# Dust-acoustic envelope solitons and rogue waves in an electron depleted
plasma

J. Akter∗,1, N. A. Chowdhury∗∗,1, and A. A. Mamun1,2 1Department of Physics,
Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
2Wazed miah science research centre, Jahangirnagar University, Savar,
Dhaka-1342, Bangladesh
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Theoretical investigation of the nonlinear propagation and modulational
instability (MI) of the dust acoustic (DA) waves (DAWs) in an unmagnetized
electron depleted dusty plasma (containing opposite polarity warm dust grains
and non-extensive positive ions) has been made by deriving a nonlinear
Schrödinger equation with the help of perturbation method. Two types of mode,
namely, fast and slow DA modes, have been found. The criteria for the
formation of bright and dark envelope solitons as well as the first-order and
second-order rogue waves have been observed. The effects of various dusty
plasma parameters (viz., dust mass, dust charge, dust and ion number
densities, etc.) on the MI of DAWs have been identified. It is found that
these dusty plasma parameters significantly modify the basic features of the
DAWs. The applications of the results obtained from this theoretical
investigation in different regions of space, viz., magnetosphere of Jupiter,
upper mesosphere, Saturn’s F-ring, and cometary tail, etc.

###### keywords:

Electron depletion , NLSE , modulational instability , envelope solitons ,
rogue waves.

††journal: “Physics”

##  1 Introduction

The existence of highly charged massive dust grains does not only occurs in
astrophysical environments, viz., supernova explosion [1], Martian atmosphere
[2], cometary tail [4, 3, 2], circumstellar clouds [5], interstellar clouds
[5], solar system [5], upper mesosphere [6, 4], magnetosphere of Jupiter [6],
and Saturn’s F-ring [7, 8] but also in many laboratory experiments, viz.,
laser-matter interaction [9], Q-machine [10], ac-discharge [10], rf-discharges
[10], fusion devices [10], etc. The investigation of nonlinear electrostatic
pulses, viz., dust-acoustic (DA) double layers (DADLs) [3, 8], DA shock waves
(DASHWs) [11, 5], DA solitary waves (DASWs) [3, 2], DA rogue waves (DARWs),
and DA Gardner solitons (DAGSs) [12] associated with DA Waves (DAWs) [13] in
multi-component plasma medium (MCPM) within theoretical framework [14, 15, 16]
is one of the most interesting topics among the plasma physicists in twentieth
century.

Non-extensive particles can be found in non-equilibrium complex systems due to
the existence of external force fields (viz., long range interactions,
gravitational forces, and Coulomb electric forces, etc.), and are also
governed by the non-extensive $q$-distribution which first introduced by Renyi
[17] and subsequently improved by Tsallis [18], and have been identified by
the GEOTAIL [19, 20]. It may be noted that the non-extensive parameter ($q$)
describes the characteristics of the non-extensive $q$-distribution [21]. A
number of authors have considered non-extensive electrons [4, 6, 7, 15, 21,
22] for interpreting the characteristics of the nonlinear electrostatic waves
in space environments, viz., cometary tail [4], upper mesosphere [4],
magnetosphere of Jupiter [6], Saturn’s F-ring [7, 21], ionosphere [15], lower
part of magnetosphere [15], and solar wind [22], etc. Emamuddin et al. [12]
considered MCPM, and studied DAGSs in presence of non-extensive plasma
particles, and found that DAGSs exhibit positive and negative solitons
according to the critical value of $q$. Roy et al. [23] examined the DASHWs in
non-extensive plasmas, and obtained that the positive DASHWs height decreases
as $q$. Bacha and Tribeche [24] demonstrated the amplitude of the pulse
increases while the width of the pulse decreases with increasing $q$ in a
three components dusty plasma.

In many space and laboratory dusty plasma situations, most of the background
electrons could stick onto the surface of dust grains during the charging
processes and as a result one might encounter a significant depletion of the
electron number density in the ambient dusty plasma [8, 25, 26]. This scenario
is relevant to a number of space dusty plasma systems, for example, planetary
rings (particularly, Saturn’s F-ring), and laboratory experiments. It should
be noted here that a complete depletion of the electrons is not possible
because the minimum value of the ratio between the electron and ion number
densities turns out to be the square root of the electron to ion mass ratio
when electron and ion temperatures are approximately equal and the grain
surface potential approaches zero [8, 25, 26]. A number of authors have
studied the propagation of the nonlinear electrostatic waves in electron
depleted dusty plasma (EDDP) [25, 26, 27, 28, 29, 30, 31, 32]. Borhanian and
Shahmansouri [26] studied three dimensional DASWs in an EDDP with two
temperature super-thermal ions, and highlighted that the super-thermality of
ions leads to increase the amplitude of the solitary waves. Tantawy and Moslem
[25] examined DASWs and DASHWs in a MCPM in presence of positive and negative
ions, and observed that the number density of the positive and negative ions
as well as dust grains are significantly modified the amplitude of DASWs and
DASHWs. Shukla and Silin [31] investigated the features of the low-frequency
dust-ion-acoustic waves in an unmagnetized EDDP medium. Tagare [32] examined
DASWs and DADLs in three components EDDP medium having cold dust grains and
two temperature Maxwellian ions, and observed that DASWs exist in a particular
region where DADLs do not exist. Ferdousi et al. [14] analyzed DASHWs in
multi-component EDDP medium having non-thermal plasma species. Sahu and
Tribeche et al. [8] reported small amplitude DADLs in a two components EDDP
with ions featuring non-extensive distribution, and observed that the non-
extensivity of the ions admits compressive as well as rarefactive DADLs in
EDDPs. Hossen et al. [3] considered three components DP medium containing
mobile opposite polarity dust grains (OPDGs) and non-extensive ions to study
DASWs, and found that the height of the positive solitary potential increases
with positive ion number density while the negative solitary potential
decreases.

The nonlinear Schrödinger equation (NLSE) is arguably the most far-reaching
and beautiful physical equation ever constructed [33, 34, 35, 36, 37, 38, 39,
40]. It describes not only the stability of DAWs but also the nature of the
energy localization and re-distribution in various plasma medium. Envelope
solitonic solutions, which precisely describe the energy localization and re-
distribution, is one of the mysterious solution of the NLSE. On the other
hand, the ambiguous appearance of the rogue waves (RWs) is also an interesting
nonlinear phenomena in inter-disciplinary science (viz., nonlinear fiber
optics, parametrically driven capillary waves, super-fluids, optical cavities,
plasmonics [44], hydrodynamics [45], biology [46], Bose-Einstein condensates,
ocean waves, and stock market [51], etc.). The RWs are investigated in a MCPM
and have been experimentally observed and also modeled by using the NLSE [47,
48, 49, 50, 51]. Zaghbeer et al. [4] investigated the MI of the DAWS in a four
components plasma medium with non-extensive electrons and ions, and observed
that the variation of the non-extensivity of electron and ion would lead to
increase the amplitude of DARWs. Eghbali et al. [15] numerically analyzed the
criterion of the MI of DIAWs in presence of non-extensive electrons in non-
planar geometry, and found that the value of maximum growth rate decreases
with increasing $q$. Moslem et al. [40] demonstrated three components plasma
medium having non-extensive electrons and ions as well as massive negatively
charged dust grains to study the stability of the DAWs, and reported that for
negative $q$, the $k_{c}$ increases with $q$. Jahan et al. [6] considered an
OPDGs to investigate the MI of the DAWs, and found that the critical wave
number ($k_{c}$) decreases with increasing the negative dust charge state when
other parameters remain constant. Chowdhury et al. [38] studied the amplitude
modulation of the nucleus-acoustic waves, and observed that the variation of
the plasma parameters causes to change the thickness but not the height of the
the envelope pulses.

Recently, Hossen et al. [5] investigated the DASWs in a three components EDDP
having inertial an OPDGs and inertialess $q$-distributed ions. Ferdousi et al.
[11] studied DASHWs in a two components EDDPs in presence of non-extensive
ions, and observed that both polarities of DASHWs can exist according to the
value of $q$. This is all very nice and certainly deserves further
examination, the purpose of this article is to extend Ferdousi et al. [11]
work by deriving a NLSE and also investigate the MI of DAWs and associated DA
electrostatic envelope solitons as well as first-order and second-order DARWs
in a three components EDDP having inertial warm positively and negatively
charged massive dust grains as well as inertialess non-extensive
$q$-distributed ions.

The outline of the paper is as follows: The governing equations describing our
plasma model are presented in Section 2. The NLSE is derived in Section 3.
Modulational instability is given in Section 4. The formation of envelope
solitons and rogue waves are, respectively, presented in Sections 5 and 6. A
brief conclusion is finally provided in Section 7.

##  2 Governing Equations

We consider an unmagnetized three components EDDP medium consisting of
inertial warm positively and negatively charged massive dust grains, and
inertialess non-extensive $q$-distributed positive ions. At equilibrium, the
quasi-neutrality condition for our plasma model can be written as
$Z_{i}n_{i0}+Z_{+}n_{+0}\approx Z_{-}n_{-0}$; where $n_{i0}$, $n_{+0}$, and
$n_{-0}$ are, respectively, the equilibrium number densities of positive ions,
warm positive and negative dust grains, and also $Z_{i}$, $Z_{+}$, and $Z_{-}$
are, respectively, the charge state of the positive ions, warm positive and
negative dust grains. Now, the normalized governing equations for studying
DAWs can be written as

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

where $n_{+}$, $n_{-}$, and $n_{i}$ are the number densities of positively
charged dust grains, negatively charged dust grains, and positive ions
normalized by their equilibrium values $n_{+0}$, $n_{-0}$, and $n_{i0}$,
respectively; $u_{+}$ and $u_{-}$ are the dust fluid speeds normalized by the
DA wave speed $C_{+}=(Z_{+}k_{B}T_{i}/m_{+})^{1/2}$ (with $T_{i}$ being the
ion temperature, $m_{+}$ being the positive dust mass, and $k_{B}$ being the
Boltzmann constant); $\phi$ is the electrostatic wave potential normalized by
$k_{B}T_{i}/e$ (with $e$ being the magnitude of single electron charge); the
time and space variables are normalized by $\omega_{P+}^{-1}=({m_{+}}/4\pi
e^{2}Z_{+}^{2}n_{+0})^{1/2}$ and $\lambda_{D}=(k_{B}T_{i}/4\pi
e^{2}Z_{+}n_{+0})^{1/2}$, respectively; $P_{+}=P_{+0}(N_{+}/n_{+0})^{\gamma}$
[with $P_{+0}$ being the equilibrium adiabatic pressure of the warm positive
dust grains and $\gamma=(N+2)/N$, where $N$ is the degree of freedom, for one
dimensional case, $N$=1 so that $\gamma$=3]; $P_{+0}=n_{+0}k_{B}T_{+}$ (with
$T_{+}$ being the temperature of the warm positive dust grains and $k_{B}$
being the Boltzmann constant); $P_{-}=P_{-0}(N_{-}/n_{-0})^{\gamma}$ [with
$P_{-0}$ being the equilibrium adiabatic pressure of the warm negative dust
grains]; $P_{-0}=n_{-0}k_{B}T_{-}$ (with $T_{-}$ being the temperature of the
warm negative dust grains); and $e_{1}=T_{+}/Z_{+}T_{i}$,
$e_{2}=T_{-}m_{+}/Z_{+}T_{i}m_{-}$, $e_{3}=Z_{-}m_{+}/Z_{+}m_{-}$, and
$e_{4}=Z_{-}n_{-0}/Z_{+}n_{+0}$. We have considered for our numerical analysis
$T_{+}>T_{-}$, $T_{i}\gg(T_{+},~{}T_{-})$, $m_{-}>m_{+}$, $Z_{-}>Z_{+}$, and
$n_{-0}>n_{+0}$. Now, the $q$ distributed positive ion number density can be
expressed in the following form [18, 4]

$\displaystyle n_{i}=[1-(q-1)\phi]^{\frac{q+1}{2(q-1)}},\ $ (6)

where the parameter $q$ is known as entropic index, and is used to explain the
effects of non-extensivity on nonlinear structure of DAWs. It is well known
that $q<1$ refers to the super-extensivity while $q>1$ refers to the sub-
extensivity, and $q=1$ indicates the Maxwellian distribution. By substituting
Eq. (6) into Eq. (5) and expanding up to third order in $\phi$, we get

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}+n_{+}-e_{4}n_{-}=1-e_{4}+S_{1}\phi$ $\displaystyle\hskip
39.83368pt+S_{2}\phi^{2}+S_{3}\phi^{3}+\cdots,\ $ (7)

where

$\displaystyle S_{1}=[(e_{4}-1)(q+1)]/2,$ $\displaystyle
S_{2}=[(1-e_{4})(q+1)(3-q)]/8,$ $\displaystyle
S_{3}=[(1-e_{4})(q+1)(3-q)(3q-5)]/48.\ $

We note that the right hand side of the Eq. (7) is the contribution of
positive ions.

##  3 Derivation of the NLSE

To study the MI of the DAWs, we want to derive the NLSE by employing the
reductive perturbation method (RPM) and for that purpose, we can write the
stretched coordinates in the form [36, 37, 38]

$\displaystyle\xi={\epsilon}(x-v_{g}t),$ (8)
$\displaystyle\tau={\epsilon}^{2}t,\ $ (9)

where $v_{g}$ is the group velocity and $\epsilon$ ($\epsilon\ll 1$) is a
small parameter. Then, we can write the dependent variables as [36, 37, 38]

$\displaystyle
n_{+}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{+l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (10) $\displaystyle
u_{+}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{+l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (11) $\displaystyle
n_{-}=1+\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}n_{-l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (12) $\displaystyle
u_{-}=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}u_{-l}^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],$ (13)
$\displaystyle\phi=\sum_{m=1}^{\infty}\epsilon^{m}\sum_{l=-\infty}^{\infty}\phi^{(m)}(\xi,\tau)~{}\mbox{exp}[il(kx-\omega
t)],\ $ (14)

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

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

Now, by substituting (8)-(16) into (1)-(4), and (7), and equating the
coefficients of $\epsilon$ for $m=l=1$, then we get the following equations

$\displaystyle ku_{+1}^{(1)}=\omega n_{+1}^{(1)},$ (17) $\displaystyle
k\phi_{1}^{(1)}+k\alpha n_{+1}^{(1)}=\omega n_{+1}^{(1)},$ (18) $\displaystyle
ku_{-1}^{(1)}=\omega n_{-1}^{(1)},$ (19) $\displaystyle k\beta
n_{-1}^{(1)}=\omega n_{-1}^{(1)}+ke_{3}\phi_{1}^{(1)},$ (20) $\displaystyle
n_{+1}^{(1)}=k^{2}\phi_{1}^{(1)}+S_{1}\phi_{1}^{(1)}+e_{4}n_{-1}^{(1)},$ (21)

where $\alpha=3e_{1}$ and $\beta=3e_{2}$. These equations reduce to

$\displaystyle n_{+1}^{(1)}=\frac{k^{2}}{A}\phi^{(1)}_{1},$ (22)
$\displaystyle u_{+1}^{(1)}=\frac{\omega k}{A}\phi^{(1)}_{1},$ (23)
$\displaystyle n_{-1}^{(1)}=\frac{e_{3}k^{2}}{C}\phi^{(1)}_{1},$ (24)
$\displaystyle u_{-1}^{(1)}=\frac{\omega ke_{3}}{C}\phi^{(1)}_{1},\ $ (25)

where $A=\omega^{2}-\alpha k^{2}$ and $C=\beta k^{2}-\omega^{2}$. Therefore,
the dispersion relation can be written as

$\displaystyle\omega^{2}=\frac{k^{2}M\pm k^{2}\sqrt{M^{2}-4GH}}{2G},\ $ (26)

Figure 1: Plot of $P/Q$ vs $k$ for different values of $e_{3}$ when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{4}=2.0$, $q=2$, and $\omega_{s}$.

where $M=1+e_{3}e_{4}+\alpha k^{2}+\alpha S_{1}+\beta k^{2}+\beta S_{1}$,
$G=k^{2}+S_{1}$, and $H=\beta+\alpha\beta k^{2}+\alpha\beta S_{1}+\alpha
e_{3}e_{4}$. In order to obtain the real positive values of $\omega$, the
condition $M^{2}>4GH$ must be satisfied. The positive and negative signs of
Eq. (26) determine two types of DA modes, i.e., the positive sign refers to
the fast ($\omega_{f}$) DA mode whereas the negative sign refers to the slow
($\omega_{s}$) DA mode. The fast DA mode corresponds to the case in which both
dust species oscillate in phase with ions. On the other hand, the slow DA mode
corresponds to the case in which only one of the inertial massive dust
components oscillates in phase with ions but the other species are in anti-
phase with them [6]. The second-order ($m=2$ with $l=1$) equations are given
by

$\displaystyle
n_{+1}^{(2)}=\frac{k^{2}}{A}\phi_{1}^{(2)}+\frac{ikB_{1}}{A^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(27) $\displaystyle u_{+1}^{(2)}=\frac{\omega
k}{A}\phi_{1}^{(2)}+\frac{iB_{2}}{A^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(28) $\displaystyle
n_{-1}^{(2)}=\frac{e_{3}k^{2}}{C}\phi_{1}^{(2)}-\frac{ike_{3}B_{3}}{C^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},$
(29) $\displaystyle u_{-1}^{(2)}=\frac{\omega
ke_{3}}{C}\phi_{1}^{(2)}-\frac{ie_{3}B_{4}}{C^{2}}\frac{\partial\phi_{1}^{(1)}}{\partial\xi},\
$ (30)

where

$\displaystyle B_{1}=2\omega kv_{g}-\omega^{2}-A-\alpha k^{2},$ $\displaystyle
B_{2}=2kv_{g}\omega^{2}-\omega^{3}-\alpha\omega k^{2}-kv_{g}A,$ $\displaystyle
B_{3}=2\omega kv_{g}-\omega^{2}+C-\beta k^{2},$ $\displaystyle
B_{4}=2kv_{g}\omega^{2}-\omega^{3}-\beta\omega k^{2}+kv_{g}C,\ $

and the group velocity of DAWs, with the compatibility condition, can be
written as

$\displaystyle v_{g}=\frac{\partial\omega}{\partial
k}=\frac{B_{5}-2A^{2}C^{2}}{2\omega k(C^{2}+e_{3}e_{4}A^{2})},\ $ (31)

where $B_{5}=e_{3}e_{4}A^{2}\omega^{2}-e_{3}e_{4}CA^{2}+e_{3}e_{4}\beta
A^{2}k^{2}+\alpha k^{2}C^{2}+AC^{2}+\omega^{2}C^{2}$. The coefficients of
$\epsilon$ for $m=2$ and $l=2$ provide the second-order harmonic amplitudes
which are found to be proportional to $|\phi^{(1)}_{1}|^{2}$

$\displaystyle n_{+2}^{(2)}=S_{4}|\phi_{1}^{(1)}|^{2},$ (32) $\displaystyle
u_{+2}^{(2)}=S_{5}|\phi_{1}^{(1)}|^{2},$ (33) $\displaystyle
n_{-2}^{(2)}=S_{6}|\phi_{1}^{(1)}|^{2},$ (34) $\displaystyle
u_{-2}^{(2)}=S_{7}|\phi_{1}^{(1)}|^{2},$ (35)
$\displaystyle\phi_{2}^{(2)}=S_{8}|\phi_{1}^{(1)}|^{2},\ $ (36)

where

$\displaystyle S_{4}=\frac{2k^{2}A^{2}S_{8}+\alpha
k^{6}+3\omega^{2}k^{4}}{2A^{3}},$ $\displaystyle S_{5}=\frac{\omega
S_{4}A^{2}-\omega k^{4}}{kA^{2}},$ $\displaystyle
S_{6}=\frac{2e_{3}k^{2}C^{2}S_{8}-\beta
e_{3}^{2}k^{6}-3\omega^{2}k^{4}}{2C^{3}},$ $\displaystyle S_{7}=\frac{\omega
S_{6}C^{2}-\omega e_{3}^{2}k^{4}}{kC^{2}},$ $\displaystyle
S_{8}=\frac{B_{6}}{B_{7}},$ $\displaystyle B_{6}=2S_{2}A^{3}C^{3}-\beta
e_{4}e_{3}^{2}A^{3}k^{6}-3e_{4}e_{3}^{2}\omega^{2}k^{4}A^{3}$
$\displaystyle-\alpha C^{3}k^{6}-3\omega^{2}k^{4}C^{3},$ $\displaystyle
B_{7}=2A^{2}k^{2}C^{3}-2e_{3}e_{4}A^{3}C^{2}k^{2}-8k^{2}A^{3}C^{3}$
$\displaystyle-2S_{1}A^{3}C^{3}.\ $

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

$\displaystyle n_{+0}^{(2)}=S_{9}|\phi_{1}^{(1)}|^{2},$ (37) $\displaystyle
u_{+0}^{(2)}=S_{10}|\phi_{1}^{(1)}|^{2},$ (38) $\displaystyle
n_{-0}^{(2)}=S_{11}|\phi_{1}^{(1)}|^{2},$ (39) $\displaystyle
u_{-0}^{(2)}=S_{12}|\phi_{1}^{(1)}|^{2},$ (40)
$\displaystyle\phi_{0}^{(2)}=S_{13}|\phi_{1}^{(1)}|^{2},\ $ (41)

where

$\displaystyle S_{9}=\frac{2\omega v_{g}k^{3}+\alpha
k^{4}+k^{2}\omega^{2}+S_{13}A^{2}}{A^{2}(v_{g}^{2}-\alpha)},$ $\displaystyle
S_{10}=\frac{v_{g}S_{9}A^{2}-2\omega k^{3}}{A^{2}},$ $\displaystyle
S_{11}=\frac{\beta e_{3}^{2}k^{4}+2\omega
v_{g}e_{3}^{2}k^{3}+e_{3}^{2}k^{2}\omega^{2}-e_{3}S_{13}C^{2}}{C^{2}(v_{g}^{2}-\beta)},$
$\displaystyle S_{12}=\frac{v_{g}S_{11}C^{2}-2\omega e_{3}^{2}k^{3}}{C^{2}},$
$\displaystyle
S_{13}=\frac{2S_{2}A^{2}C^{2}(v_{g}^{2}-\alpha)(v_{g}^{2}-\beta)+B_{8}}{B_{9}A^{2}C^{2}+A^{2}C^{2}(v_{g}^{2}-\beta)},$
$\displaystyle B_{8}=e_{4}e_{3}^{2}A^{2}(v_{g}^{2}-\alpha)(\beta k^{4}+2\omega
v_{g}k^{3}+k^{2}\omega^{2})$ $\displaystyle-C^{2}(v_{g}^{2}-\beta)(\alpha
k^{4}+2\omega v_{g}k^{3}+k^{2}\omega^{2}),$ $\displaystyle
B_{9}=e_{4}e_{3}(v_{g}^{2}-\alpha)-S_{1}(v_{g}^{2}-\alpha)(v_{g}^{2}-\beta).\
$

Finally, the third harmonic modes ($m=3$) and ($l=1$) gives a set of
equations, which can be reduced to the following NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,$
(42)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. In equation (42), $P$ is the
dispersion coefficient which can be written as

$\displaystyle P=\frac{B_{10}-A^{3}C^{3}}{2\omega
k^{2}AC(C^{2}+e_{3}e_{4}A^{2})},\ $

where

$\displaystyle B_{10}=e_{3}e_{4}A^{3}(4\beta\omega
v_{g}k^{3}+4v_{g}k\omega^{3}+\beta k^{2}C+k^{2}v_{g}^{2}C$
$\displaystyle-4k^{2}\omega^{2}v_{g}^{2}-2\beta
k^{2}\omega^{2}-\beta^{2}k^{4}-\omega^{4})$ $\displaystyle-C^{3}(4\alpha\omega
v_{g}k^{3}+4v_{g}k\omega^{3}+\alpha k^{2}A+k^{2}v_{g}^{2}A$
$\displaystyle-4k^{2}\omega^{2}v_{g}^{2}-2\alpha
k^{2}\omega^{2}-\alpha^{2}k^{4}-\omega^{4}),\ $

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

$\displaystyle
Q=\frac{2S_{2}A^{2}C^{2}(S_{8}+S_{13})+3S_{3}A^{2}C^{2}-B_{11}}{2\omega
k^{2}(C^{2}+e_{3}e_{4}A^{2})},\ $

where

$\displaystyle B_{11}=\alpha k^{4}S_{4}C^{2}+\alpha k^{4}S_{9}C^{2}+\beta
e_{3}e_{4}k^{4}S_{6}A^{2}$ $\displaystyle+2e_{3}e_{4}\omega
k^{3}S_{12}A^{2}+k^{2}\omega^{2}S_{4}C^{2}+k^{2}\omega^{2}S_{9}C^{2}$
$\displaystyle+2\omega S_{5}k^{3}C^{2}+2\omega
k^{3}S_{10}C^{2}+2e_{3}e_{4}\omega k^{3}S_{7}A^{2}$ $\displaystyle+\beta
e_{3}e_{4}k^{4}S_{11}A^{2}+e_{3}e_{4}S_{6}k^{2}\omega^{2}+e_{3}e_{4}S_{11}k^{2}\omega^{2}.\
$

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

Figure 2: Plot of $P/Q$ vs $k$ for different values of $e_{3}$ when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{4}=2.0$, $q=1$, and $\omega_{f}$. Figure 3:
Plot of $P/Q$ vs $k$ for different values of $e_{3}$ when $e_{1}=0.07$,
$e_{2}=0.007$, $e_{4}=2.0$, $q=2$, and $\omega_{f}$.

##  4 Modulational instability

The stable and unstable parametric regimes of the DAWs are organized by the
sign of the dispersion ($P$) and nonlinear ($Q$) coefficients of the standard
NLSE (42) [36, 37, 38, 39, 40, 41, 42, 43]. When $P$ and $Q$ have same sign
(i.e., $P/Q>0$), the evolution of the DAWs 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 DAWs are modulationally
stable. The plot of $P/Q$ against $k$ yields stable and unstable parametric
regimes for the DAWs. The point, at which transition of $P/Q$ curve intersects
with $k$-axis, is known as threshold or critical wave number $k$ ($=k_{c}$).

Figure 1 shows the variation of $P/Q$ with $k$ for different values of $e_{3}$
and also for DA slow mode ($\omega_{s}$). It is clear from this figure that
(a) the stable parametric regime of DAWs increases with increasing
(decreasing) the mass of the positive (negative) dust grains while $Z_{+}$ and
$Z_{-}$ remain constant; (b) an increase in the value of negative dust charge
state causes to increase the stable parametric regime of DAWs whereas an
increase in the value of positive dust charge state causes to increase the
unstable parametric regime for a fixed value of $m_{+}$ and $m_{-}$ (via
$e_{3}$).

Figure 4: Plot of $\mbox{Re}(\Phi)$ vs $\xi$ for bright envelope soliton when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$, $e_{4}=2.0$, $q=2$, $\tau=0$,
$\psi_{0}=0.002$, $U=0.4$, $\Omega_{0}=0.4$, $k=0.5$, and $\omega_{f}$. Figure
5: Plot of $\mbox{Re}(\Phi)$ vs $\xi$ for dark envelope soliton when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$, $e_{4}=2.0$, $q=2$, $\tau=0$,
$\psi_{0}=0.002$, $U=0.4$, $\Omega_{0}=0.4$, $k=0.1$, and $\omega_{f}$. Figure
6: Plot of $|\Phi|$ vs $\xi$ for different values of $e_{4}$ when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$, $q=2$, $\tau=0$, $\psi_{0}=0.002$,
$U=0.4$, $\Omega_{0}=0.4$, $k=0.5$, and $\omega_{f}$. Figure 7: Plot of
$|\Phi|$ vs $\xi$ for different values of $e_{4}$ when $e_{1}=0.07$,
$e_{2}=0.007$, $e_{3}=0.6$, $q=2$, $\tau=0$, $\psi_{0}=0.002$, $U=0.4$,
$\Omega_{0}=0.4$, $k=0.1$, and $\omega_{f}$.

The criteria for the formation of dark envelope solitons associated with
stable region (i.e., $P/Q<0$) of DAWs as well as bright envelope solitons and
DARWs associated with unstable region (i.e., $P/Q>0$) of DAWs is shown in Fig.
2 by depicting the variation of $P/Q$ with $k$ for the values of $e_{3}$ and
for Maxwellian ions (i.e., $q=1$). It is obvious from this figure that (a) the
modulationally stable parametric regime of DAWs reduces (enhances) as we
increase the value of positive (negative) dust mass for a fixed value of
$Z_{-}$ and $Z_{+}$; (b) the modulationally unstable parametric regime
increases (decreases) with an increase of $Z_{-}$ ($Z_{+}$) for the fixed
values of positive and negative dust mass (via $e_{3}$).

Figure 3 indicates that how the mass and charge state of the positive and
negative dust grains of a three components EDDP can organize the stability
criterion for the DAWs. The $k_{c}$ as well as the stable parametric regime
(i.e., $P/Q<0$) of DAWs decreases with an increase in the value of $e_{3}$ for
a sub-extensive limit of ions (i.e., $q=2$). It can be deduced from Figs. 2-3
that the direction of the variation of $k_{c}$ is independent to the two
limits of $q$ (i.e., $q>0$ and $q<0$) but dependent to the variation of the
mass and charge state of the positive and negative dust grains.

##  5 Envelope solitons

The bright and dark envelope solitons can be written as [36, 37, 38, 39]

$\displaystyle\Phi(\xi,\tau)=\left[\psi_{0}~{}\mbox{sech}^{2}\left(\frac{\xi-U\tau}{W}\right)\right]^{\frac{1}{2}}$
$\displaystyle\times\exp\left[\frac{i}{2P}\left\\{U\xi+\left(\Omega_{0}-\frac{U^{2}}{2}\right)\tau\right\\}\right],$
(43)
$\displaystyle\Phi(\xi,\tau)=\left[\psi_{0}~{}\mbox{tanh}^{2}\left(\frac{\xi-U\tau}{W}\right)\right]^{\frac{1}{2}}$
$\displaystyle\times\exp\left[\frac{i}{2P}\left\\{U\xi-\left(\frac{U^{2}}{2}-2PQ\psi_{0}\right)\tau\right\\}\right],\
$ (44)

where $\psi_{0}$ indicates the envelope amplitude, $U$ is the travelling speed
of the localized pulse, $W$ is the pulse width which can be written as
$W=(2P\psi_{0}/Q)^{1/2}$, and $\Omega_{0}$ is the oscillating frequency at
$U=0$. We have depicted bright envelope solitons in Figs. 4 and 6 by using Eq.
(43), and also dark envelope solitons in Figs. 5 and 7 by using Eq. (44). The
characteristics of bright envelope solitons can be observed from Fig. 6, and
it is clear from this figure that (a) the width of the bright envelope
solitions associated with DAWs enhances with decreasing (increasing) number
density of negative (positive) dust when the charge state of both dust are
remain invariant (via $e_{4}$); (b) the height of the bright envelope soliton
remains constant with the variation of different plasma parameters such as the
number density and charge state of OPDGs of an EDDP, and this result is a good
agreement with the work of Chowdhury et al. [38].

Figure 7 illustrates the effects of the negative dust number density
($n_{-0}$) and positive dust number density ($n_{+0}$) as well as the number
of charges residing on dust grains on the formation of dark envelope solitons
associated with DAWs, and it is obvious from this figure that (a) the increase
in the value of $e_{4}$ causes to change the width of the dark envelope
solitons but does not cause any change in the magnitude of the amplitude of
the dark envelope solitons; (b) the magnitude of the amplitude of the dark
envelope solitons does not depend on any plasma parameters such as number
density and charge state of positive and negative dust grains (via $e_{4}$),
and this result is a good agreement with the work of Chowdhury et al. [38].

##  6 Rogue waves

The NLSE (42) has a variety of solutions, among them there is a hierarchy of
rational solutions that are located on a nonzero background and 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 [53]. Thus, these waves were nicknamed “waves that appear from
nowhere and disappear without a trace [54]”. The first-order rational solution
of NLSE (42) is given as [52, 53, 54]

$\displaystyle\Phi_{1}(\xi,\tau)=\sqrt{\frac{2P}{Q}}\Big{[}\frac{4+16i\tau
P}{1+4\xi^{2}+16\tau^{2}P^{2}}-1\Big{]}\mbox{exp}(2i\tau P).$ (45)

The nonlinear superposition of the two or more first-order RWs gives rise
higher-order RWs and form a more complicated nonlinear structure with high
amplitude. The second-order rational solution is expressed as [52, 53, 54]

$\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).\ $ (46)

where

$\displaystyle
G_{2}(\xi,\tau)=\frac{-\xi^{4}}{2}-6(P\xi\tau)^{2}-10(P\tau)^{4}$
$\displaystyle\hskip 5.69046pt-\frac{3\xi^{2}}{2}-9(P\tau)^{2}+\frac{3}{8},$
$\displaystyle
M_{2}(\xi,\tau)=-P\tau\Big{[}\xi^{4}+4(P\xi\tau)^{2}+4(P\tau)^{4}$
$\displaystyle\hskip 5.69046pt-3\xi^{2}+2(P\tau)^{2}-\frac{15}{4}\Big{]},$
$\displaystyle
D_{2}(\xi,\tau)=\frac{\xi^{6}}{12}+\frac{\xi^{4}(P\tau)^{2}}{2}+\xi^{2}(P\tau)^{4}$
$\displaystyle\hskip
5.69046pt+\frac{\xi^{4}}{8}+\frac{9(P\tau)^{4}}{2}-\frac{3(P\xi\tau)^{2}}{2}$
$\displaystyle\hskip
5.69046pt+\frac{9\xi^{2}}{16}+\frac{33(P\tau)^{2}}{8}+\frac{3}{32}.\ $

The solutions (45) and (46) represent the profile of the first-order and
second-order RWs, which concentrate a significant amount of energy into a
relatively small area, within the modulationally unstable parametric regime.
We have numerically analyzed Eq. (45) in Figs. 8-10 to understand the effects
of non-extensivity of inertialess ions on the shape of first-order DARWs
associated with DAWs in the modulationally unstable parametric regime, and it
is clear from these figures that the nonlinearity as well as the height and
thickness of the first-order DARWs decrease with $q$ for three possible ranges
of $q$ (i.e., $q=$ negative, $q=$ positive but less than 1, and $q=$ positive
but grater than 1). It may be noted here that a number of authors [1, 11, 3,
2, 5, 8] have considered non-extensive ions, and have also used three possible
ranges of $q$ (i.e., $q=$ negative, $q=$ positive but less than $1$, and $q=$
positive but grater than 1) for numerical study of the nonlinear electrostatic
structures associated with DAWs in an EDDP, and have also identified that the
possible existence of such kind of non-extensive ions along with dust grains
in cometary tail [4, 3, 2], upper mesosphere [6, 4], magnetosphere of Jupiter
[6], and Saturn’s F-ring [7, 8], etc.

Figure 8: Plot of $|\Phi_{1}|$ vs $\xi$ for different values of $q$ when
$e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$, $e_{4}=2.0$, $\tau=0$, $k=0.5$, and
$\omega_{f}$. Figure 9: Plot of $|\Phi_{1}|$ vs $\xi$ for different values of
$q$ when $e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$, $e_{4}=2.0$, $\tau=0$,
$k=0.5$, and $\omega_{f}$. Figure 10: Plot of $|\Phi_{1}|$ vs $\xi$ for
different values of $q$ when $e_{1}=0.07$, $e_{2}=0.007$, $e_{3}=0.6$,
$e_{4}=2.0$, $\tau=0$, $k=0.5$, and $\omega_{f}$.

(a)  (b)

Figure 11: Profile of the (a) first-order rational solution; (b) second-order
rational solution when other parameters are $e_{1}=0.07$, $e_{2}=0.007$,
$e_{3}=0.6$, $e_{4}=2.0$, $q=2$, $k=0.5$, and $\omega_{f}$. Figure 12: Profile
of the first-order (dashed green curve) and second-order (solid blue curve)
rational solutions when other parameters are $e_{1}=0.07$, $e_{2}=0.007$,
$e_{3}=0.6$, $e_{4}=2.0$, $\tau=0$, $q=2$, $k=0.5$, and $\omega_{f}$.

The space and time evolution of the first-order and second-order rational
solution of the NLSE (42) can be observed from Figs. 11(a) and 11(b),
respectively. Figure 12 indicates the first-order and second-order solution at
$\tau=0$, and it is clear form this figures that (a) second-order rational
solution has double structure compared with first-order rational solution; (b)
the height of the second-order rational solution is always greater than the
first-order rational solution; (c) the potential profile of the second-order
rational solution becomes 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 RWs has already been confirmed
experimentally [47, 48, 49, 50] and theoretically [51]. The second-order
solution was experimentally observed by Chabchoub et al. [48] in a “Water Wave
Tank” and the experimental result regarding the amplification is a nice
agreement with the theoretical result. Bailung et al. [49] demonstrated an
experiment in a multi-component plasma 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
[50].

##  7 Conclusion

In our present article, we have investigated the characteristics of the
amplitude modulation of DAWs by using a NLSE, which are successfully derived
by employing the standard RPM in a EDDP composed of non-extensive ions,
negatively and positively charged warm dust grains. In the formation and
propagation of DAWs, the moment of inertia is provided by the mass of the
heavier components (adiabatic warm positive and negative dust grains) and
restoring force is provided by the thermal pressure of the lighter component
(non-extensive ion) of the plasma medium. So, each of the plasma components of
the plasma medium provides a great contribution to the formation and
propagation of the DAWs in three components EDDP medium. The observational
data have disclosed the ubiquitous existence of non-extensive ions as well as
positive and negative dust grains in astrophysical environments such as
cometary tail [4, 3, 2], Saturn’s F-ring [7, 8], upper mesosphere [6, 4], and
magnetosphere of Jupiter [6] as well as laboratory situations such as laser-
matter interaction [9]. Hence, the implications of our results should be
useful to understand the mechanism of MI of the DAWs and the formation of
DARWs as well as envelope solitons in the modulationally unstable parametric
regime, which is determined by the sign of P and Q of the standard NLSE, in a
three components EDDP medium such as cometary tail [4, 3, 2], Saturn’s F-ring
[7, 8], upper mesosphere [6, 4], and magnetosphere of Jupiter [6] as well as
laboratory situations such as laser-matter interaction [9]. It may be noted
here that the gravitational and magnetic field effects are very important but
beyond the scope of our present work. In future and for better understanding,
someone can study the nonlinear propagation in a three components EDDP medium
by considering the gravitational and magnetic field effects.

## References

  * [1] B. Sahu and M. Tribeche, Astrophys. Space Sci. 338, 259 (2012).
  * [2] M. M. Hossen, et al., Phys. Plasmas 23, 023703 (2016).
  * [3] M. M. Hossen, et al., Eur. Phys. J. D 70, 252 (2016).
  * [4] S. K. Zaghbeer, et al., Astrophys. Space Sci. 353, 493 (2014).
  * [5] M. M. Hossen, et al., High Energy Density Phys. 24, 9 (2017).
  * [6] S. Jahan, et al., Commun. Theor. Phys. 71, 327 (2019).
  * [7] A. S. Bains, et al., Astrophys. Space Sci. 343, 621 (2013).
  * [8] B. Sahu and M. Tribeche, Astrophys. Space Sci. 341, 573 (2012).
  * [9] M. Shahmansouri and H. Alinejad, Phys. Plasmas 20, 033704 (2013).
  * [10] P. K. Shukla and A. A. Mamun, Introduction to Dusty Plasma Physics (Institute of Physics, Bristol, 2002).
  * [11] M. Ferdousi, et al., Astrophys. Space Sci. 43, 360 (2015).
  * [12] M. Emamuddin, et al., Phys. Plasmas 20, 043705 (2013).
  * [13] A. Barkan, et al., Phys. Plasmas 2, 3563 (1995).
  * [14] M. Ferdousi, et al., Eur. Phys. J. D 71, 102 (2017).
  * [15] M. Eghbali, et al., Pramana J. Phys. 88, 15 (2017).
  * [16] N. N. Rao, et al., Planet. Space Sci. 38, 543 (1990).
  * [17] A. Renyi, Acta Math. Acad. Sci. Hung. 6, 285 (1955).
  * [18] C. Tsallis, J. Stat. Phys. 52, 479 (1988).
  * [19] T. E. Eastman, et al., et. al., Geophys. Res. Lett. 103, 23503 (1998).
  * [20] G. P. Pavlos, et al., Physica A 390, 2819 (2011).
  * [21] M. H. Rahman, et al., Phys. Plasmas 25, 102118 (2018).
  * [22] M. Bacha, et al., Physica A 466, 199 (2017).
  * [23] K. Roy, et al., Astrophys. Space Sci. 350, 599 (2014).
  * [24] M. Bacha and M. Tribeche, Astrophys. Space Sci. 337, 253 (2012).
  * [25] S. A. El-Tantawy, et al., Astrophys. Space Sci. 337, 209 (2012).
  * [26] J. Borhanian et al., Phys. Plasmas 20, 013707 (2013).
  * [27] M. Tribeche, et al., Phys. Plasmas 7, 4013 (2000).
  * [28] M. Tribeche, et al., Phys. Plasmas 11, 3001 (2004).
  * [29] B. Tadsen, et al., Phys. Plasmas 22, 113701 (2015).
  * [30] M. Steckiewicz, et al., et al., Geophys. Res. Lett. 42, 8877 (2015).
  * [31] P. K. Shukla and V. P. Silin, Phys. Scr. 45, 508 (1992).
  * [32] S. G. Tagare, Phys. Plasmas 04, 3167 (1997).
  * [33] N. A. Chowdhury, et al., Phys. plasmas 24, 113701 (2017).
  * [34] M. H. Rahman, et al., Chinese J. Phys. 56, 2061 (2018).
  * [35] N. A. Chowdhury, et al., Contrib. Plasma Phys. 58, 870 (2018).
  * [36] I. Kourakis and P. K. Shukla, Phys. Plasmas 10, 3459 (2003).
  * [37] I. Kourakis et al., Nonlinear Proc. Geophys. 12, 407 (2005).
  * [38] N. A. Chowdhury, et al., Vacuum 147, 31 (2018).
  * [39] R. Fedele, Phys. Scr. 65, 502 (2002).
  * [40] W. M. Moslem, et al., Physical Review E 84, 066402 (2011).
  * [41] N. A. Chowdhury, et al., Plasma Phys. Rep. 45, 459 (2019).
  * [42] N. Ahmed, et al., Chaos 28, 123107 (2018).
  * [43] N. A. Chowdhury, et al., Chaos 27, 093105 (2017).
  * [44] R. E. Tolba, et al., Phys. Plasmas 22, 043707 (2015).
  * [45] T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967).
  * [46] A. M. Turing, Philos. Trans. R. Soc. London B 237, 37 (1952).
  * [47] A. Chabchoub, et al., Phys. Rev. Lett. 106, 204502 (2011).
  * [48] A. Chabchoub, et al., Phys. Rev. X 2, 011015 (2012).
  * [49] H. Bailung, et al., Phys. Rev. Lett. 107, 255005 (2011).
  * [50] B. Kibler, et al., Nat. Phys. 6, 790 (2010).
  * [51] Shalini, et al., Phys. Plasmas 22, 092124 (2015).
  * [52] S. Guo, et al., Ann. Phys. 332, 38 (2013).
  * [53] A. Ankiewicz, et al., J. Phys. A 43, 12002 (2010).
  * [54] A. Ankiewicz, et al., Phys. Lett. A 373, 3997 (2009).


# CROSS-MODAL SPECTRUM TRANSFORMATION NETWORK FOR ACOUSTIC SCENE
CLASSIFICATION

###### Abstract

Convolutional neural networks (CNNs) with log-mel spectrum features have shown
promising results for acoustic scene classification tasks. However, the
performance of these CNN based classifiers is still lacking as they do not
generalise well for unknown environments. To address this issue, we introduce
an acoustic spectrum transformation network where traditional log-mel
spectrums are transformed into imagined visual features (IVF). The imagined
visual features are learned by exploiting the relationship between audio and
visual features present in video recordings. An auto-encoder is used to encode
images as visual features and a transformation network learns how to generate
imagined visual features from log-mel. Our model is trained on a large dataset
of Youtube videos. We test our proposed method on the scene classification
task of DCASE and ESC-50, where our method outperforms other spectrum
features, especially for unseen environments.

Index Terms—  scene classification, adversarial learning, image
reconstruction, audio-visual cross-modal learning

##  1 Introduction

Acoustic scene classification is a popular task that aims to classify scenes
with audio recordings into different classes. It has many applications in
surveillance of sound event detection [1]. However, it is difficult to
classify acoustic scenes for unseen environments and unknown recording
equipment due to the lack of appropriate labelled data-sets. Current state of
the art approaches uses various augmentation methods to address this issue.
However, over-fitting happens frequently when a large number of augmentation
data is used [2].

In this work, we introduce an acoustic scene transformation algorithm to
address the unseen environment problem for acoustic scene classification. The
goal is to generate imagined visual features from short audio recordings and
to minimise the divergence between the imagined visual features and real
visual features. Real visual features are calculated from images obtained from
training videos using an auto-encoder. A transformation network is used to
learn the mapping between log-mel [3] features and visual features. In order
to decrease the complexity of learning, a vector quantisation based codebook
is applied and shared between the auto-encoder and the transformation network.
Since the videos used for training are recorded using different equipment in
various cities, the imagined visual features are more robust than the
traditional features for unseen cities. For classifying the acoustic scene, a
classifier is used to predict the category of audio recordings based on the
imagined visual features. The auto-encoder and transformation network are
trained on the Youtube 8M dataset [4]. The classifier is trained on the target
dataset, such as DCASE [5] and ESC-50 [6].

Fig. 1: Proposed model and training pipeline. In sub-figure (a), Audio-visual
pairs of the Youtube 8M dataset are used to train the encoder, decoder and
transformation network (Blue lines). In sub-figure (b), The audio recording of
DCASE dataset is used to train a classifier (Green lines) which predicts the
category of the audio recording when the transformation network is fixed.

##  2 Related Work

###  2.1 Acoustic scene classification

Many audio classification methods use deep learning networks, such as
convolutional neural networks (CNNs) [7] and recurrent neural networks (RNNs)
[1]. Suh et al. [8] introduce trident ResNet with the categorical focal loss
to address unseen devices problem. Hu et al. [9] propose a novel two-stage
network with two CNNs, classifying the acoustic input according to high-level
categories, and then low-level categories, respectively. Koutini et al. [7]
propose a novel frequency-aware CNN layer and adaptive weighting for
addressing the noisy labels problem, which achieves the top accuracy for
unseen cities in the 2019 DCASE challenge. These CNN based methods commonly
use the log-mel spectrum of audio recordings as input and output the class
probability of an acoustic scene in either frame-level or video-level.
However, the spectral content of recordings belonging to the same class can
vary greatly between different cities, especially when the recording equipment
is not matched [7].

###  2.2 Audio-visual cross-modal learning

Audio-visual data offer a wealth of resources for knowledge transfer between
different modalities [10]. Our work is closely related to the reconstruction
from audio recordings. For example, Oh et al. [11] and Athanasiadis et al.
[12] have used speech to reconstruct the speakers’ face. Chen et al. [13] Hao
et al. [14] use conditional GANs and cycle GAN to achieve the cross-modal
audio-visual generation of musical performances. However, these works focus on
special cases, such as faces or musical instruments, where the visual object
is located at the centre of the images. For less constrained test data, the
reconstructed images may be noisy, due to large variation. Arandjelovic et al.
[15] propose a generic audio-visual model to classify if a given video frame
and a short audio clip correspond to each other, where the fusion layers only
select the common features of the recording and images. Aytar et al. [10]
propose a student-teacher training procedure where a pre-trained visual
recognition model transfers the knowledge from the visual modality to the
sound modality. Compared to Aytar et al. [10], our proposed method our
proposed method does not need any pre-trained visual recognition model as the
teachers of the audio model.

##  3 Spectrum transformation Network for Classification

In this section, we introduce the spectrum transformation network and discuss
how to train its objective function along with implementation details. The
transformation network with classifier is shown in Fig. 1.

###  3.1 Transformation network

The acoustic spectrum transformation network aims to learn a mapping function
between acoustic features and visual features from unlabelled videos. Although
unlabelled videos do not contain the category information, they provide a
large number of audio-visual pairs $\\{\bm{I},\bm{A}\\}$, where $\bm{A}$ is
the log-mel of a fixed short length audio recording and $\bm{I}$ is randomly
selected frame of the corresponding video. The main challenge for training the
acoustic spectrum transformation network is that the distributions of the
audio samples and visual samples are different.

The visual features of $\bm{I}$ are calculated by an auto-encoder based on VQ-
VAE [16], which is a reconstruction system. As shown in Fig. 1, the audo-
encoder has an encoder and an decoder where the encoder maps the input image
$\bm{I}$ to a sequence of discrete latent variables and the decoder
reconstruct a image ${\bm{I}}^{\prime}$ from the latent variables. A codebook
is applied between the encoder and decoder. To be more specific, the image
$\bm{I}$ is mapped to an embedding $\bm{E}(\bm{I})$ by the encoder. This
embedding is then quantized based on the distance to the prototype vectors of
the codebook, $\bm{B}=\\{\bm{b}_{k}\\}$, $k\in\\{1,...,K\\}$. Then each
embedding in $\bm{E}(\bm{I})$ is replace by the nearest prototype embedding in
the codebook $\bm{B}$. We call the quantized embedding “visual features”
$\bm{Q}^{n}$ where $n$ is the index of the nearest prototype vector,
$\bm{Q}^{n}=\bm{b}_{k}$, where,

$n=\arg\min_{j}\left\|\bm{E}(\bm{I})-\bm{b}_{j}\right\|$ (1)

The set of index $n$ is called as estimated index vector, $\bm{N}$. The
decoder reconstructs the image ${\bm{I}}^{\prime}$ from the corresponding
vectors in the codebook with another non-linear function. To learn these
mappings, the gradient of the reconstruction error is then back-propagated
through the decoder and the encoder using a straight-through gradient
estimator [17]. The training objective is:

$\mathbb{L}_{E}=\frac{\left\|\bm{I}-\bm{I}^{\prime}\right\|_{2}^{2}}{{\delta_{\bm{I}}}}+\left\|\operatorname{sg}\left[\bm{E}(\bm{I})\right]-\bm{b}_{k}\right\|_{2}^{2}+\beta\left\|\bm{E}(\bm{I})-\operatorname{sg}[\bm{b}_{k}]\right\|_{2}^{2}$
(2)

where $\beta$ is the loss weight and $\delta_{I}$ is the diagonal co-variance
of training images and $\operatorname{sg}$ represents the stop-gradient
operator that is defined as an identity at the forward computation time and
has zero partial derivatives.

The imagined visual features of $\bm{A}$ are calculated by the acoustic
spectrum transformation network, which is a non-linear mapping from $\bm{A}$
to an estimated index vector $n=\bm{G}(\bm{A})$ for $\bm{B}$. Therefore, we
define the quantized vector as “imagined visual features”
$\bm{Q}^{\bm{G}(\bm{A})}$. Since the index vectors have far fewer elements
than the quantized vectors, the acoustic spectrum transformation network is
easy to generate, which is also verified by our experiments. To help guide the
transformation network to produce imagined visual features (IVF) from audio
signals that closely match visual features produced from real images, we
further add a discriminator network to distinguish between the real features
$\bm{Q}^{\bm{G}(\bm{A})}$ and $\bm{Q}^{n}$, where $n$ is calculated from
$\bm{E}(\bm{I})$ via Eq. (2). The training objective is designed based on
WGAN-GP [18]:

$\displaystyle\mathbb{L}_{D}$
$\displaystyle=\mathbb{E}[\bm{D}(\bm{Q}^{n})-\bm{D}(\bm{Q}^{\bm{G}(\bm{A})})]$
(3)
$\displaystyle-\lambda\mathbb{E}\left[\left(\left\|\nabla\bm{D}(\bm{C})\right\|_{2}-1\right)^{2}\right]$

where

$\bm{C}=w\bm{Q}^{\bm{G}(\bm{A})}+(1-w)\bm{Q}^{n}$ (4)

$\lambda$ is weight of discriminator loss, $\bm{D}$ is the discriminator with
cross-entropy loss, $\nabla$ is the gradient operator and $w$ is random value
from 0 to 1. $\bm{C}$ represents the penalty sampled uniformly along straight
lines between pairs of points sampled from the real feature distribution and
the fake feature distribution. Our full objective of transformation network
is: $\mathbb{L}=\mathbb{L}_{E}+\mathbb{L}_{D}$.

###  3.2 Classifier network

The classifier is trained on audio recordings ${\bm{A}}$ of the target
dataset, such as DCASE, where the previous transformation network $\bm{G}$ and
codebook $\bm{B}$ are fixed. The goal of the classifier network is to predict
the category of the audio recording in the audio dataset, such as DCASE
dataset. It consists of eight convolution layers, four mean-pooling and a max-
pooling layer, the output dimension is the number of the category with a
cross-entropy training loss.

###  3.3 Implementation detail

To save the computation cost, all frames in the training Youtube 8M dataset
are resized to $512\times 512$ pixels. For the audio part of Youtube 8M, DCASE
and ESC-50, the audio waveform is resampled at 16k Hz. The Spectrums are
computed by taking STFT with 16 FFT frequency bands. The training and test
sets of DCASE include 14.4K and 5.8K audio recordings. Our network is
implemented in TensorFlow and optimized by ADAM with $\beta_{1}$ = 0.5,
$\beta_{2}$ = 0.0002, and the learning rate of 0.001 and the batch size of 16
for 9000 epochs. The network was trained for 223 hours on 4 NVIDIA 2080 Ti.

##  4 Experimentation Results

###  4.1 Dataset

We train our model on a subset of Youtube 8M dataset, DCASE datasets [5] and
ESC-50 dataset [6]. The subset Youtube dataset consists of ten kinds of
scenes: Airport, Bus, Street pedestrian, Park, Metro, Street traffic, Shopping
mall, Public Square, Metro Station and Tram, which matches to DCASE dataset.
The image size is $512*512$. For the audio data, the length of recording is
$10$ seconds for Youtube 8M and DCASE and $5$ seconds for ESC-50. The audio
waveform is re-sampled at 16 kHz. Training, validation and testing data
include 14K videos, 2k videos and 1k videos. The total length is 472 hours.
Our network is implemented in Keras and TensorFlow and optimized by ADAM with
$\beta_{1}$ = 0.5, $\beta_{2}=0.0002$, and the learning rate of $0.0002$ and
the batch size of $640$ for $5000$ epochs. The network was trained for $436$
hours on a workstation hosting $4$ NVIDIA 2080-Ti.

###  4.2 Evaluate imagined visual features

Table 1: Acoustic Scene Classification Accuracy using different features on DCASE. No. | Method | Seen | Unseen
---|---|---|---
1 | CQT [19] | 72.3 | 64.6
2 | Gammatone [20] | 77.5 | 68.1
3 | MFCC [21] | 64.4 | 59.6
4 | log-mel | 73.4 | 62.5
5 | log-mel + delta | 76.5 | 65.1
6 | log-mel+deltas+delta-deltas | 77.2 | 66.5
7 | IVF (without VQ) | 78.3 | 73.2
8 | IVF (Q) | 81.5 | 79.3
9 | IVF | 83.7 | 82.5

To understand the influence of the acoustic spectrum transformation network of
the propose classification algorithm, we compare the the visual feature with
other baseline features, which is widely used in DCASE challenge. Apart from
that, to show the influence of the VQ layer of the transformation model, we
modify the transformation model and evaluate the performance of the ablated
model in Table 1. In the table, model (1-3) shows the performance of CQT
(Constant-Q transform) [19], Gammatone [20] and MFCC [21]. Models (4-6) show
the performance of the Mel family Spectrum, which gives higher accuracy than
models (1-3). Model (4) uses the log-mel Spectrum, while Model (5) uses the
log-mel + delta. In Model (6), the log-mel and its deltas and delta-deltas
[22] without padding are considered. For mel Spectrum family, “log-
mel+deltas+delta-deltas” [22] provides the highest accuracy. Model (7) with
log-mel shows the performance of the model without the VQ layer, which
decreases performance. The reason maybe that the VQ layer and generating the
index can decrease the complexity of the imagined visual features. Model (8)
shows the performance of the model where the output of the transformation
network is the quantized embedding $\bm{Q}^{n}$. Model (9) shows the
performance of the full proposed model where the output of the transformation
network is the estimated index of codebook $\bm{B}$. Our proposed method
achieves the high accuracy on the seen and unseen environment.

###  4.3 Comparison with the state-of-the-art methods

In this section, we compare our proposed classification method with other the
state-of-the-art methods on the DCASE and ESC-50 dataset. Tables 2 and 3 show
the accuracy of different methods tested on DCASE and ESC-50 dataset
respectively. As the length of audio recording is different in DCASE and
ESC-50, their classifiers are separately trained.

On both benchmarks, we convincingly beat the previous state-of-the-art
methods. For DCASE, our method improves the accuracy by 7 $\%$ for the seen
cities and 10 $\%$ for unseen cities, since our proposed is trained on YouTube
videos recorded in different cities. The results of our proposed model and Suh
et al. [8] both indicate that the GAN networks can improve the accuracy of the
acoustic scene classification by capturing visual information from video
datasets. For evaluating the effect of the Youtube 8M dataset, the accuracy of
Suh et al. [8] trained on a labelled version Youtube 8M and DCASE (Suh et al.
[8] \+ 8M) is 82.4 for seen cities and 76.2 for unseen cities, which is lower
than that of our method. Youtube 8M dataset has some wrong labels data, which
may affect the performance of other state of the art methods. It is important
to note that our transformation network is trained without label information.
The accuracy is further improved 8% by fine-tuning with a target dataset.

For ESC-50, we have 2$\%$ higher accuracy than human performance and have a
marginally better performance to the transfer learning method [23]. These
results show that a large number of video data can improve the performance of
classification. Note that since the size of the Youtube 8M dataset is huge, we
only train our network with a subset for efficiency, so it is possible that
further gains can be achieved by using all the available training data.

Table 2: Acoustic Scene Classification Accuracy at the seen cities and unseen cities on DCASE. Method | Seen Cities | Unseen Cities
---|---|---
SoundNet [10] | 75.8 | 67.5
Gao et al. [24] | 77.0 | 73.1
Hu et al. [9] | 77.5 | 74.7
Suh et al. [8] | 78.1 | 74.6
Suh et al.+ 8M | 82.4 | 76.2
Ours | 83.7 | 82.5
Table 3: Acoustic Scene Classification Accuracy on ESC-50. Method | Accuracy
---|---
SVM-MFCC [6] | 39.6
Autoencoder [10] | 39.9
Random Forest [6] | 44.3
Piczak ConvNet [25] | 64.5
SoundNet [10] | 74.2
AV learning [15] | 79.3
Human [6] | 81.3
Transfer learning [23] | 83.5
Ours | 83.7

##  5 Conclusion

We have proposed a spectrum transformation network in order to address the
unseen cities problem in the scene classification task. The proposed network
is able to learn the correlation between audio and visual features and
transform a log-mel signal to imagined visual features. The imagined visual
features are used to classify audio recordings as other traditional spectra.
The transformation network can be trained on unlabelled datasets while the
classifier only needs a limited number of target data. The experiments on
DCASE dataset and ESC-50 dataset show we can achieve higher accuracy than the
state-of-the-art methods, especially for unseen cities. Our future research
will focus on reconstructing scenes with detailed geometric features from
audio recordings.

## References

  * [1] Qiuqiang Kong, Turab Iqbal, Yong Xu, Wenwu Wang, and Mark D Plumbley,  “DCASE 2018 challenge baseline with convolutional neural networks,”  arXiv preprint arXiv:1808.00773, 2018.
  * [2] Joseph Lemley, Shabab Bazrafkan, and Peter Corcoran,  “Smart augmentation learning an optimal data augmentation strategy,”  Ieee Access, vol. 5, pp. 5858–5869, 2017.
  * [3] Steven Davis and Paul Mermelstein,  “Comparison of parametric representations for monosyllabic word recognition in continuously spoken sentences,”  IEEE transactions on acoustics, speech, and signal processing, vol. 28, no. 4, pp. 357–366, 1980.
  * [4] Sami Abu-El-Haija, Nisarg Kothari, Joonseok Lee, Paul Natsev, George Toderici, Balakrishnan Varadarajan, and Sudheendra Vijayanarasimhan,  “Youtube-8m: A large-scale video classification benchmark,”  arXiv preprint arXiv:1609.08675, 2016.
  * [5] Annamaria Mesaros, Toni Heittola, and Tuomas Virtanen,  “Acoustic scene classification in DCASE 2019 challenge: closed and open set classification and data mismatch setups,”  in Proc. CVRP, 2019.
  * [6] Karol J Piczak,  “ESC: Dataset for environmental sound classification,”  in Proc. the ACM international conference on Multimedia. ACM, 2015, pp. 1015–1018.
  * [7] Khaled Koutini, Hamid Eghbal-zadeh, and Gerhard Widmer,  “CP-JKU submissions to DCASE’19: Acoustic scene classification and audio tagging with receptive-field-regularized cnns,”  in Proc. DCASE, 2019.
  * [8] Sangwon Suh, Sooyoung Park, Youngho Jeong, and Taejin Lee,  “Designing acoustic scene classification models with cnn variants,”  Proc. DCASE, 2020.
  * [9] Hu Hu, Chao-Han Huck Yang, Xianjun Xia, Xue Bai, Xin Tang, Yajian Wang, Shutong Niu, Li Chai, Juanjuan Li, Hongning Zhu, et al.,  “Device-robust acoustic scene classification based on two-stage categorization and data augmentation,”  arXiv preprint arXiv:2007.08389, 2020.
  * [10] Yusuf Aytar, Carl Vondrick, and Antonio Torralba,  “Soundnet: Learning sound representations from unlabeled video,”  in Advances in NIPS, 2016, pp. 892–900.
  * [11] Tae-Hyun Oh, Tali Dekel, Changil Kim, Inbar Mosseri, William T Freeman, Michael Rubinstein, and Wojciech Matusik,  “Speech2face: Learning the face behind a voice,”  in Proc. CVRP, 2019, pp. 7539–7548.
  * [12] Christos Athanasiadis, Enrique Hortal, and Stylianos Asteriadis,  “Audio-visual domain adaptation using conditional semi-supervised generative adversarial networks.,”  Neurocomputing, 2019.
  * [13] Lele Chen, Sudhanshu Srivastava, Zhiyao Duan, and Chenliang Xu,  “Deep cross-modal audio-visual generation,”  in Proc. the Conference on Thematic Workshops of ACM Multimedia 2017. ACM, 2017, pp. 349–357.
  * [14] Wangli Hao, Zhaoxiang Zhang, and He Guan,  “Cmcgan: A uniform framework for cross-modal visual-audio mutual generation,”  in Proc. the Conference on Artificial Intelligence, 2018.
  * [15] Relja Arandjelovic and Andrew Zisserman,  “Look, listen and learn,”  in Proc. the Conference on Computer Vision, 2017, pp. 609–617.
  * [16] Aaron van den Oord, Oriol Vinyals, et al.,  “Neural discrete representation learning,”  in Advances in Neural Information Processing Systems, 2017, pp. 6306–6315.
  * [17] Yoshua Bengio, Nicholas Léonard, and Aaron Courville,  “Estimating or propagating gradients through stochastic neurons for conditional computation,”  arXiv preprint arXiv:1308.3432, 2013.
  * [18] Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville,  “Improved training of wasserstein GANs,”  in Advances in neural information processing systems, 2017, pp. 5767–5777.
  * [19] Christian Schörkhuber and Anssi Klapuri,  “Constant-q transform toolbox for music processing,”  in Sound and Music Computing Conference, Barcelona, Spain, 2010, pp. 3–64.
  * [20] Daniel PW Ellis,  “Gammatone-like spectrograms,”  web resource: http://www. ee. columbia. edu/dpwe/resources/matlab/gammatonegram, 2009.
  * [21] Min Xu, Ling-Yu Duan, Jianfei Cai, Liang-Tien Chia, Changsheng Xu, and Qi Tian,  “Hmm-based audio keyword generation,”  in Pacific-Rim Conference on Multimedia. Springer, 2004, pp. 566–574.
  * [22] Ron J Weiss, Jan Chorowski, Navdeep Jaitly, Yonghui Wu, and Zhifeng Chen,  “Sequence-to-sequence models can directly translate foreign speech,”  arXiv preprint arXiv:1703.08581, 2017.
  * [23] Anurag Kumar, Maksim Khadkevich, and Christian Fügen,  “Knowledge transfer from weakly labeled audio using convolutional neural network for sound events and scenes,”  in Proc. CASSP. IEEE, 2018, pp. 326–330.
  * [24] Wei Gao, Mark D McDonnell, and STEM UniSA,  “Acoustic scene classification using deep residual networks with focal loss and mild domain adaptation,”  Proc. DCASE, 2020.
  * [25] Karol J Piczak,  “Environmental sound classification with convolutional neural networks,”  in IEEE International Workshop on Machine Learning for Signal Processing (MLSP). IEEE, 2015, pp. 1–6.