These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Sudipta Nandy These authors contributed equally to this work.

1]Department of Physics, Cotton University, Panbazar, Guwahati, 781001, Assam,
India

# Soliton Management for ultrashort pulse: dark and anti-dark solitons of
Fokas-Lenells equation with a damping like perturbation and a gauge equivalent
spin system

Riki Dutta<EMAIL_ADDRESS>Gautam K. Saharia<EMAIL_ADDRESS>Sagardeep Talukdar<EMAIL_ADDRESS><EMAIL_ADDRESS>[

###### Abstract

We investigate the propagation of an ultrashort optical pulse using Fokas-
Lenells equation (FLE) under varying dispersion, nonlinear effects and
perturbation. Such a system can be said to be under soliton management (SM)
scheme. At first, under a gauge transformation, followed by shifting of
variables, we transform FLE under SM into a simplified form, which is similar
to an equation given by Davydova and Lashkin for plasma waves, we refer to
this form as DLFLE. Then, we propose a bilinearization for DLFLE in a non-
vanishing background by introducing an auxiliary function which transforms
DLFLE into three bilinear equations. We solve these equations and obtain dark
and anti-dark one-soliton solution (1SS) of DLFLE. From here, by reverse
transformation of the solution, we obtain the 1SS of FLE and explore the
soliton behavior under different SM schemes. Thereafter, we obtain dark and
anti-dark two-soliton solution (2SS) of DLFLE and determine the shift in phase
of the individual solitons on interaction through asymptotic analysis. We
then, obtain the 2SS of FLE and represent the soliton graph for different SM
scheme. Thereafter, we present the procedure to determine N-soliton solution
(NSS) of DLFLE and FLE. Later, we introduce a Lax pair for DLFLE and through a
gauge transformation we convert the spectral problem of our system into that
of an equivalent spin system which is termed as Landau-Lifshitz (LL) system.
LL equation (LLE) holds the potential to provide information about various
nonlinear structures and properties of the system.

###### keywords:

Fokas-Lenells Equation, Soliton Management, Soliton, Lax Pair, Landau-Lifshitz
Equation

##  1 Introduction

In optics, soliton is a nonlinear optical pulse that propagates through a
waveguide without getting distorted and hence can acts as a good information
carrier in digital and quantum communication devices. So the research on
soliton has always been in great demand. For the soliton to exist the system
must balance the dispersion and nonlinear effects. Nonlinear Schrödinger
equation (NLSE) is the most basic equation that addresses both of the two
effects and solving it results in a soliton solution [16, 17, 32, 21, 37, 4].
Beyond optical soliton, NLSE (popularly known by the name Gross–Pitaevskii
equation [36]) also describes the propagation of matter-wave in Bose-Einstein
condensates, where the dispersion is represented by kinetic energy of the
bosons and nonlinear effect arises due to intra and inter bosonic
interactions. In optics, besides NLSE, there are other equations taking into
account some higher order effects. Some of these well-studied equations are
Kaup–Newell equation (KNE) [22], Chen–Lee–Liu equation (CLLE) [14, 45],
Tiriki-Biswas equation (TBE) [40], sasa-satsuma equation (SSE) [30, 20, 44],
etc. It may be interesting to point out that KNE, CLLE and TBE are also
referred to as byproducts of derivative NLSE (DNLSE). However, a high
intensity light source can produce ultrashort (femtosecond) pulses. To
describe such a pulse, one must account for the higher order effects namely,
spatio-temporal dispersion (STD) and nonlinear dispersion (ND) effect along
with the group velocity dispersion (GVD) and Kerr nonlinear effects (KNLE).
Fokas-Lenells equation (FLE) [29, 28] is a nonlinear equation that addresses
all these effects. The dimensionless form of FLE is presented as

$\displaystyle iU_{t}+a_{1}U_{xx}-a_{2}U_{xt}+b|U|^{2}\ (U+i\ a_{2}U_{x})$
$\displaystyle=0$ (1)

where $U$ is the field function that describes the complex waveform of an
ultrashort pulse. The suffix $x$ and $t$ denote the partial differentiations
of $U$ with respect to $x$ and $t$ respectively. $U_{t}$ is the temporal
evolution of the pulse, $U_{xx}$, $U_{xt}$, $|U|^{2}U$ and $|U|^{2}U_{x}$
represents GVD, STD, KNLE and ND respectively. Eqn. (1) is relatively a new
equation and the study on this is going on in full swing. Many analytical
approaches have been employed to derive soliton solutions of FLE [3, 19, 41,
1, 5, 6, 35, 42, 13, 9, 10, 23, 34, 33, 39]. In general the works on FLE is
done for constant dispersion and nonlinear coefficients. Only a few papers
have been published with variable coefficients in FLE [25, 31, 43]. In this
manuscript, however, our purpose is to study a system of an ultrashort pulse
having time-varying dispersion, nonlinear effects and under a damping like
perturbation. Under the SM scheme, Eqn. (1) modified accordingly and rewritten
as

$\displaystyle iU_{t}+a_{1}D(t)\ U_{xx}-a_{2}U_{xt}+bR(t)|U|^{2}\ (U+i\
a_{2}U_{x})$ $\displaystyle=\Gamma(t)\ (U_{x}-i\ n\ U)$ (2)

here the time-variable coefficient parameters $D(t)$ and $R(t)$ represents
variation in dispersion and nonlinear effects respectively and $\Gamma(t)$ is
the gain parameter. The form of $\Gamma(t)$ in this manuscript is considered
as $\Gamma(t)=\frac{R(t)D^{\prime}(t)-R^{\prime}(t)D(t)}{2D(t)R(t)}$ and the
symbol prime (′) represents the derivative with respect to $t$. The first and
second terms on the right hand side of Eqn. (2) are for damping and gain (or
loss) of the medium. From hereon throughout the manuscript wherever we mention
FLE, we are referring to Eqn. (2). The two main objectives of our manuscript
are 1.) at first, we introduce a suitable bilinear scheme to obtain dark and
anti-dark soliton solutions of Eqn. (2). Thereafter, explore the soliton
solution under different SM schemes and provide a systematic procedure to
evaluate higher order soliton solutions. 2.) later, we are going to transform
an already gauge transformed form of Eqn. (2) which we call as DLFLE (Eqn.
(5)), into an equation of equivalent spin system called LLE. To the best of
our knowledge this work had not been published before for such a system.

To address the first objective, we at first, through a gauge transformation
followed by change in variables, convert Eqn. (2) into a simplified form
(DLFLE), then we propose a bilinear scheme and introduce an auxiliary function
to make the bilinear process more convenient. Our proposed scheme transforms
the higher order nonlinear equation into three bilinear forms. We obtain dark
and anti-dark 1SS of DLFLE. Now, by reverse transforming the obtained 1SS, we
get the 1SS of FLE. Then we mention the criteria upon which the nature of
soliton becomes dark or anti-dark and also provide the graphical
representation under different SM schemes. We also obtain 2SS of DLFLE and
determine the shift in phase of the two individual solitons upon interaction.
Eventually, we obtain the 2SS of FLE. In our previous paper [8], we had
implemented an analogous scheme to realize soliton solution of FLE (DLFLE form
to be precise), but for constant coefficients ($D(t)=R(t)=1$) and now we
extend the scheme to realize the soliton of FLE under SM (Eqn. (2)).

Again, it is observed that various nonlinear equations shows gauge equivalence
with spin system equations (referred as LLE) [38, 24, 12, 11], even DLFLE
having $D(t)=R(t)=1$ shows gauge equivalence LLE [8]. LLE can provide
interesting features about a system which are important from physical and
mathematical point of view [26, 15]. We now introduce a Lax pair for DLFLE and
from that to accomplish our second objective, we are going to search for a
gauge transformation which transforms the spectral problem of DLFLE into a
spectral problem of LLE.

This manuscript is arranged as: in the following section we consider a gauge
transformation followed by change in variables that will convert Eqn. (2) into
DLFLE (Eqn. (5)). Then using Hirota bilinearization, we derive dark and anti-
dark 1SS of DLFLE and eventually 1SS of FLE and present the graphs under
various SM scheme. Then, derive the multi-soliton solutions and perform
asymptotic analysis. In the third section we propose a Lax pair for DLFLE and
through a gauge transformation we transform the system into a gauge equivalent
LLE. The fourth section will conclude the manuscript.

##  2 Bilinearization of FLE with non-vanishing background

The transformation of Eqn. (2) with respect to the variables ($x$ and $t$)
gives us the liberty to choose the constant coefficients ($a_{1}$, $a_{2}$,
$b$ and $n$) as per our convenience since the coefficients can be absorbed in
the process of variable transformation. Assuming $n=\frac{1}{a_{2}}$,
$m=\frac{a_{1}}{a_{2}}>0$ and positive $b$, consider the gauge transformation

$\displaystyle U=\sqrt{\frac{m}{b}}ne^{i(nx+2mnt)}u$ (3)

followed by the transformation of variables

$\displaystyle\xi=2(x+mt),\quad\tau=-\frac{mn^{2}}{2}t$ (4)

and for the time-varying parameters

$\displaystyle D(t)\rightarrow D(\tau),\quad R(t)\rightarrow
R(\tau),\quad\Gamma(t)\rightarrow\Gamma(\tau)$

we get the following equation

$\displaystyle u_{\xi\tau}-D(\tau)\ u+2i\ R(\tau)\ |u|^{2}u_{\xi}$
$\displaystyle=\Gamma(\tau)\ u_{x}$ (5)

here $u$ is the field function corresponding to the new transformed system.
For constant dispersion and nonlinear effects, i.e. constant coefficients
($D(t)=R(t)=1$), Eqn. (5) is the first negative hierarchy of DNLSE and is
similar to an equation governing the dynamics of short-wavelength ion-
cyclotron waves in plasma [7, 27]. Davydova and Lashkin first realized this
and hence in case of constant coefficients, Eqn. (5) can be termed as Davydova
Lashkin FLE (DLFLE). In this manuscript, we refer Eqn. (5) as DLFLE, even
though $D(t)$ and $R(t)$ are not constants. For a constant
$\Gamma(\tau)=\Gamma$, the right hand side of Eqn. (5) takes the form of a
perturbation representing linear damping [2].

For a soliton solution of DLFLE, we assume a non-vanishing background
condition $u\rightarrow\rho\ e^{i\ (\kappa\xi+\omega(\tau))}$ as
$\xi\rightarrow\pm\infty$. Under this situation, we expect dark and anti-dark
soliton solutions of Eqn. (5) through bilinearization.

To write Eqn. (5) in the bilinear form let us assume

$\displaystyle u$ $\displaystyle=\sqrt{\frac{D(\tau)}{R(\tau)}}\ \frac{g}{f}$
(6)

where $g$ and $f$ are two complex functions of ($\xi,\tau$). Consequently,
Eqn. (5) becomes

$\frac{1}{f^{2}}(D_{\xi}D_{\tau}-D(\tau))g.f-\frac{g}{f^{3}}D_{\xi}D_{\tau}(f.f)+\frac{2i\
|g|^{2}}{f^{3}f^{*}}D(\tau)D_{\xi}(g.f)+\frac{g\lambda
f.f}{f^{3}}-\frac{\lambda g.f}{f^{2}}+\\\
\frac{s|g|^{2}}{f^{3}}-\frac{s|g|^{2}f^{*}}{f^{3}f^{*}}=0$ (7)

where $D_{\xi}$, $D_{\tau}$ are Hirota derivatives [18] and are defined as

$\displaystyle
D_{\xi}^{m}D_{\tau}^{n}g(\xi,\tau).f(\xi,\tau)=(\frac{\partial}{\partial\xi}-\frac{\partial}{\partial\xi^{\prime}})^{m}(\frac{\partial}{\partial\tau}-\frac{\partial}{\partial\tau^{\prime}})^{n}g(\xi,\tau).f(\xi^{\prime},\tau^{\prime})\Bigg{|}_{(\xi=\xi^{\prime})(\tau=\tau^{\prime})}$
(8)

Notice that the last two terms in Eqn. (7) contain an auxiliary function $s$
which is introduced so that the multilinear Eqn. (7) can be cast into three
bilinear equations given below (Eqns. (9) - (11)). Here, the constant
$\lambda$ will also be determined while solving the three equations. The
bilinear equations in terms of $g$, $f$ and $s$ are

$\displaystyle(D_{\xi}D_{\tau}-D(\tau)-\lambda)g.f$ $\displaystyle=0$ (9)
$\displaystyle(D_{\xi}D_{\tau}-\lambda)f.f$ $\displaystyle=sg^{*}$ (10)
$\displaystyle 2i\ D(\tau)D_{\xi}(g.f)$ $\displaystyle=sf^{*}$ (11)

Now to obtain soliton solution, the expansions of $g$, $f$ and $s$ with
respect to an arbitrary parameter $\epsilon$ are as follows

$\displaystyle g$ $\displaystyle=g_{0}\
(1+\epsilon^{2}g_{2}+\epsilon^{4}g_{4}+...),\quad\quad\quad
f=1+\epsilon^{2}f_{2}+\epsilon^{4}f_{4}+...$ (12) $\displaystyle s$
$\displaystyle=s_{0}\ (1+\epsilon^{2}s_{2}+\epsilon^{4}s_{4}+...)$ (13)

###  2.1 Dark and Anti-Dark 1SS

To obtain the 1SS of DLFLE, we drop the terms of order greater than or equal
to $\epsilon^{3}$ in $g$, $f$ and $s$. Thus, Eqn. (6) can be represented as

$\displaystyle u$ $\displaystyle=\sqrt{\frac{D(\tau)}{R(\tau)}}\
\frac{g_{0}(1+\epsilon^{2}g_{2}^{(1)})}{1+\epsilon^{2}f_{2}^{(1)}}\Big{|}_{\epsilon=1}$
(14)

Let us consider the expressions for $g_{0}$, $s_{0}$, $g_{2}^{(1)}$,
$s_{2}^{(1)}$ and $f_{2}^{(1)}$ are as follows

$\displaystyle g_{0}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega(\tau))}$
(15) $\displaystyle s_{0}$ $\displaystyle=\rho_{s}\ e^{i\
(\kappa\xi+\omega(\tau))}$ (16) $\displaystyle g_{2}^{(1)}$
$\displaystyle=K_{1}\ e^{\theta_{1}+\theta_{1}^{*}}$ (17) $\displaystyle
s_{2}^{(1)}$ $\displaystyle=M_{1}\ e^{\theta_{1}+\theta_{1}^{*}}$ (18)
$\displaystyle f_{2}^{(1)}$ $\displaystyle=T_{1}\
e^{\theta_{1}+\theta_{1}^{*}}$ (19)

where $\theta_{1}$ = $p_{1}\ \xi+\Omega_{1}\int D(\tau)d\tau$. $p_{1}$,
$\Omega_{1}$, $K_{1}$, $M_{1}$, $T_{1}$ are complex parameters. Let us
consider, $p_{1}$ = $p_{1r}+i\ p_{1i}$ and $T_{1}$ = $T_{1r}+i\ T_{1i}$ where
$p_{1r}$, $p_{1i}$, $T_{1r}$ and $T_{1i}$ are real. On substituting the above
expressions into Eqns. (9) - (11) yield the following

$\displaystyle\rho_{s}$ $\displaystyle=-2\ \kappa\rho$ (20)
$\displaystyle\lambda$ $\displaystyle=2\ \kappa\rho^{2}$ (21)
$\displaystyle\omega(\tau)$ $\displaystyle=(-\frac{1}{\kappa}-2\ \rho^{2})\int
D(\tau)d\tau$ (22) $\displaystyle\Omega_{1}$
$\displaystyle=\frac{h_{1}}{p_{1}}$ (23) $\displaystyle M_{1}$
$\displaystyle=\frac{T_{1}^{2}}{K_{1}^{*}}$ (24) $\displaystyle K_{1}$
$\displaystyle=\gamma_{1}\ T_{1}^{*}$ (25)

where $h_{1}$ is real and $\gamma_{1}$ is a complex of absolute value 1, and
these two are represented as

$\displaystyle\gamma_{1}$
$\displaystyle=\frac{(p_{1}+p_{1}^{*})T_{1}-i(T_{1}-T_{1}^{*})\kappa}{(p_{1}+p_{1}^{*})T_{1}^{*}-i(T_{1}-T_{1}^{*})\kappa}$
(26) $\displaystyle h_{1}$ $\displaystyle=-\frac{|p_{1}|^{2}\
(-1+\gamma_{1})^{2}\ \ \kappa\rho^{2}}{(p_{1}+p_{1}^{*})^{2}\gamma_{1}}$ (27)

and the system obeys the constraint

$\displaystyle p_{1r}^{2}\ |T_{1}|^{2}+2p_{1r}\ \kappa\ (1+\kappa\rho^{2})\
T_{1r}\ T_{1i}+\kappa^{2}\ (1+\kappa\rho^{2})\ T_{1i}^{2}$ $\displaystyle=0$
(28)

now keeping either $T_{1r}$ or $T_{1i}$ fixed, we can calculate the other from
Eqn. (28). And this will also lead to a restriction $\ |p_{1r}|\ \leq\
\sqrt{\kappa^{3}\rho^{2}\ (1+\kappa\rho^{2})}$. It is interesting to point out
that in case of $D(t)=R(t)=1$, the obtained expressions for $g$, $f$ align
with the results already deduced for DLFLE having constant coefficients [8],
which is expected.

As already mentioned, the soliton obtained will always have a background of
$\rho$, and the nature of soliton being dark (dip in the background) or anti-
dark (bulge on the background) depends upon the parameters $p_{1r}$, $\kappa$
and $T_{1i}$. For positive $p_{1r}$, when $\kappa$ and $T_{1i}$ have different
signs we get a dark soliton and for the same sign of $\kappa$ and $T_{1i}$ the
soliton becomes anti-dark. For the case of negative $p_{1r}$ the criteria
become vice versa. This statement is supported by the expression of the
amplitude ($A_{1}$) given as

$\displaystyle A_{1}$ $\displaystyle=\rho\
\Big{|}\sqrt{\frac{D(\tau)}{R(\tau)}}\ \frac{T_{1}+\gamma_{1}\
|T_{1}|}{T_{1}+|T_{1}|}\Big{|}$ (29)

For the mentioned criteria of anti-dark soliton in the above paragraph,
$A_{1}$ is greater than $\rho$ and for the dark soliton criteria, $A_{1}$ is
less than $\rho$. Now, the obtained 1SS $u$ (Eqn. (14)) is a solution of
DLFLE). By replacing $\xi$ and $\tau$ in terms of $x$ and $t$ from Eqn. (4)
and putting the expression of $u$ inside of $U$ in Eqn. (3), we get the 1SS of
FLE. One important and obvious detail we like to mention at this point that
all the anti-dark 1SS of FLE presented in Figs. (1) - (6) have parameters
$p_{1r}=-10$, $\kappa=5$, $T_{1r}=-6$ and for all the dark 1SS graphs, we have
$p_{1r}=10$, $\kappa=-5$, $T_{1r}=6$. $T_{1r}$ is calculated using Eqn. (28).
And for all the graphs, we fix $\rho=2$, $m=0.2$, $n=4$, $b=1$,
$a_{2}=\frac{1}{n}$, $a_{1}=m\ a_{2}$.

Again the role of $D(\tau)$, $R(\tau)$ and $\Gamma(\tau)$ very much impact the
structure and propagation of the solitons. The existence of $\Gamma(\tau)$
also affects the background of the system. Depending upon the relationship
between $D(\tau)$ and $R(\tau)$, $\Gamma(\tau)$ either vanishes or exists. In
the following subsections (2.1.1) - (2.1.5) we are going to drive into some of
the different SM scheme.

####  2.1.1 $D(t)=R(t)=1$ (or some other constants) and $\Gamma(t)$ vanishes

$D(t)$ and $R(t)$ are linearly dependent functions, hence $\Gamma(t)=0$. The
background in this case remains unaffected since the gain parameter is zero.
Fig. (1(a)) demonstrates this condition for anti-dark 1SS in a 3D plot. Fig.
(1(b)) demonstrates the same but for dark 1SS.

(a) Anti-Dark 1SS

(b) Dark 1SS

Figure 1: 3D plot representation of 1SS under SM scheme: $D(t)=R(t)=1$. (a)
represents anti-dark soliton and (b) represents dark soliton.

####  2.1.2 $D(t)=1$, $R(t)=1+\sigma cos(kt)$ and $\Gamma(t)$ is non-vanishing

$D(t)$ and $R(t)$ are linearly independent functions, hence $\Gamma(t)\neq 0$.
The background in this case becomes sinusoidal for the contribution of cosine
function in $R(t)$ and the gain parameter being non-zero. Figs. (2(a)) -
(2(b)) demonstrate this condition. We keep the value of $\sigma$ below 1 to
avoid the scenario $R(t)\to 0$ and eventually $u,U\to\infty$, which can be
problematic in graphical presentation. If $D(t)$ and $R(t)$ interchange their
values than the sinusoidal nature of the graph reserves (ups and downs get
interchanged) as demonstrated in Fig. (2(c)). And for the same expressions of
$D(t)$ and $R(t)$ (= $1+\sigma cos(kt)$), implies $\Gamma(t)=0$, then for a
small value of $\sigma$, the graph will become similar to that of Figs. (1(a))
and (1(b)).

(a) Anti-Dark 1SS

(b) Dark 1SS

(c) Anti-Dark 1SS

Figure 2: 3D plot representation of 1SS under SM scheme: $D(t)=1$,
$R(t)=1+\sigma cos(kt)$ with $\sigma=0.1$ and $k=10\pi$. (a) represents anti-
dark soliton and (b) represents dark soliton. (c) represents anti-dark soliton
but under SM scheme $D(t)=1+\sigma cos(kt)$, $R(t)=1$.

####  2.1.3 $D(t)=1$, $R(t)=1+\sigma e^{(-kt^{2})}$ and $\Gamma(t)$ is non-
vanishing

$D(t)$ and $R(t)$ are linearly independent functions, hence $\Gamma(t)\neq 0$.
In this case the background has a depth for the Gaussian term $\sigma
e^{(-kt^{2})}$ in $R(t)$ and the gain parameter being non-zero. Figs. (3(a)) -
(3(b)) demonstrate this condition. If $D(t)$ and $R(t)$ interchange their
values than the background will have a bulge instead of a depth as
demonstrated in Fig. (3(c)). Again for the same expressions of $D(t)$ and
$R(t)$ (= $1+\sigma e^{(-kt^{2})}$), $\Gamma(t)=0$, then for a small value of
$\sigma$, the graph will become similar to that of Figs. (1(a)) and (1(b)).

(a) Anti-Dark 1SS

(b) Dark 1SS

(c) Anti-Dark 1SS

Figure 3: 3D plot representation of 1SS under SM scheme: $D(t)=1$,
$R(t)=1+\sigma e^{(-kt^{2})}$ with $\sigma=0.25$ and $k=100$. (a) represents
anti-dark soliton and (b) represents dark soliton. (c) represents anti-dark
soliton but under SM scheme: $D(t)=1+\sigma e^{(-kt^{2})}$, $R(t)=1$.

####  2.1.4 $D(t)=R(t)=e^{(\sigma t)}\ cos(kt)$ and $\Gamma(\tau)$ vanishes

$D(\tau)$ and $R(\tau)$ are linearly dependent functions, hence
$\Gamma(\tau)=0$. In this case the background is unaffected. $D(\tau)$ and
$R(\tau)$ are periodic functions and this makes the propagation of the soliton
harmonic from mean position ($x=0$) along the time ($t$) axis. Figs (4(a)) and
(4(b)) demonstrate this case. The amplitude remains constant along the
propagation of the soliton.

(a) Anti-Dark 1SS

(b) Dark 1SS

Figure 4: 3D plot representation of 1SS under SM scheme: $D(t)=R(t)=e^{(\sigma
t)}\ cos(kt)$ with $\sigma=0.25$ and $k=\pi$. (a) represents anti-dark soliton
and (b) represents dark soliton.

####  2.1.5 $D(t)=e^{(\sigma t)}\ cos(kt)$ and $R(t)=cos(kt)$ and $\Gamma(t)$
is non-vanishing

$D(\tau)$ and $R(\tau)$ are linearly independent functions and the gain
parameter is a non-zero constant, $\Gamma(\tau)=\frac{\sigma}{2}$. In this
case the background has a constant gain (or lose) of $\frac{\sigma}{2}$
factor. Just like the previous case (subsection (2.1.4)), here also $D(\tau)$
and $R(\tau)$ are periodic functions and thus the propagation of soliton is
harmonic from mean position along the time axis.Figs (5(a)) and (5(b))
demonstrate this case. Amplitude amplification is clearly visible from the
graphs unlike the soliton in Figs. (4) where the amplitude remains constant.

(a) Anti-Dark 1SS

(b) Dark 1SS

Figure 5: 3D plot representation of 1SS under SM scheme: $D(t)=e^{(\sigma t)}\
cos(kt)$ and $R(t)=cos(kt)$ with $\sigma=0.25$ and $k=\pi$. (a) represents
anti-dark soliton and (b) represents dark soliton.

Now, exploring the last two cases, i.e. subsections (2.1.4) and (2.1.5), we
find that $\sigma$ controls the deviation of the soliton from the mean
position and $k$ can control the frequency of the soliton and can also keep in
check the deviation. In the last subsection (2.1.5) which is of a non-
vanishing constant gain ($\Gamma(t)=\frac{\sigma}{2}$), $\sigma$ can increase
(or decrease) the amplitude of the soliton and also affect the background of
the system. Comparing Figs. (6(a)) and (6(b)), we can see increasing $\sigma$
from $0.1$ to $0.3$, increases the deviation of soliton from mean position.
Again, increasing $k$ from $\pi$ to $2\pi$, increases the frequency of
deviating from mean along $t$ axis and also reduces the magnitude of deviation
for same time propagation as we can see comparing Figs. (6(c)) with (6(b)).
The same are represented as density plots in Figs. (6(d)) - (6(f)) for a more
clear view. Keeping in mind the distance required for the soliton to travel,
one can fix the gain parameter in such a way that the background change is
bearable and the soliton also gain some desirable amplification. With this
knowledge while constructing such a SM FLE system, the soliton structure and
the propagation behavior can be tuned according to one’s requirements. This is
a big advantage that the SM scheme provides to a FLE system.

(a) Anti-Dark 1SS

(b) Dark 1SS

(c) Anti-Dark 1SS

(d) Anti-Dark 1SS

(e) Dark 1SS

(f) Anti-Dark 1SS

Figure 6: Graphical representation of 1SS under SM scheme: $D(t)=e^{(\sigma
t)}\ cos(kt)$ and $R(t)=cos(kt)$. (a), (b) and (c) are 3D plot representation
and (d), (e) and (f) are density plot representation. (a) and (d): anti-dark
soliton with $\sigma=0.05$ and $k=10\pi$, (b) and (e): dark soliton with
$\sigma=0.5$ and $k=10\pi$ and (c) and (f): anti-dark soliton with
$\sigma=0.5$ and $k=40\pi$.

###  2.2 Dark and Anti-Dark 2SS

In this subsection, we attempt to obtain the 2SS of DLFLE and eventually 2SS
of FLE. For this we drop the terms of order greater than or equal to
$\epsilon^{5}$ in $g$, $f$ and $s$. Hence, Eqn. (6) for 2SS can be rewritten
as

$\displaystyle u$
$\displaystyle=\sqrt{\frac{D(\tau)}{R(\tau)}}\frac{g_{0}(1+\epsilon^{2}g_{2}^{(2)}+\epsilon^{4}g_{4}^{(2)})}{1+\epsilon^{2}f_{2}^{(2)}+\epsilon^{4}f_{4}^{(2)}}\Big{|}_{\epsilon=1}$
(30)

Let us consider the expressions for $g_{2}^{(2)}$,
$g_{4}^{(2)}$,$s_{2}^{(2)}$, $s_{4}^{(2)}$, $f_{2}^{(2)}$ and $f_{4}^{(2)}$ in
2SS are as follows

$\displaystyle g_{2}^{(2)}$ $\displaystyle=K_{1}\
e^{\theta_{1}+\theta_{1}^{*}}+K_{2}\ e^{\theta_{2}+\theta_{2}^{*}}$ (31)
$\displaystyle g_{4}^{(2)}$ $\displaystyle=K_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (32) $\displaystyle
s_{2}^{(2)}$ $\displaystyle=M_{1}\ e^{\theta_{1}+\theta_{1}^{*}}+M_{2}\
e^{\theta_{2}+\theta_{2}^{*}}$ (33) $\displaystyle s_{4}^{(2)}$
$\displaystyle=M_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (34) $\displaystyle
f_{2}^{(2)}$ $\displaystyle=T_{1}\ e^{\theta_{1}+\theta_{1}^{*}}+T_{2}\
e^{\theta_{2}+\theta_{2}^{*}}$ (35) $\displaystyle f_{4}^{(2)}$
$\displaystyle=T_{12}\
e^{\theta_{1}+\theta_{1}^{*}+\theta_{2}+\theta_{2}^{*}}$ (36)

where $\theta_{j}$ = $p_{j}\ \xi+\Omega_{j}\ \tau$. $p_{j}$, $\Omega_{j}$,
$K_{j}$, $M_{j}$, $T_{j}$, $K_{12}$, $M_{12}$, $T_{12}$ are complex parameters
($j=1,2$). We assume $p_{j}$ = $p_{jr}+i\ p_{ji}$ and $T_{j}$ = $T_{jr}+i\
T_{ji}$ where $p_{jr}$, $p_{ji}$, $T_{jr}$ and $T_{ji}$ are real. Now,
substituting the above equations into Eqns. (9) - (11), yields the following
expressions

$\displaystyle\Omega_{j}$ $\displaystyle=\frac{h_{j}}{p_{j}}$ (37)
$\displaystyle M_{j}$ $\displaystyle=\frac{T_{j}^{2}}{K_{j}^{*}}$ (38)
$\displaystyle M_{12}$ $\displaystyle=\frac{T_{12}^{2}}{K_{12}^{*}}$ (39)
$\displaystyle K_{j}$ $\displaystyle=\gamma_{j}\ T_{j}^{*}$ (40)
$\displaystyle K_{12}$ $\displaystyle=\gamma_{1}\ \gamma_{2}\ T_{12}^{*}$ (41)
$\displaystyle T_{12}$ $\displaystyle=c_{12}\ T_{1}\ T_{2}$ (42)

the expressions for $\gamma_{j}$’s and $h_{j}$’s are similar to Eqns. (26) and
(27) respectively as

$\displaystyle\gamma_{j}$
$\displaystyle=\frac{(p_{j}+p_{j}^{*})T_{j}-i(T_{j}-T_{j}^{*})\kappa}{(p_{j}+p_{j}^{*})T_{j}^{*}-i(T_{j}-T_{j}^{*})\kappa}$
(43) $\displaystyle h_{j}$ $\displaystyle=-\frac{|p_{j}|^{2}\
(-1+\gamma_{j})^{2}\ \ \kappa\rho^{2}}{(p_{j}+p_{j}^{*})^{2}\gamma_{j}}$ (44)

and $c_{12}$ is real, expressed as

$\displaystyle c_{12}$ $\displaystyle=\frac{(p_{1r}\ T_{1r}\ T_{2i}-p_{2r}\
T_{1i}\ T_{2r})^{2}+(p_{1r}-p_{2r})^{2}\ T_{1i}^{2}\ T_{2i}^{2}}{(p_{1r}\
T_{1r}\ T_{2i}-p_{2r}\ T_{1i}\ T_{2r})^{2}+(p_{1r}+p_{2r})^{2}\ T_{1i}^{2}\
T_{2i}^{2}}$ (45)

the 2SS system is under two constraints (for $j=1,2$)

$\displaystyle p_{jr}^{2}\ |T_{j}|^{2}+2p_{jr}\ \kappa\ (1+\kappa\rho^{2})\
T_{jr}\ T_{ji}+\kappa^{2}\ (1+\kappa\rho^{2})\ T_{ji}^{2}$ $\displaystyle=0$
(46)

for $T_{jr}$’s and $T_{ji}$’s to be real, there must be restriction of $\
|p_{jr}|\ \leq\ \sqrt{\kappa^{3}\rho^{2}\ (1+\kappa\rho^{2})}$. The amplitude
($A_{j}$) of the jth soliton is given as

$\displaystyle A_{j}$ $\displaystyle=\rho\ \sqrt{\frac{D(\tau)}{R(\tau)}}\
\Big{|}\frac{T_{j}+\gamma_{j}\ |T_{j}|}{T_{j}+|T_{j}|}\Big{|}$ (47)

this is similar to Eqn. (29), and thus we infer that the nature of the jth
soliton to be dark or anti-dark is dependent upon the same criteria of
$\kappa$, $p_{jr}$, $T_{ji}$ as that in the 1SS case.

####  2.2.1 Asymptotic analysis of 2SS

Soliton has a unique characteristics which is that soliton upon interaction
with other soliton, retains all of its characteristics like shape, size, etc.
with a shift in its phase. To study the amplitudes and the phase shift of the
individual solitons of 2SS, we are going to perform asymptotic analysis. The
individual solitons of 2SS when asymptotically apart, are essentially two
separated 1SS. In the limit of before interaction $\tau\to-\infty$, fixing
$\theta_{1}$ to a finite value implies $|e^{\theta_{1}}|$ is finite and
$|e^{\theta_{2}}|\to\infty$. In this case, Eqn. (30) takes the form

$\displaystyle u_{1-}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega(\tau))}\
\sqrt{\frac{D(\tau)}{R(\tau)}}\ \frac{K_{2}\ (1+c_{12}\ K_{1}\
e^{\theta_{1}+\theta_{1}^{*}})}{T_{2}\ (1+c_{12}\ T_{1}\
e^{\theta_{1}+\theta_{1}^{*}})}$ (48) $\displaystyle\Rightarrow u_{1-}$
$\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega(\tau))}\
\sqrt{\frac{D(\tau)}{R(\tau)}}\ e^{i\phi_{1}}\ \frac{(1+K_{1}^{\prime}\
e^{\theta_{1}+\theta_{1}^{*}})}{(1+T_{1}^{\prime}\
e^{\theta_{1}+\theta_{1}^{*}})}$ (49)

here $K_{1}^{\prime}=c_{12}\ K_{1}$ and $T_{1}^{\prime}=c_{12}\ T_{1}$. Eqn.
(49) is simply an expression for 1SS with an additional phase $\phi_{1}$. The
fractional term $\frac{K_{2}}{T_{2}}$ in Eqn. (48) has a magnitude of 1 but a
phase difference of $\phi_{1}$ which results the phase term $e^{i\phi_{1}}$ in
Eqn. (49).

In the limit of after interaction $\tau\to\infty$ and for a fixed finite value
of $\theta_{1}$, we have a finite $|e^{\theta_{1}}|$ and $|e^{\theta_{2}}|\to
0$. This time, the corresponding asymptotic form of Eqn. (30) becomes

$\displaystyle u_{1+}$ $\displaystyle=\rho\ e^{i\ (\kappa\xi+\omega(\tau))}\
\sqrt{\frac{D(\tau)}{R(\tau)}}\ \frac{(1+K_{1}\
e^{\theta_{1}+\theta_{1}^{*}})}{(1+T_{1}\ e^{\theta_{1}+\theta_{1}^{*}})}$
(50)

we see Eqn. (50) is also in the form of 1SS just like Eqn. (49) with a
difference in phase. This phase shift $\phi_{1}$ is given as

$\displaystyle\phi_{1}$ $\displaystyle=tan^{-1}\Big{[}-\frac{2\kappa\
T_{i_{2}}^{2}\ (\kappa\ T_{r_{2}}\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})}{(\kappa\
(T_{r_{2}}-T_{i_{2}})\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})\ (\kappa\
(T_{r_{2}}+T_{i_{2}})\ T_{i_{2}}+p_{r_{2}}|T_{2}|^{2})}\Big{]}$ (51)

The amplitudes of both the Eqns. (49) and (50) are same and can be expressed
through Eqn. (29). This case represents a purely elastic collision between two
solitons since each of them retains all their characteristics with a slight
change in phase. We can perform the same analysis by keeping $\theta_{2}$
finite. This time also, the amplitude before and after interaction remains the
same and the phase shift can be represented in a similar way as in Eqn. (51)
but by replacing the components of $p_{2}$ and $T_{2}$ with the respective
components of $p_{1}$ and $T_{1}$. With this, we can infer that our system
DLFLE with a non-vanishing background condition supports elastic soliton
interaction.

Finally, just like in 1SS case, here also we get the 2SS of FLE by putting
Eqn. (30) in Eqn. (3). Figs. (7(a)) - (7(d)) demonstrate 2SS under SM scheme:
$D(t)=R(t)=1$ and hence $\Gamma(t)=0$. This implies that the background is
unaffected. We can observe from the two density plots (Figs. (7(c)) - (7(d)))
that the individual solitons even upon interaction with each other, propagates
with the same characteristics as was before interaction, but only a shift
occurs in their respective phases.

(a) Anti-dark-Anti-dark 2SS

(b) Dark-Dark 2SS

(c) Anti-dark-Anti-dark 2SS

(d) Dark-Dark 2SS

Figure 7: Graphical representation of 2SS under SM scheme: $D(t)=R(t)=1$. (a)
and (b) are 3D plot representation and (c) and (d) are density plot
representation. (a) and (c) represent anti-dark-anti-dark soliton with
$p_{1r}=-10$, $p_{2r}=-20$, $\kappa=5$, $T_{1r}=-6$, $T_{2r}=-9$ and (b) and
(d) represent dark-dark soliton with $p_{1r}=10$, $p_{2r}=20$, $\kappa=-5$,
$T_{1r}=6$, $T_{2r}=9$. $T_{1i}$ and $T_{2i}$ are calculated using Eqn. (46).
In all the four graphs, we fix $\rho=2$, $m=0.2$, $n=4$, $b=1$,
$a_{2}=\frac{1}{n}$, $a_{1}=m\ a_{2}$.

Now, for the SM scheme as demonstrated in subsection (2.1.5): $D(t)=e^{(\sigma
t)}\ cos(kt)$, $R(t)=cos(kt)$ and hence $\Gamma(t)=\frac{\sigma}{2}$, we
compare two 1SS represented in Figs. (8(a)) and (8(b)) with 2SS represented in
Fig. (8(c)). We notice that upon interactions, the individual solitons of 2SS
in Fig. (8(c)) repeatedly alternating their phases, thereby maintaining their
original trajectories as they would have without interactions for 1SS case, as
seen in Figs. (8(a)) and (8(b)). At a first glance of Fig. (8(c)), it might
appear that the two individual solitons are not interacting with each other
but in reality they are interacting with repeated phase shifts that their
trajectories appear to be of non interaction . Figs. (8(d)) - (8(f)) represent
the same in density plot.

(a) Anti-Dark 1SS

(b) Anti-Dark 1SS

(c) Anti-Dark 2SS

(d) Anti-Dark 1SS

(e) Anti-Dark 1SS

(f) Anti-Dark 2SS

Figure 8: Graphical representation of 2SS under SM scheme: $D(t)=e^{(\sigma
t)}\ cos(kt)$ and $R(t)=cos(kt)$ and $\Gamma(t)=\frac{\sigma}{2}$ with
$\sigma=0.25$ and $k=\pi$. (a), (b) and (c) are 3D plot representation and
(d), (e) and (f) are density plot representation. (a) and (d) represent anti-
dark 1SS with $p_{1r}=-10$, $T_{1r}=-6$, $T_{1i}$ from Eqn. (28), (b) and (e)
represent anti-dark 1SS with $p_{1r}=-20$, $T_{1r}=-9$, $T_{1i}$ from Eqn.
(28) and (c) and (f) represent anti-dark-anti-dark 2SS with $p_{1r}=-10$,
$p_{2r}=-20$, $T_{1r}=-6$, $T_{2r}=-9$, $T_{1i}$ and $T_{2i}$ from Eqn. (46).
In all the six graphs, we fix $\rho=2$, $\kappa=5$, $m=0.2$, $n=4$, $b=1$,
$a_{2}=\frac{1}{n}$, $a_{1}=m\ a_{2}$.

###  2.3 Scheme for obtaining NSS

In this subsection, we represent the scheme to obtain NSS of DLFLE and FLE.
For this, we have to drop the terms greater than or equal to $\epsilon^{2N+1}$
in $g$, $f$ and $s$. Eqn. (6) becomes

$\displaystyle u$ $\displaystyle=\sqrt{\frac{D(\tau)}{R(\tau)}}\
\frac{g^{(N)}}{f^{(N)}}$ (52)

the expansions of $g^{(N)}$ and $f^{(N)}$ are as follow

$\displaystyle g^{(N)}$ $\displaystyle=g_{0}\ \Big{(}1+\sum_{j=1}^{N}\ K_{j}\
e^{\theta_{j}+\theta_{j}^{*}}+\frac{1}{2}\sum_{j,l=1;\ j\neq l}^{N}K_{jl}\
e^{\theta_{j}+\theta_{j}^{*}+\theta_{l}+\theta_{l}^{*}}+...+K_{12...N}\ e^{\
\sum\limits_{j=1}^{N}\theta_{j}+\theta_{j}^{*}}\Big{)}$ (53) $\displaystyle
f^{(N)}$ $\displaystyle=1+\sum_{j=1}^{N}\ T_{j}\
e^{\theta_{j}+\theta_{j}^{*}}+\frac{1}{2}\sum_{j,l=1;\ j\neq l}^{N}T_{jl}\
e^{\theta_{j}+\theta_{j}^{*}+\theta_{l}+\theta_{l}^{*}}+...+T_{12...N}\ e^{\
\sum\limits_{j=1}^{N}\theta_{j}+\theta_{j}^{*}}$ (54)

where $\theta_{j}=p_{j}\ \xi+\Omega_{j}\ \tau$. $p_{j}$, $\Omega_{j}$,
parameters within $g^{(N)}$ (i.e. $K^{\prime}s$) and within $f^{(N)}$ (i.e.
$T^{\prime}s$) are complex parameters (subscript $j=1$, … , $N$). These
parameters can be calculated using the bilinear Eqns. (9) - (11). Just like
2SS has two constraints (Eqn. (46)), the NSS consists of N number of
constraints similar to that expressed in Eqn. (46) but for $j=1$, … , $N$. The
individual jth soliton of NSS has the same amplitude expression $A_{j}$ from
Eqn. (47) and obey the same criteria for dark and anti-dark soliton nature
with respect to the parameters $\kappa$, $p_{jr}$, $T_{ji}$ as already
discussed for 1SS and 2SS. And eventually to obtain NSS of FLE, we have to put
Eqn. (52) into Eqn. (3).

##  3 Gauge Connected Landau-Lifshitz equation

In this section we are going to establish the gauge connection between Eqn.
(5) with its counterpart LLE. To approach this, we exploit the equivalence of
the Lax pairs of the two systems (DLFLE and LLE). Let us consider the Lax pair
($L$, $M$) for Eqn. (5)

$\displaystyle\Psi_{\xi}=L\Psi\quad\quad\quad\Psi_{\tau}=M\Psi$ (55)

here $\Psi$ is the Jost function corresponding to the field function $u$ in
which ($L$, $M$) are given by

$\displaystyle L$
$\displaystyle=\frac{-i\zeta^{2}}{2}\Sigma+\sqrt{\frac{R(\tau)}{D(\tau)}}\
\zeta q_{\xi}$ (56) $\displaystyle M$
$\displaystyle=\frac{i}{2\zeta^{2}}D(\tau)\Sigma-\frac{i}{\zeta}\sqrt{D(\tau)\
R(\tau)}\ \Sigma q+iR(\tau)\ \Sigma q^{2}$ (57)

$\zeta$ is the spectral parameter, $\Sigma$ and $q$ are $2\times 2$ matrices
given as follow

$\quad\Sigma=\left(\begin{tabular}[]{c c}1&0\\\ 0&-1\\\
\end{tabular}\right),\quad\quad q=\left(\begin{tabular}[]{c c}0&u\\\
$-u^{*}$&0\\\ \end{tabular}\right)$
from Eqn. (55) follows the compatibility condition which is

$\displaystyle L_{\tau}-M_{\xi}+[L,M]=0$ (58)

Eqn. (58) is also called zero curvature equation. Under a local gauge
transformation we write a matrix $g$ such as
$g({\xi},{\tau},\zeta_{0})=\Psi({\xi},{\tau},\zeta)|_{\zeta=\zeta_{0}}$ and
the Jost function $\Psi$ transforms to $\Phi$ as
$\quad\Psi\rightarrow\Phi({\xi},{\tau},\zeta,\zeta_{0})=g({\xi},{\tau},\zeta_{0})^{-1}\Psi({\xi},{\tau},\zeta)$.
$\Phi$ is the Jost function corresponding to the spin field function $S$ of
the LL system. The new Lax pair ($L^{\prime}$, $M^{\prime}$) associated with
$\Phi$ written as

$\displaystyle\Phi_{\xi}=L^{\prime}\Phi\quad\quad\quad\Phi_{\tau}=M^{\prime}\Phi$
(59)

and the relation between ($L^{\prime}$, $M^{\prime}$) and ($L$, $M$) expressed
as

$\displaystyle L^{\prime}=g^{-1}Lg-g^{-1}g_{\xi}$ (60) $\displaystyle
M^{\prime}=g^{-1}Mg-g^{-1}g_{\tau}$ (61)

where

$\displaystyle g_{\xi}g^{-1}=L|_{\zeta=\zeta_{0}},\quad\quad\quad
g_{\tau}g^{-1}=M|_{\zeta=\zeta_{0}}$ (62)

the compatibility condition Eqn. (58) for the new system rewritten as

$\displaystyle L^{\prime}_{\tau}-M^{\prime}_{\xi}+[L^{\prime},M^{\prime}]=0$
(63)

Eqn. (63) is corresponding to the spin field function ($S$). $S$ is expressed
as

$\displaystyle S=g^{-1}\Sigma g$ (64)

and $S^{2}=I$. This leads to the expressions for the derivatives of $S$ namely
$S_{\xi}$, $S_{\tau}$ and $S_{\tau\xi}$ as

$\displaystyle S_{\xi}=g^{-1}[\Sigma,L_{0}]g\Rightarrow
S_{\xi}=2\zeta_{0}\sqrt{\frac{R(\tau)}{D(\tau)}}\ g^{-1}\Sigma q_{\xi}g$ (65)
$\displaystyle S_{\tau}=g^{-1}[\Sigma,M_{0}]g\Rightarrow
S_{\tau}=-\frac{2i}{\zeta_{0}}\sqrt{D(\tau)R(\tau)}\ g^{-1}qg$ (66)
$\displaystyle S_{\tau\xi}=2iR(\tau)(g^{-1}q_{\xi}\ qg-g^{-1}q\
q_{\xi}g)-\frac{2i}{\zeta_{0}}\sqrt{D(\tau)R(\tau)}\ g^{-1}q_{\xi}g$ (67)

using the above equations, the compatibility condition Eqn. (63) yields

$\displaystyle D(\tau)\ S_{\xi}+\frac{\zeta_{0}^{2}}{2i}\
[S,S_{{\tau}{\xi}}]=0$ (68)

This is the gauge connected LLE for the Eqn. (5). One important thing to note
is that Eqn. (68) is consistent for the system of Eqns. (59), (60), (61) in
any matrix $S$ of arbitrary dimension as long as $S$ satisfies $S^{2}=I$.

##  4 Conclusion

We have transformed FLE under SM into DLFLE and using Hirota bilinearization
we obtained 1SS and 2SS of DLFLE and under reverse transformation we obtained
1SS and 2SS of FLE. Under different SM scheme, we have explored the soliton
using graphical representation. We have also mentioned that under certain SM
scheme the soliton can be amplified accompanied by a certain amount of
affecting the background. This information can be utilized to construct a
desirable FLE system for ultrashort pulse with customized properties.
Thereafter, we have determined the shift in phase of the individual solitons
suffered upon interaction and also showed that the interaction was elastic. We
believe that the derived soliton solutions will be a useful information
carrier in digital and quantum communication where higher order effects like
STD and ND are taken into account along with GVD, KLNE. The proposed method
may be extended to other nonlinear models such as coupled FLE under SM to
obtain soliton solutions in multi-mode waveguide. This will be one of our
future objectives. Secondly, we have obtained a gauge equivalence between
DLFLE and LL spin system with the explicit construction of the new equivalent
Lax pair. LLE exhibits a variety of nonlinear structures like spin waves,
solitary wave, soliton, spatio-temporal patterns, etc. The derived LLE can be
useful to study analogous properties of FLE in a spin system.

Acknowledgments

R. Dutta and S. Talukdar acknowledge Department of Science & Technology, Govt.
of India for INSPIRE fellowship award. Corresponding award numbers DST/INSPIRE
Fellowship/2020/IF200303 and DST/INSPIRE Fellowship/2020/IF200278.

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## Statements & Declarations

Funding

R. Dutta and S. Talukdar receive fellowship from Department of Science &
Technology, Govt. of India under INSPIRE programme. Corresponding fellowship
award numbers DST/INSPIRE Fellowship/2020/IF200303 and DST/INSPIRE
Fellowship/2020/IF200278. Other than this, the authors did not receive any
fund support from any organization for the submitted work.

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Authors Contribution

Conceptualization: Riki Dutta, Sudipta Nandy; Formal Analysis: Riki Dutta;
Supervision: Sudipta Nandy; Validation: Sudipta Nandy, Sagardeep Talukdar,
Gautam K. Saharia; Writing - original draft: Riki Dutta; Writing - review &
editing: All the authors.