{)}\Big{)}=0\end{split}$
(2.9)

(2.9) can be rewritten in the form

$\displaystyle\dot{r}^{2}-2\dot{r}Z(r)\sqrt{Z^{-1}(r)+Z^{-2}(r)}+Z(r)=0\hskip
5.69054pt\longrightarrow\hskip 5.69054pt\dot{r}=\sqrt{1+Z(r)}\pm 1$ (2.10)

where the overdot refers to the derivative with respect to the Schwarzschild
time. The negative sign refers to an incoming geodesic while the positive sign
stands for an outgoing geodesic. It is assumed that the pair production occurs
just inside the event horizon at $r_{in}\approx r_{+}$.

A couple of points are in order. If $\Omega$ is the energy of the particle
that is created during the pair production then the mass of the black hole
after emission would be $M-\Omega(>0)$. Since $r_{in}$ is the radius of the
event horizon when the particle is produced just inside it, while $r_{out}$ is
the radius of the event horizon when the particle has tunneled across and
emitted out of the black hole, these are determined from solutions of $Z(r)=0$

$\displaystyle\begin{split}r_{in}&=l\sqrt{M}\\\
r_{out}&=l\sqrt{M-\Omega}\end{split}$ (2.11)

where $r_{out}<r_{in}$, with $r_{in}$ in conformity with the constraint for a
positive mass mentioned earlier. With this background we proceed to calculate
the tunneling probability of Hawking radiation.

The region between $r_{in}$ and $r_{out}$ separating the two points acts as a
potential barrier for the tunneling particle to overcome. For the particle
produced during pair production, its energy would be lower than the acting
barrier thus mimicking a classically forbidden region. In this region the
action $\zeta$ is imaginary and we can profitably use the WKB approximation,
as a semiclassical way, to estimate the tunneling probability [17, 25]. See
the works in [31, 32, 33, 34, 35, 36] where the issue of tunneling probability
has also been addressed. Indeed writing for the imaginary part of the action

$\mbox{Im}\zeta=\mbox{Im}\int_{r_{in}}^{r_{out}}p_{r}dr=\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}dp_{r}dr$
(2.12)

and noting from Hamilton’s equation the relationship

$dp_{r}=\displaystyle\frac{dH}{\dot{r}}$ (2.13)

where $H$ takes the values $M$ and $M-\Omega$ corresponding to $p_{r}=0$ and
$p_{r}=p_{r}$ respectively, we project the integral (2.12) to read

$\mbox{Im}\zeta=\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{M}^{M-\Omega}\displaystyle\frac{dH}{\dot{r}}dr$
(2.14)

Then, substituting the value of $\dot{r}$ from (2.10) and concentrating on the
outgoing geodesic we are led to

$\mbox{Im}\zeta=\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{M}^{M-\Omega}\displaystyle\frac{dH}{1+\sqrt{1+Z(r)}}dr$
(2.15)

Let $\mathcal{M}_{in}(=M)$ and $\mathcal{M}_{out}(=M-\Omega)$ be the
respective mass of the black hole before and after the pair production. Since
the emitted particle has a very negligible mass, we can approximate

$\displaystyle\begin{split}\mathcal{M}&=\mathcal{M}_{\mbox{in}}\approx\mathcal{M}_{\mbox{out}}\\\
d\mathcal{M}&\approx-d\Omega\end{split}$ (2.16)

The integral limits in (2.15) translate to $\Omega=0$ when $\mathcal{M}=M$ and
$\Omega=\Omega$ when $\mathcal{M}=M-\Omega$ with $dH$ replaced by $-d\Omega$.
As such, the expression (2.15) becomes

$\mbox{Im}\zeta=-\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{\Omega}\displaystyle\frac{d\Omega}{1+\sqrt{1+Z^{(}r)}]}dr$
(2.17)

From the consideration that the emitted particle has a very small energy of
the order of $\Omega$, binomial expansion in the vicinity of the event
horizon, where the quantity $Z$ is supposed to be small, gives a simplified
representation

$\mbox{Im}\zeta=-\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{\Omega}\displaystyle\frac{2d\Omega}{4+Z(r)}dr$
(2.18)

to first order in $Z$. Replacing $M$ by $M-\Omega$ and substituting (2.2) in
(2.18) we obtain

$\mbox{Im}\zeta=-\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{\Omega}\displaystyle\frac{2d\Omega}{4+\left[-M+\Omega+\displaystyle\frac{r^{2}}{l^{2}}\right]}dr$
(2.19)

To tackle the above integral we substitute
$-M+\Omega+\displaystyle\frac{r^{2}}{l^{2}}=-4\alpha$ implying
$d\Omega=-4d\alpha$. Further, the limits $\Omega=0\rightarrow\alpha=\alpha(0)$
and $\Omega=\Omega\rightarrow\alpha=\alpha(\Omega)$ also hold. As a
consequence, $\mbox{Im}\zeta$ is convertible to the form

$\mbox{Im}\zeta=2\hskip
2.84526pt\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{\alpha(0)}^{\alpha(\Omega)}\displaystyle\frac{d\alpha}{1-\alpha}dr$
(2.20)

We now use the residue theorem of complex analysis to estimate the integral
(2.20). Noticing that there is a simple pole at $\alpha=1$ and that the
residue at this point is $-1$ it follows that

$\mbox{Im}\zeta=-4\pi(r_{out}-r_{in})=4\pi(r_{in}-r_{out})$ (2.21)

This gives straightforwardly the following result for the tunneling
probability

$\Gamma(\Omega)\approx\exp(-2\mbox{Im}\zeta)=\exp(-8\pi\rho)$ (2.22)

where $\rho=r_{in}-r_{out}=\displaystyle\frac{\Omega
l}{2\sqrt{M}}+\mathcal{O}(\Omega^{2})$. A comparison of the right side of
(2.21) with the estimate made in [21] for the rotational uncharged BTZ black
hole shows that ours is a factor of 2 too large. More precisely, while (2.22)
shows the dependence of $\exp(-\frac{1}{\sqrt{M}})$, the $(3+1)$-dimensional
calculation of Parikh and Wilczek [17] depicts a slightly different behaviour
of $\exp(-M)$. A similar form like (2.21) was obtained in the massive gravity
case [25] but the difference of $r_{in}$ and $r_{out}$ depended on a
complicated function of the mass. It is interesting to note that if rotational
contributions are discarded then the noncommutative result of [12] matches
with ours including the dependence of $\Gamma$ on $\exp(-\frac{1}{\sqrt{M}})$.

##  III Tunneling for charged nonrotating BTZ black hole

The charged counterpart of the nonrotating BTZ metric [8, 9, 29, 10] can be
written as:

$ds^{2}=Y(r)dt^{2}-Y^{-1}(r)dr^{2}-r^{2}d\phi^{2}$ (3.1)

$Y(r)=-M+\displaystyle\frac{r^{2}}{l^{2}}-\displaystyle\frac{\pi}{2}Q^{2}\mbox{ln}r$
(3.2)

We follow a similar prescription as in the previous section to calculate the
tunneling probability of this black hole. The radius of the event horizon
before and after emission are given by .

$\begin{split}r_{in}=\displaystyle\frac{\pi
Q^{2}l^{2}}{4}+\displaystyle\frac{1}{4}\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}M}\qquad\quad\\\ r_{out}=\displaystyle\frac{\pi
Q^{2}l^{2}}{4}+\displaystyle\frac{1}{4}\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}(M-\Omega)}\end{split}$ (3.3)

This is obtained by setting $Y(r)=0$ similar to the vanishing criterion of
$Z(r)$ undertaken in the previous section to determine the corresponding
$r_{in}$ and $r_{out}$.
The imaginary action can then be expressed as:

$\mbox{Im}\zeta=-\mbox{Im}\int_{r_{in}}^{r_{out}}\int_{0}^{\Omega}\displaystyle\frac{2d\Omega}{4+[-(M-\Omega)+\displaystyle\frac{r^{2}}{l^{2}}-\displaystyle\frac{\pi}{2}Q^{2}\mbox{ln}r]}dr$
(3.4)

After suitable substitution and using the residue theorem to solve the above
integral, the final tunneling probability for the charged BTZ black hole is
determined to be

$\Gamma(\Omega)\approx\exp(-8\pi\sigma)$ (3.5)

where

$\sigma=r_{in}-r_{out}=\displaystyle\frac{2l^{2}\Omega}{\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}M}}+\mathcal{O}(\Omega^{2})$ (3.6)

and $\pi^{2}Q^{4}l^{4}+16Ml^{2}>8\pi Q^{2}l^{2}$. Comparing with the result
(2.22) we see that the tunneling probability for the charged nonrotating black
hole is smaller than the uncharged one. Further, we remark that the dependence
on the mass M of the argument of the above exponential is different from the
$(3+1)$-dimensional case [17], where it was found that for a black hole with
charge Q the transmission probability went like
$\approx\exp\left[-4\pi(2\Omega(M-\frac{\Omega}{2}))-(M-\Omega)\sqrt{(M-\Omega)^{2}-Q^{2}}+M\sqrt{M^{2}-Q^{2}}\right]$.

##  IV Thermodynamical aspects

The influential works of Bekenstein, Hawking, Bardeen et al [37, 16, 38] in
the 1970s paved the way for active research in black hole thermodynamics. In
particular, Bekenstein [37] postulated that the area of the event horizon was
proportional to its entropy. Further works by Bardeen et al [38] laid the
foundations of the four laws of black hole thermodynamics by drawing an
analogy from the conventional ones of thermodynamics.

In the present context we set up our results against the standard ones of [39,
40, 41]. According to Hawking’s theory [16] the so-called information paradox
could be resolved by the emission of particles from a black hole due to
quantum pair-production near the event horizon. One denotes by $T_{H}$ as
characteristic temperature when the radiation occurs. The value of $\rho$
obtained for the uncharged black hole when substituted in (2.22) gives the
Boltzmann factor $\exp(-{\frac{\mathcal{E}}{T_{H}}})$ where
$\mathcal{E}=\Omega$ is the energy of the emitted particle. Similarly for the
charged case we employ the estimate of $\sigma$ made in (3.6). This provides
us with the following expression of Hawking temperature ($\mathcal{T}$) of the
(2+1)-dimensional uncharged (uc) and charged (c) black hole:

$\mathcal{T}^{uc}=\displaystyle\frac{\sqrt{M}}{4\pi l}$ (4.1)
$\mathcal{T}^{c}=\displaystyle\frac{\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}M}}{16\pi l^{2}}$ (4.2)

Since the surface gravity $\kappa$ in both cases is $2\pi\mathcal{T}$ we at
once deduce straightforwardly

$\displaystyle\kappa^{uc}=\frac{\sqrt{M}}{2l}$ (4.3)
$\displaystyle\kappa^{c}=\displaystyle\frac{\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}M}}{8l^{2}}$ (4.4)

We know from Bekenstein’s work [37] that the area of a black hole event
horizon is proportional to its entropy. This means that just like entropy can
never decrease, the area of the event horizon of a black hole can only
increase. In (3+1)-dimensions the surface of a black hole is the area of a
sphere ($4\pi r^{2}$). In $(2+1)$ dimensions the analogue can be identified
with its perimeter $A=2\pi r_{out}$. A simple derivation yields for the
respective areas of uncharged and charged black holes

$\displaystyle A^{uc}=2\pi l\sqrt{M-\Omega}$ (4.5) $\displaystyle
A^{c}=\displaystyle\frac{\pi^{2}Q^{2}l^{2}}{2}+\displaystyle\frac{\pi}{2}\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}(M-\Omega)}$ (4.6)

These relations imply for the entropy relations

$\displaystyle S^{uc}=\displaystyle\frac{A}{4}=\displaystyle\frac{\pi
l}{2}\sqrt{M-\Omega}$ (4.7) $\displaystyle
S^{c}=\displaystyle\frac{A}{4}=\displaystyle\frac{\pi^{2}Q^{2}l^{2}}{8}+\displaystyle\frac{\pi}{8}\sqrt{\pi^{2}Q^{4}l^{4}-8\pi
Q^{2}l^{2}+16l^{2}(M-\Omega)}$ (4.8)

Note that the functional forms of $S^{c}$ and $S^{uc}$ are different.
Evidently, because of the influence of the charge $Q$, $S^{c}$ is higher than
$S^{uc}$.

##  V Summary

In this paper we introduced a set of complex coordinate transformations to
reinterpret the BTZ metric in terms of Painlevé coordinates. Thus the basic
equations were set up that give conditions for the incoming and outgoing
geodesics. Following Parikh and Wilczek’s approach, we then evaluated the
transmission probability of Hawking radiation for both uncharged and charged
nonrotating black holes in the framework of WKB approximation. The related
issue of black hole thermodynamics was also discussed and estimates of typical
thermodynamical parameters, like the surface gravity and entropy, were made.

##  VI Acknowledgment

One of us (SS) thanks Ricardo Troncoso and Tajron Jurić for enlightening
correspondences. He also acknowledges support from Shiv Nadar University for a
research fellowship. Both of us are indebted to the anonymous reviewer for
carefully going through our manuscript and advising a number of constructive
changes.

## References

  * [1] E. T. Newman and A. I. Janis, J. Math. Phys. 6, 915 (1965).
  * [2] E. T. Newman, E. Couch, R. Chinnapared, A. Exton, A. Prakash, and R. Torrence, J. Math. Phys. 6, 918 (1965).
  * [3] T. Adamo and E. T. Newman, Scholarpedia 9(10), 31791 (2014).
  * [4] M. Bañados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849 (1992).
  * [5] M. Bravo-Gaete and M. Hassaine, Phys. Rev. D 90, 024008 (2014).
  * [6] N. R. Pantoja, H. Rago and R. O. Rodriguez, J. Math. Phys. 45, 1994 (2004).
  * [7] G. Clément, Phys. Lett. B. 367, 70 (1996).
  * [8] J. Sadeghi and V. R. Shajiee, Eur. Phys. J. Plus 132, 1-5 (2017).
  * [9] C. Martinez, C. Teitelboim, and J. Zanelli, Phys. Rev. D 61, 104013 (2000).
  * [10] M. Akbar et al, Phys. Rev. D 83, 084031 (2011).
  * [11] S. Carlip, Class. Quant. Grav. 12, 2853 (1995).
  * [12] K. S. Gupta, T. Jurić and A. Samsarov, J. High Eng. Phys. 2017, 107 (2017).
  * [13] K. S. Gupta, E. Harikumar, T. Jurić, S. Meljanac and A. Samsarov, Adv. High Eng. Phys. 2014, 139172 (2014).
  * [14] T. Jurić and A. Samsarov, Phys. Rev. D 93, 104033 (2016).
  * [15] S. H. Hendi, B.E. Panah, and S. Panahiyan, J. High Energy Phys. 2016, 029 (2016).
  * [16] S. W. Hawking, Commun. Math. Phys. 43, 220 (1975); Nature 248, 5443 (1974).
  * [17] M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000).
  * [18] H. Kim, Phys. Lett. B 703, 94 (2011).
  * [19] A. Yale, Phys. Lett. B 697, 398 (2011).
  * [20] K. Nozari and S. H. Mehdipour, Class. Quantum Grav. 25, 175015 (2008).
  * [21] S-Q Wu and Q-Q Jiang, J. High Eng. Phys. 03, 079 (2006).
  * [22] H. C. Rosu, Int. J. Mod. Phys. D03, 543 (1994).
  * [23] R. Babar, W. Javed and A. Övgün, Mod. Phys. Lett. 35, 2050104 (2020).
  * [24] R. Ali, K. Bamba and S. A. A. Shah, Symmetry 11, 631 (2019).
  * [25] S. Chougule, S. Dey, B. Pourhassan, and M. Faizal, Eur. Phys. J. C 78, 685 (2018).
  * [26] H. Kim, Phys. Rev. D 59, 064002 (1999).
  * [27] S. W. Hawking, C. J. Hunter, and M. M. Taylor-Robinson, Phys. Rev. D 59, 064005 (1999).
  * [28] Y. Hu, J. Zhang and Z. Zhao, Int. J. Mod. Phys. 16, 847 (2007).
  * [29] P. Prasia and V. C. Kuriakose, Eur. Phys. J. 77C, 27 (2017).
  * [30] N. R. Pantoja, H. Rago and R. O. Rodríguez, J. Math. Phys. 45, 1994 (2004).
  * [31] Z. W. Feng, H. L. Li, X. T. Zu, S. Z. Yang, European Physical Journal C76, 212 (2016).
  * [32] Z. W. Feng, Q. C. Ding, S. Z. Yang, European Physical Journal C 79, 445 (2019).
  * [33] Z. W. Feng, X. Zhou, S. Q. Zhou, D. D. Feng, Annals of Physics, 416, 168144 (2020).
  * [34] E. E Flanagan, Phys. Rev. Lett. 127, 041301.
  * [35] R. Li, J. Wang, Phys. Rev. D 104, 026011 (2021).
  * [36] C. H. Fleming, Hawking radiation as tunneling, http://www.physics.umd.edu/grt/taj/776b/fleming.pdf.
  * [37] J. D. Bekenstein Nuovo Cim. Lett. 4, 737-740 (1972); Phys. Rev. D 7, 2333-2346 (1973); Phys. Rev. D 9, 3292 (1974).
  * [38] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31, 161-170 (1973).
  * [39] S. Carlip, Phys. Rev. D 51, 632 (1995).
  * [40] D. Kubizňák, R. B. Mann, and M. Teo, Class. Quantum Grav. 34, 063001 (2017).
  * [41] A. M. Frassino, R. B. Mann, and J. R. Mureika, Phys. Rev. D 92, 124069 (2015).