# Electrostatic wave interaction via asymmetric vector solitons as precursor
to rogue wave formation in non-Maxwellian plasmas

N. Lazarides  Khalifa University of Science and Technology, Department of
Mathematics, Abu Dhabi, P.O. Box 127788, United Arab Emirates
<EMAIL_ADDRESS>Giorgos P. Veldes  University of Thessaly,
Department of Physics, Lamia 35100, Greece  D. J. Frantzeskakis  Department of
Physics, National and Kapodistrian University of Athens, GR-15784 Zografou,
Athens, Greece  Ioannis Kourakis  Khalifa University of Science and
Technology, Department of Mathematics, Abu Dhabi, P.O. Box 127788, United Arab
Emirates  Space $\&$ Planetary Science Center, Khalifa University of Science
and Technology, Abu Dhabi, P. O. Box 127788, United Arab Emirates  Department
of Physics, National and Kapodistrian University of Athens, GR-15784 Zografou,
Athens, Greece  Hellenic Space Center, Leoforos Kifissias 178, Chalandri,
Athens, GR-15231, Greece

###### Abstract

An asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been
derived through a multiscale perturbation method applied to a plasma fluid
model, in which two wavepackets of distinct (carrier) wavenumbers ($k_{1}$ and
$k_{2}$) and amplitudes ($\Psi_{1}$ and $\Psi_{2}$) are allowed to co-
propagate and interact. The original fluid model was set up for a non-
magnetized plasma consisting of cold inertial ions evolving against a
$\kappa-$distributed electron background in one dimension. The reduction
procedure resulting in the CNLS equations has provided analytical expressions
for the dispersion, self-modulation and cross-coupling coefficients in terms
of the two carrier wavenumbers. These coefficients present no symmetry
whatsoever, in the general case (of different wavenumbers).

The possibility for coupled envelope (vector soliton) solutions to occur has
been investigated. Although the CNLS equations are asymmetric and non-
integrable, in principle, the system admits various types of vector soliton
solutions, physically representing nonlinear, localized electrostatic plasma
modes, whose areas of existence is calculated on the wavenumbers’ parameter
plane. The possibility for either bright (B) or dark (D) type excitations for
either of the (2) waves provides four (4) combinations for the envelope pair
(BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the
soliton parameters (maximum amplitude, width) are also calculated for each
type of vector soliton solution, in its respective area of existence.  The
dependence of the vector soliton characteristics on the (two) carrier
wavenumbers and on the spectral index $\kappa$ characterizing the electron
distribution has been explored. In certain cases, the (envelope) amplitude of
one component may exceed its counterpart (second amplitude) by a factor 2.5 or
higher, indicating that extremely asymmetric waves may be formed due to
modulational interactions among copropagating wavepackets.

As $\kappa$ decreases from large values, modulational instability occurs in
larger areas of the parameter plane(s) and with higher growth rates. The
distribution of different types of vector solitons on the parameter plane(s)
also varies significantly with decreasing $\kappa$, and in fact dramatically
for $\kappa$ between $3$ and $2$. Deviation from the Maxwell-Boltzmann picture
therefore seems to favor modulational instability as a precursor to the
formation of bright (predominantly) type envelope excitations and freak waves.

###### keywords:

Plasma fluid model, Multiscale perturbative reduction, Coupled nonlinear
Schrödinger equations, Modulational instability, Kappa distribution, Vector
Solitons

##  1 Introduction

A pair of nonlinearly coupled nonlinear Schrödinger equations (hereafter
referred to as the CNLS system of equations) arises as a prototype model of
mathematical physics, which occurs in various physical contexts [1, 2, 3, 4],
including water waves [5, 6], left-handed (negative refraction index)
transmission lines [7], optical pulse propagation in birefringence fibers [8,
9, 10], and in optical nonlinear media [11], vector solitons in left-handed
metamaterials [12], polarized pulse pair propagation in anisotropic dispersive
media [13], in electrically driven graphene multilayer mediums [14], pulse
propagation in isotropic Kerr media with chromatic dispersion [15], and even
breathers and rogue waves in optical fibers [16]. Formally similar systems of
equations have been used to model light (beam) propagation [17, 18, 19, 20]
and electrostatic or electromagnetic wave propagation in plasmas [21, 22, 23,
24, 25, 26, 27, 28]. Independently from a physical context, various studies of
vector solitons and rogue waves have been carried out, based on general CNLS
models [29, 30, 31, 32, 6, 33, 34], and variants of CNLS equations such as
coupled derivative nonlinear Schrödinger equations [35, 36], vector
($N-$component) CNLS [37, 38, 39], nonlocal CNLS [40], CNLS equations with
variable coefficients [41], coherently coupled CNLS equations [42], and
coupled high-order nonlinear Schrödinger equations [43], among others, have
been investigated with respect to vector solitons and rogue waves.

A pair of nonlinearly coupled CNLS equations was recently [27, 28] derived
from a plasma model consisting of a cold inertial ion fluid evolving against
an electron background. Generalizing an earlier study involving a thermal
(Maxwellian) electron distribution [27], the formalism has been recently [28]
adopted to electron population(s) that follows a kappa distribution [44, 45,
46, 47, 48, 49, 50, 51]. The kappa (family of) distribution functions (DF) is
characterized by a spectral index $\kappa$, and exhibits a high-energy tail in
the large (suprathermal) range of electron velocities. The analytical
expression of the kappa DF converges to the Maxwell-Boltzmann distribution for
infinite $\kappa$. Such distributions are a common occurrence in Space plasma
observations, e.g. in the solar wind [44, 45, 52] and in planetary
magnetospheres [53]. The original plasma fluid model and its lengthy algebraic
reduction to a pair of CNLS equations for the envelopes of two modulated
electrostatic wavepackets, by using a Newell type multi-scale perturbation
technique, is described in great detail in Refs. [27, 28], so the details will
be omitted in the following. The main outcome of that study, in the form of
the (six) coefficients involved in the resulting CNLS equations, will be
presented here, for completeness, in terms of the wavenumbers of the two
interacting waves and the spectral index $\kappa$ which characterizes the
electron distribution. Based on the system of CNLS equations obtained in those
earlier studies, our ambition in the paper at hand is to investigate the
existence of coupled localized envelope modes (vector solitons), from first
principles, and to explore their dependence on the intrinsic plasma
parameters, namely the two wavenumbers ($k_{1}$, $k_{2}$) and $\kappa$.

It must be pointed out that the CNLS system of equations that forms the basis
of our study is _not_ amenable to the widely studied (integrable) Manakov
system [54], unless identical carrier waves ($k_{1}=k_{2}$) are considered.
Indeed, for arbitrary wavenumbers of the two co-propagating waves, the
coefficients of the CNLS equations do not exhibit any known symmetry, hence
the system is rendered non-symmetric. For the same reason, any attempt for
reduction of the number of the coefficients of these CNLS equations cannot
give less than four coefficients [55, 56].

The CNLS system of equations in its general form admits vector soliton
solutions, rogue waves, and breathers, which can be obtained either
analytically or numerically. In this paper, within the context of the plasma
fluid model considered for electrostatic waves, we have obtained four
different types of vector solitons, whose components are actually combinations
of bright and dark type envelope solitons, i.e. reminiscent of solutions of
the single nonlinear Schrödinger equation. We have derived a set of analytical
conditions for the existence of such vector solitons, in terms of the various
coefficients (assumed to take arbitrary values), and we have subsequently
explored their parametric dependence on the relevant plasma parameters: the
two wavevectors and the spectral index $\kappa$. In each of these existence
regions, on the ($k_{1},k_{2}$) plane, we have also calculated the vector
soliton parameters, i.e., the envelope amplitudes and their (common) width.
These (amplitude and width) obviously vary, upon a variation of either of the
wavenumbers or the spectral index $\kappa$. Several illustrative examples are
shown in the following, in which a structural transition between a particular
type of a vector soliton (and its parameters) and another can be observed.
These transition may either be smooth, or take place through a divergence of
the width and the amplitudes at a transition point (boundary between different
regions). In certain cases, one of the components of a vector soliton may
acquire a very high amplitude with respect to that of its sister component. We
may characterize these solutions as extremely asymmetric vector solitons, a
configuration which has not been discussed before.

##  2 Asymmetric Nonlinear Schrödinger Equations and Coefficients

A plasma fluid model was considered in earlier work [28], as a basis to
describe electrostatic (ion-acoustic) waves. A non-magnetized plasma was
considered, consisting of a cold inertial ion fluid evolving against a
“thermalized" (highly energetic) electron background, in a one-dimensional
geometry. Given the large mass disparity between the electrons and the massive
ions, the former were assumed to be inertia-less, thus characterized by an
equilibrium configuration, modeled as a $\kappa-$type velocity distribution.
Two co-propagating wavepackets were considered, with wavevectors $k_{1}$ and
$k_{2}$, respectively. A Newell type multiple scale perturbation technique
led, after a tedious calculation [28], to the following pair of CNLS
equations:

$\displaystyle i\left(\frac{\partial\Psi_{1}}{\partial
t_{2}}+v_{g,1}\frac{\partial\Psi_{1}}{\partial
x_{2}}\right)+P_{1}\frac{\partial^{2}\Psi_{1}}{\partial
x_{1}^{2}}+\left(Q_{11}|\Psi_{1}|^{2}+Q_{12}|\Psi_{2}|^{2}\right)\Psi_{1}=0,$
(1) $\displaystyle i\left(\frac{\partial\Psi_{2}}{\partial
t_{2}}+v_{g,2}\frac{\partial\Psi_{2}}{\partial
x_{2}}\right)+P_{2}\frac{\partial^{2}\Psi_{2}}{\partial
x_{1}^{2}}+\left(Q_{21}|\Psi_{1}|^{2}+Q_{22}|\Psi_{2}|^{2}\right)\Psi_{2}=0\,,$
(2)

where

$P_{j}=-\frac{3}{2}\frac{c_{1}k_{j}}{(k_{j}^{2}+c_{1})^{5/2}}=\frac{1}{2}\frac{\partial^{2}\omega_{j}}{\partial
k_{j}^{2}},\qquad
v_{g,j}=\frac{c_{1}}{(k_{j}^{2}+c_{1})^{3/2}}=\frac{\partial\omega_{j}}{\partial
k_{j}},\qquad\omega_{j}=\frac{k_{j}}{\sqrt{k_{j}^{2}+c_{1}}}$ (3)

are the (linear) dispersion coefficients, the group velocities, and the
frequency dispersion relations, respectively. Note that
$P_{j}=\frac{1}{2}\frac{\partial^{2}\omega_{j}}{\partial k_{j}^{2}}$, in a way
formally analogous to the group-velocity-dispersion (GVD) terms known in
nonlinear optics. $Q_{11}$ and $Q_{22}$ are self-modulation coefficients and
$Q_{12}$ and $Q_{21}$ are cross-coupling coefficients. For the exact
expressions providing the (four) coefficients $Q_{ij}$ (for $i,j=$ 1 or 2) as
functions of $k_{1}$, $k_{2}$ and $\kappa$ (via the constants $c_{1}$,
$c_{2}$, and $c_{3}$), see the Supplementary Information part accompanying
this article. We emphasize that, unless $k_{1}=k_{2}$ (a stringent condition
that won’t be satisfied, in general), the above pairs of coefficients take
different values, viz. $P_{1}\neq P_{2}$, $Q_{11}\neq Q_{22}$ and $Q_{12}\neq
Q_{21}$.

The constants $c_{1}$, $c_{2}$, and $c_{3}$, incorporating the effect of
$\kappa$, are given as by [57]

$\displaystyle c_{1}=\frac{\kappa-\frac{1}{2}}{\kappa-\frac{3}{2}},\qquad
c_{2}=\frac{\left(\kappa-\frac{1}{2}\right)\left(\kappa+\frac{1}{2}\right)}{2!\left(\kappa-\frac{3}{2}\right)^{2}},\qquad
c_{3}=\frac{\left(\kappa-\frac{1}{2}\right)\left(\kappa+\frac{1}{2}\right)\left(\kappa+\frac{3}{2}\right)}{3!\left(\kappa-\frac{3}{2}\right)^{3}}.$
(4)

The above expressions are the first three coefficients in the Mc Laurin
expansion $n_{e}\simeq 1+c_{1}\phi+c_{2}\phi^{2}+c_{3}\phi^{3}+\cdots$, of the
electron number density

$n_{e}=\left(1-\frac{\phi}{\kappa-\frac{3}{2}}\right)^{-\left(\kappa-\frac{1}{2}\right)},$
(5)

that is obtained upon integrating the kappa distribution [44, 58, 52],
actually a straightforward replacement for the Maxwell-Boltzmann distribution
when dealing space and astrophysical plasmas [47, 59]. The nature and
composition of the plasmas allows for extracting the spectral index $\kappa$
in each particular case. Indeed, kappa distributions with $2<\kappa<6$ have
been found to fit the observations and satellite data in the solar wind [60],
among other successful examples of data fitting in space plasmas, making the
kappa distribution an ubiquitous paradigm in Space science [52].

The CNLS equations (1) and (2) can be transformed to

$\displaystyle
i\left(\frac{\partial\Psi_{1}}{\partial\tau}+\delta\frac{\partial\Psi_{1}}{\partial\xi}\right)+P_{1}\frac{\partial^{2}\Psi_{1}}{\partial\xi^{2}}+\left(Q_{11}|\Psi_{1}|^{2}+Q_{12}|\Psi_{2}|^{2}\right)\Psi_{1}=0,$
(6) $\displaystyle
i\left(\frac{\partial\Psi_{2}}{\partial\tau}-\delta\frac{\partial\Psi_{2}}{\partial\xi}\right)+P_{2}\frac{\partial^{2}\Psi_{2}}{\partial\xi^{2}}+\left(Q_{21}|\Psi_{1}|^{2}+Q_{22}|\Psi_{2}|^{2}\right)\Psi_{2}=0,$
(7)

after a change of the independent variables $x$ and $t$ (in which the
subscripts have been dropped) through $\xi=x-vt$ and $\tau=t$, with
$v=(v_{g,1}+v_{g,2})/2$ and $\delta=(v_{g,1}-v_{g,2})/2$ being is the half-sum
and the half-difference of the group velocities, respectively.

Note that the “walk-off” parameter $\delta$ is, by its definition, clearly a
function of both $k_{1}$ and $k_{2}$ (in addition to $\kappa$), through the
group velocities $v_{g,j}$ ($j=1,2$) of the two interacting wavepackets in the
plasma. Generally speaking, a large $\delta$ would be able to cause dynamic
instabilities in the system and eventually prevent the formation of various
types of vector solitons (to be discussed later in this article). However,
from the expression of the group velocities $v_{g,j}$ in Eq. (3), we may
observe that their values are limited in the interval
$[-1/\sqrt{c_{1}},+1/\sqrt{c_{1}}]$, where $c_{1}=c_{1}(\kappa)$ – defined in
(4) above – exceeds unity. It follow that, in the Maxwellian case
($\kappa=100$), $c_{1}=1$ and the extremal values of $\delta$ are $\pm 0.5$
(assuming co-propagating wavepackets), while for lower values of $\kappa$, the
corresponding extremal values of $\delta$ are (in absolute value) even
smaller, all the way down to zero (attained for $\kappa=3/2$). Therefore, the
walk-off parameter acquires small values, and is not expected to prevent
soliton formation.

Then, by applying the transformation
$\Psi_{1}=\bar{\Psi}_{1}\,\exp\left[i\left(\frac{\delta^{2}}{4P_{1}}\tau-\frac{\delta}{2P_{1}}\xi\right)\right]$
and
$\Psi_{2}=\bar{\Psi}_{2}\,\exp\left[i\left(\frac{\delta^{2}}{4P_{2}}\tau+\frac{\delta}{2P_{2}}\xi\right)\right]$
to Eqs. (6) and (7) we obtain the more familiar form

$\displaystyle
i\frac{\partial\bar{\Psi}_{1}}{\partial\tau}+P_{1}\frac{\partial^{2}\bar{\Psi}_{1}}{\partial\xi^{2}}+\left(Q_{11}|\bar{\Psi}_{1}|^{2}+Q_{12}|\bar{\Psi}_{2}|^{2}\right)\bar{\Psi}_{1}=0,$
(8) $\displaystyle
i\frac{\partial\bar{\Psi}_{2}}{\partial\tau}+P_{2}\frac{\partial^{2}\bar{\Psi}_{2}}{\partial\xi^{2}}+\left(Q_{21}|\bar{\Psi}_{1}|^{2}+Q_{22}|\bar{\Psi}_{2}|^{2}\right)\bar{\Psi}_{2}=0,$
(9)

where $\bar{\Psi}_{j}$ are complex functions of the new variables $\xi$ and
$\tau$.

##  3 Modulational Instability: Compatibility Condition and Growth Rate

Modulational instability (MI) analysis for two co-propagating plane-wave
solutions of the CNLS equations (6) and (7) can be performed using the
procedure described e.g., in Refs. [61]. The plane waves

$\Psi_{j}=\Psi_{j,0}\,e^{i\tilde{\omega}_{j}\tau},$ (10)

where $\Psi_{j,0}$ ($j=1,2$) is a constant real amplitude and
$\tilde{\omega}_{j}$ is an internal frequency, are solutions of the CNLS
equations for $\tilde{\omega}_{1}=Q_{11}\Psi_{1,0}^{2}+Q_{12}\Psi_{2,0}^{2}$
and $\tilde{\omega}_{2}=Q_{21}\Psi_{1,0}^{2}+Q_{22}\Psi_{2,0}^{2}$, and
constitute nonlinear modes that may become modulationally unstable in the
presence of a small amplitude perturbation of wavenumber $K$ and perturbation
frequency $\Omega$. By following the standard approach, we find that the above
nonlinear modes are unstable whenever the compatibility condition

$\left[(\Omega-\delta K)^{2}-\Omega_{1}^{2}\right]\left[(\Omega+\delta
K)^{2}-\Omega_{2}^{2}\right]=\Omega_{c}^{4},$ (11)

where $\Omega_{j}^{2}=P_{j}K^{2}\left(P_{j}K^{2}-2Q_{jj}\Psi_{j,0}^{2}\right)$
and $\Omega_{c}^{4}=4P_{1}P_{2}Q_{12}Q_{21}\Psi_{1,0}^{2}\Psi_{2,0}^{2}K^{4}$,
has at least one pair of complex conjugate roots ($j=1,2$). Then, the
(positive) imaginary part of these roots provides the growth rate $\Gamma$ of
the modulationally unstable modes. When Eq. (11) has two pairs of complex
conjugate roots, then the largest of their imaginary parts provide the growth
rate of the modulationally unstable modes, i.e.,

$\Gamma={\rm max}\\{{\rm Im}(\Omega_{r})\\},$ (12)

where $\Omega_{r}$ denotes one of the four roots of the compatibility
condition above.

The growth rate $\Gamma$ is calculated numerically for the CNLS equations by
finding the roots of the fourth degree (in $\Omega$) polynomial resulting from
the compatibility condition, Eq. (11). Then $\Gamma$ is mapped on the
$k_{1}-k_{2}$ parameter plane for several values of the perturbation
wavenumber $K$, and two values of the spectral index $\kappa$, i.e., for
$\kappa=2$ and $\kappa=3$. The amplitudes of the nonlinear wave modes in these
calculations are $\Psi_{1,0}=\Psi_{2,0}=0.1$.

Figure 1:  Maps of the growth rate $\Gamma$ on the $k_{1}-k_{2}$ plane for
$\kappa=2$, $\Psi_{1,0}=\Psi_{2,0}=0.1$, and perturbation wavenumber (a)
$K=0.1$; (b) $K=1.3$; (c) $K=1.6$; (d) $K=3.4$. The values of $K$ are chosen
so that they illustrate the variability of the growth rate patterns in the
best possible way.  Figure 2:  Maps of the growth rate $\Gamma$ on the
$k_{1}-k_{2}$ plane for $\kappa=3$, $\Psi_{1,0}=\Psi_{2,0}=0.1$, and
perturbation wavenumber (a) $K=0.1$; (b) $K=0.4$; (c) $K=1.9$; (d) $K=3.4$.
The values of $K$ are chosen so that they illustrate the variability of the
growth rate patterns in the best possible way.

In Figs. 1 and 2, maps of the growth rate $\Gamma$ are shown on the
$k_{1}-k_{2}$ parameter plane of the wavenumbers of the two co-propagating
(carrier) waves for spectral index $\kappa=2$ and $\kappa=3$, respectively.
These values of $\kappa$ were selected to be into the range of physically
acceptable values for space plasmas that have been observed to be in the
interval from $\kappa=2$ to $\kappa=6$. Moreover, they are more or less
symmetrically arranged around the value $\kappa\simeq 2.5$. In Fig. 1, in
particular, for $\kappa=2$, the system is modulationally unstable for low
values of the perturbation wavenumber $K$, for all the parameter plane shown
(Fig. 1(a)). At around $K\approx 1.2$, the first modulationally stable island
becomes visible, which acquires substantial area roughly in the middle of the
$k_{1}-k_{2}$ parameter plane for $K=1.3$ (blue color, Fig. 1(c)). Note that
there are also two narrow modulationally stable (blue) areas develop around
the curves where $Q_{12}$ and $Q_{21}$ are zero. For $K$ larger than $1.3$,
the modulationally stable areas grow and merge together into a single large
one, which continues to grow with increasing $K$ (Figs. 1(c)-(d)). For $K$
larger than $3.4$, almost all of the $k_{1}-k_{2}$ parameter plane shown in
Fig. 1 becomes modulationally stable. Note the strong modulational instability
in Fig. 1(d) that is limited at low $k_{1}$ and low $k_{2}$, where the
nonlinear coupling coefficients $Q_{12}$ and $Q_{21}$ acquire very large
values.

In Fig. 2, for $\kappa=3$, the system possess a modulationally stable area
even for low values of $K$, as e.g., can be observed in Fig. 2(a) (blue
color), which is close and around the diagonal $k_{1}=k_{2}$. Already at
$K=0.4$, the modulationally unstable area has grown significantly (Fig. 2(b)),
while it continues to grow with increasing value of $K$ (Figs. 2(c)-(d)). For
$K$ larger than $3.4$ almost all of the parameter plane shown is
modulationally stable. Again, the strongly unstable areas are limited in the
low $k_{1}$ and low $k_{2}$ areas of the plane.

##  4 Vector Solitons: Existence and their Parameters

The CNLS equations Eqs. (8) and (9) admit several types of vector soliton
solutions that are combinations of bright and dark soliton solutions of the
single NLS equation. Indeed, as we shall show below, four types of vector
solitons, i.e., bright-bright (BB), bright-dark (BD), dark-bright (DB), ans
dark-dark (DD), may exist in the CNLS system Eqs. (8) and (9). Their
parameters, i.e., their amplitudes $A_{1}$ and $A_{2}$, their (common) width
$b$, and their internal frequencies $\nu_{1}$ and $\nu_{2}$ are related
through simple mathematical expressions to the coefficients $P_{j}$ and
$Q_{ij}$ of the CNLS equations (8) and (9). Each component of these vector
solitons represent modulated electrostatic wavepackets which are moving in the
plasma and interact strongly and nonlinearly with strengths $Q_{12}$ and
$Q_{21}$.

The four types of vector solitons that may exist in the system of CNLS
equations are given below along with their parameters, i.e., their amplitudes
$A_{1}$ and $A_{2}$ and their width $b$, as a function of the CNLS
coefficients $P_{j}$ and $Q_{ij}$ ($j=1,2$) which are in turn functions of the
wavenumbers $k_{1}$ and $k_{2}$. Note that in the vector solitons expressions
below one of the parameters may take arbitrary values. Here, we choose the
amplitude $A_{1}$ as the free parameter and fix it to $A_{1}=0.1$ in what
follows.

Case I: Bright-bright (BB) vector solitons.
We seek for bright-bright (BB) vector solitons in the form

$\displaystyle\bar{\Psi}_{1}=A_{1}\,{\rm
sech}\left(b\xi\right)e^{-i\nu_{1}\tau},\qquad\bar{\Psi}_{2}=A_{2}\,{\rm
sech}\left(b\xi\right)e^{-i\nu_{2}\tau}.$ (13)

By substitution of Eqs. (13) into Eqs. (8) and (9) we obtain

$\left({A_{2}}/{A_{1}}\right)^{2}=-\alpha,\qquad\left({b}/{A_{1}}\right)^{2}=-\beta,$
(14)

and $\nu_{1}=-b^{2}P_{1}$, $\nu_{2}=-b^{2}P_{2}$, where

$\displaystyle\alpha=\frac{Q_{21}P_{1}-Q_{11}P_{2}}{P_{1}Q_{22}-P_{2}Q_{12}},\qquad\beta=\frac{1}{2}\frac{Q_{11}Q_{22}-Q_{21}Q_{12}}{P_{2}Q_{12}-P_{1}Q_{22}}.$
(15)

Case II: Bright-dark (BD) vector solitons.
We seek for bright-dark (BB) vector solitons in the form

$\displaystyle\bar{\Psi}_{1}=A_{1}\,{\rm
sech}\left(b\xi\right)e^{-i\nu_{1}\tau},\qquad\bar{\Psi}_{2}=A_{2}\,\tanh\left(b\xi\right)e^{-i\nu_{2}\tau}.$
(16)

By substitution of Eqs. (16) into Eqs. (8) and (9) we obtain

$\left({A_{2}}/{A_{1}}\right)^{2}=+\alpha,\qquad\left({b}/{A_{1}}\right)^{2}=-\beta,$
(17)

and $\nu_{1}=-b^{2}P_{1}-Q_{12}A_{2}^{2}$, $\nu_{2}=-Q_{22}A_{2}^{2}$, where
$\alpha$ and $\beta$ are given by Eqs. (15).

Case III: Dark-bright (DB) vector solitons.
We seek for dark-bright (DB) vector solitons in the form

$\displaystyle\bar{\Psi}_{1}=A_{1}\,\tanh\left(b\xi\right)e^{-i\nu_{1}\tau},\qquad\bar{\Psi}_{2}=A_{2}\,{\rm
sech}\left(b\xi\right)e^{-i\nu_{2}\tau}.$ (18)

By substitution of Eqs. (18) into Eqs. (8) and (9) we obtain

$\left({A_{2}}/{A_{1}}\right)^{2}=+\alpha,\qquad\left({b}/{A_{1}}\right)^{2}=+\beta,$
(19)

and $\nu_{1}=-\left(Q_{11}A_{1}^{2}+Q_{12}A_{2}^{2}\right)$,
$\nu_{2}=-\left(b^{2}P_{2}+Q_{21}A_{1}^{2}\right)$, where $\alpha$ and $\beta$
are given by Eqs. (15).

Case IV: Dark-dark (DD) vector solitons.

We seek for dark-bright (DB) vector solitons in the form

$\displaystyle\bar{\Psi}_{1}=A_{1}\,\tanh\left(b\xi\right)e^{-i\nu_{1}\tau},\qquad\bar{\Psi}_{2}=A_{2}\,\tanh\left(b\xi\right)e^{-i\nu_{2}\tau}.$
(20)

By substitution of Eqs. (20) into Eqs. (8) and (9) we obtain

$\left({A_{2}}/{A_{1}}\right)^{2}=-\alpha,\qquad\left({b}/{A_{1}}\right)^{2}=+\beta,$
(21)

and $\nu_{1}=-\left(Q_{11}A_{1}^{2}+Q_{12}A_{2}^{2}\right)$,
$\nu_{2}=-\left(Q_{21}A_{1}^{2}+Q_{22}A_{2}^{2}\right)$, where $\alpha$ and
$\beta$ are given by Eqs. (15).

Thus, the vector soliton parameters $A_{j}$, $b$, and $\nu_{j}$ are actually
themselves functions of the wavenumbers $k_{1}$ and $k_{2}$ of the two co-
propagating wavepackets through the CNLS coefficients $P_{j}$ and $Q_{ij}$
($j=1,2$). The simplest way to calculate them is to first fix one of them (one
of the amplitudes selected as reference, say $A_{1}$), and then calculate
$A_{2}$ and $b$ (e.g., from Eqs. (21) in Case IV), and last the frequencies
$\nu_{j}$.

For BB, BD, DB and DD vector solitons to exist in the CNLS system that we have
obtain from the particular plasma fluid model considered here, the right-hand-
sides of Eqs. (14), (17), (19), and (21), respectively, must be greater than
zero (so that $A_{2}$ and $b$ are real; $A_{1}$ is of course fixed to a real
number). This condition for the existence of the four types of vector solitons
can be expressed simply as

$\displaystyle\alpha<0,\,\,\,\beta<0,\,\,\,$ $\displaystyle{\rm
Case\leavevmode\nobreak\ I,\leavevmode\nobreak\ BB\leavevmode\nobreak\
vector\leavevmode\nobreak\ solitons,}$ (22)
$\displaystyle\alpha>0,\,\,\,\beta<0,\,\,\,$ $\displaystyle{\rm
Case\leavevmode\nobreak\ II,\leavevmode\nobreak\ BD\leavevmode\nobreak\
vector\leavevmode\nobreak\ solitons,}$ (23)
$\displaystyle\alpha>0,\,\,\,\beta>0,\,\,\,$ $\displaystyle{\rm
Case\leavevmode\nobreak\ III,\leavevmode\nobreak\ DB\leavevmode\nobreak\
vector\leavevmode\nobreak\ solitons,}$ (24)
$\displaystyle\alpha<0,\,\,\,\beta>0,\,\,\,$ $\displaystyle{\rm
Case\leavevmode\nobreak\ IV,\leavevmode\nobreak\ DD\leavevmode\nobreak\
vector\leavevmode\nobreak\ solitons\,.}$ (25)

Figure 3:  Existence areas for bright-bright (BB, black color), bright-dark
(BD, red color), dark-bright (DB, blue color), and dark-dark (DD, green
color), on the $k_{1}-k_{2}$ parameter plane, for $A_{1}=0.1$ and (a)
$\kappa=2$; (b) $\kappa=3$; (c) $\kappa=5$; (d) $\kappa=100$.  Figure 4:  The
same as Fig. 3 for values of $\kappa$ between $2$ and $3$. Existence areas for
bright-bright (BB, black color), bright-dark (BD, red color), dark-bright (DB,
blue color), and dark-dark (DD, green color), on the $k_{1}-k_{2}$ space, for
$A_{1}=0.1$, and (a) $\kappa=2.2$; (b) $\kappa=2.4$; (c) $\kappa=2.6$; (d)
$\kappa=2.8$.

In Fig. 3, we have identified on the $k_{1}-k_{2}$ plane those areas in which
each type of vector soliton may exist, for several values of the spectral
index $\kappa$ spanning a very wide range from $2$ to $100$ (for the latter,
the kappa distribution has practically converged to a Maxwell-Boltzmann one).
In Fig. 3(a), the black color that corresponds to BB vector solitons is
dominant, and occupies a large part of the plane except that for low $k_{1}$
and $k_{2}$ where BD (red color) and DB (blue color) vector solitons may
exist. Note that between the areas of existence of BB and DB, as well as
between BB and BD, DD vector solitons exist (green color) in two separate
narrow areas. Interestingly, the latter are grown around the path where
$Q_{12}$ and $Q_{21}$ are zero on the $k_{1}-k_{2}$ plane. For $\kappa=3$ or
greater, different patterns of existence of vector solitons appear (Fig.
3(b)), which typically consist of two large areas of existence of BD and DB
vector solitons (red and blue color, respectively) while in between of these
two there is a green area in which DD vector solitons exist. A small black
area in which BB solitons may exist also appears at the upper right corner of
the plane. This pattern appears for all larger values of $\kappa$, with only
slight quantitative differences observed in Figs. 3(c)-3(d)). Obviously, the
strongest variation of the existence patterns of vector solitons on the
$k_{1}-k_{2}$ plane occurs between $\kappa=2$ and $\kappa=3$. In order to
analyze this pattern variability, we present in Fig. 4 a series of patterns
for four (4) values of $\kappa$ between $2$ and $3$. Here we see how the
dominant (black) area of BB vector solitons gradually shrinks against both the
existence areas for BD and DB vector solitons (red and blue area,
respectively).

Figure 5:  Existence areas for bright-bright (BB, black color), bright-dark
(BD, red color), dark-bright (DB, blue color), and dark-dark (DD, green
color), as a function of $\kappa$ and $\delta$ for $A_{1}=0.1$ and (a)
$k_{1}=0.1$; (b) $k_{1}=0.4$; (c) $k_{1}=0.9$. The values of $\delta$ are
obtained by varying the wavenumber $k_{2}$ from $0$ to $2$ and the
corresponding fixed $k_{1}$.

It may be interesting to display the existence area for various types of
vector solitons on a plane in which one of the coordinates is the “walk-off”
parameter $\delta$. In Fig. 5, these existence areas are shown on the
$\kappa-\delta$ plane, for specific indicative values of the carrier
wavenumbers $k_{1}$ and $k_{2}$. The color code for vector solitons is the
same as that used in Figs. 3 and 4. The walk-off parameter is calculated by
setting $k_{1}$ to a fixed value, and then varying $k_{2}$ from $0$ to $2$. In
Fig. 5(a), dark-bright (DB, blue color) vector solitons are dominant in the
whole $\kappa$ interval. Small areas in which dark-dark (DD, green color) and
bright-dark (BD, red color) however exist are visible for small positive
values of $\delta$. Much richer patterns appear in Figs. 5(b) and (c), for
larger wavenumber $k_{1}$, in which all four types of vector solitons may
exist in substantial parts of the $\kappa-\delta$ plane. In both figures,
bright-bright (BB, black color) vector solitons exist in small areas of the
plane at low $\kappa$ ($\kappa<2.5$).

##  5 Illustrative Examples of Vector Solitons and their Parameters

It is tempting to investigate how the vector soliton characteristics vary when
one of the wavenumbers of the carrier waves. e.g., $k_{1}$ varies (while the
rest of the parameters as well as the amplitude $A_{1}$ remain fixed). In Fig.
6, the amplitude $A_{2}$ is plotted as a function of $k_{1}$ for two values of
the $\kappa$, i.e., for $\kappa=2$ (upper panels) and $\kappa=3$ (lower
panels), and for three different values of the wavenumber $k_{2}$ ($k_{2}=1$,
$1.5$, $1.95$). For the upper panels, the three values of $k_{2}$ correspond
to three “cuts” (sections) at $k_{2}=1$, $1.5$, $1.95$ of the $k_{1}-k_{2}$
plane in Fig. 4. As obvious in this figure, for all three values, DB type
vector solitons occur for low $k_{1}$, while upon increasing $k_{1}$ they
successively turn to DD and then BB soliton pairs.

Fig. 6(a)-(c) depicts the variation of the magnitude of (the amplitude)
$A_{2}$ as well as its behavior when boundaries of areas of existence of
different types of vector solitons are crossed. The three plots exhibit
similar behavior: the amplitude $A_{2}$ takes very high values for low $k_{1}$
(DB vector solitons) which gradually decreases with higher $k_{1}$ and almost
vanishes at the boundary between DB (blue) and DD (green) vector solitons
(existence regions). As soon as this boundary is crossed, the amplitude
$A_{2}$ increases again with increasing $k_{1}$, it reaches a maximum slightly
above $A_{1}=0.1$ and then decreases again slightly. Moreover, the second
boundary crossing, i.e. between DD and BB vector solitons, is a smooth and
continuous process as evident from the continuity of the green into the black
part(s) of the plotted curve. interestingly, for $k_{1}>0.5$, the two
amplitudes $A_{1}$ and $A_{2}$ are of the same order of magnitude.

The corresponding curves in the lower panels, obtained for $\kappa=3$, also
exhibit a vanishing amplitude at the boundary between the areas of existence
of DD to BD vector solitons. However, a different type of behavior is now
witnessed, as the amplitude $A_{2}$ diverges at the boundary separating BD-
from DD-type vector soliton regions (areas of existence): see Fig. 6(d). In
the former case, the variation of the amplitude $A_{2}$ through the
corresponding boundary is smooth, while in the latter it is not. In the same
figure, one also witnesses a smooth variation of $A_{2}$ through the boundary
between DB (blue) and BD (red) vector solitons, as well as a divergence of the
amplitude when crossing the boundary between the areas of existence of BD
(red) and DD (green) vector solitons. Similar remarks can be made for Figs.
6(e) and (f). In the latter figure, only one boundary crossing is visible for
the $k_{1}$ interval shown.

Figure 6:  The amplitudes of the vector solitons components $A_{1}$ and
$A_{2}$ as a function of the wavenumber $k_{1}$, calculated from the first of
Eqs. (21). The fixed amplitude $A_{1}=0.1$ is indicated by the horizontal
orange dashed line. (a) $\kappa=2$, $k_{2}=1$; (b) $\kappa=2$, $k_{2}=1.5$;
(c) $\kappa=2$, $k_{2}=1.95$; (d) $\kappa=3$, $k_{2}=1$; (e) $\kappa=3$,
$k_{2}=1.5$; (f) $\kappa=3$, $k_{2}=1.95$. Note that the same color code as in
Figs. 3 and 4 has been adopted in all curves shown, i.e. black/red/blue/green
color represents values prescribing BB/BD/DB/DD vector solitons, respectively.
Figure 7:  The width $b$ of the vector soliton components as a function of the
wavenumber $k_{1}$, calculated from the second of Eqs. (21). The fixed
amplitude $A_{1}=0.1$ is indicated by the horizontal orange dashed line. (a)
$\kappa=2$, $k_{2}=1$; (b) $\kappa=2$, $k_{2}=1.5$; (c) $\kappa=2$,
$k_{2}=1.95$; (d) $\kappa=3$, $k_{2}=1$; (e) $\kappa=3$, $k_{2}=1.5$; (f)
$\kappa=3$, $k_{2}=1.95$. Note that the same color code as in Figs. 3 and 4
has been adopted in all curves shown, i.e. black/red/blue/green color
represents values prescribing BB/BD/DB/DD vector solitons, respectively.

The (common) width $b$ of the vector soliton components associated to the
amplitudes $A_{2}$ (Fig. 6) are presented in Fig. 7. The width $b$ in all the
sub-panels in Fig. 7 takes very high values at low $k_{1}$, up to $k_{1}\simeq
0.1$, suggesting a very extended (spatially) – i.e. little localized –
solution, but with high amplitude $A_{2}$. (Recall that $A_{1}$ is fixed in
this figure.)

The behavior of $b$ with $k_{1}$ increasing beyond $0.1$ is then diversified
for the two values of the spectral index shown: $\kappa=2$ (upper panels) and
$\kappa=3$ (lower panels). Consider the former case first, i.e., that with
$\kappa=2$. For $k_{1}$ increasing above $0.1$, the width $b$ decreases almost
linearly but abruptly, passing smoothly through a boundary from DB (blue) to
DD (green) vector soliton existence areas. At approximately $k_{1}\simeq 0.3$,
the width $b$ reaches a very low value of the order of $0.05$, while another
boundary crossing between DD (green) to BB (black) vector solitons takes place
at that point. For further increasing $k_{1}$, the width $b$ increases again,
its exact behavior however this time depends on the selected value of the
second wavenumber $k_{2}$. In Fig. 7(a), for $k_{2}=1$, the width $b$ after a
linear increase it saturates to a value that is approximately constant around
$b\simeq 0.5$. Then, for $k_{1}$ increasing above $\leavevmode\nobreak\ 0.8$,
the width $b$ increases parabolically. For the two other values of $k_{2}$,
i.e., for $k_{2}=1.5$ and $k_{2}=1.95$ shown in Figs. 7(b) and (c),
respectively, the width $b$ as a function of $k_{1}$ does not form a plateau
but instead it reaches a maximum, then a shallow minimum, and finally it start
increasing parabolically upon further increasing $k_{1}$. Both the maximum and
minimum values of $b$ are higher for $k_{2}=1.95$, as compared to the case
with $k_{2}=1.5$.

Figure 8:  Variation of the vector soliton components as the spectral index
$\kappa$ takes the values (a) $\kappa=2$; (b) $\kappa=3$; (c) $\kappa=5$; (d)
$\kappa=100$, for a fixed point on the $k_{1}-k_{2}$ plane, i.e., for
$k_{1}=0.25$ and $k_{2}=1.5$. The envelops $\bar{\Psi}_{1}$ and
$\bar{\Psi}_{2}$ are plotted as a function of $\xi$.

For $\kappa=3$, as shown in Figs. 7(d)-(f), the width $b$ of the vector
solitons also decreases when $k_{1}$ increases further above $0.1$ but not as
abruptly as in the corresponding case with $\kappa=2$. In the $k_{1}$ interval
shown in Figs. 7(d), (e), and (f), we observe $3$, $3$, and $1$ boundary
crossings, respectively (for $k_{2}=1$, $1.5$, and $1.95$). Specifically, for
$k_{2}=1$ (Fig. 7(d)), the first boundary crossing appears at about
$k_{1}\simeq 0.8$ where from DB (blue) vector solitons one transits into BD
(red) ones smoothly (i.e. the curve representing the width $b$ is continuous)
at very low values. The second crossing, from BD (red) to DD (green) type
vector solitons occurs very close to the first one and is not smooth in this
case, in the sense that $b$ diverges at the boundary. A third crossing occurs
at $k_{1}\simeq 1.25$ where again the transition of $b$ is smooth with very
low values of $b$ at the crossing point. The latter crossing is from DD
(green) to BD (red) vector solitons. For $k_{2}=1.5$ (Fig. 7(e)), a crossing
between existence areas from DB (blue) to DD (green) vector solitons occurs at
approximatelly $k_{1}\simeq 1.25$ which is not smooth in $b$, since it
diverges there. A second, smooth crossing occurs at around $k_{1}\simeq 1.7$
in which $b$ has low values and separates existence areas of DD (green) and BB
(black) vector solitons. For that value of $k_{2}$, BB (black) vector solitons
exist only for a very short interval of $k_{1}$, and the third crossing in
this figure occurs at approximately $k_{1}\simeq 1.75$ between existence areas
of BB (black) to BD (red) vector solitons. This crossing is smooth and
continuous in the width $b$. For $k_{2}=1.95$ (Fig. 7(f)), a single crossing
occurs between existence areas of DB (blue) and BB (black) vector solitons.
The behavior of $b$ as a function of $k_{1}$ has the same features as that of
the amplitude $A_{2}$ shown in Fig. 6.

The divergence of the amplitude $A_{2}$ and the width $b$ for low $k_{1}$ in
Figs. 6 and 7 is due to the corresponding divergence of the coefficient
$Q_{11}$, which appears in the numerator of both $\alpha$ and $\beta$ in Eq.
(15). However, there is also a divergence at relatively large $k_{1}$ for
$\kappa=3$, signaling the transition from one type of vector soliton to
another. Specifically, in Figs. 6(d) and 7(d) a divergence in $A_{2}$ and $b$
sets the boundary between a bright-dark (BD) and a dark-dark (DD) vector
soliton at $k_{1}=0.86$. That divergence occurs because both $\alpha$ and
$\beta$ change sign through their denominator crossing the zero line. When
this happens, a transition of a BD vector soliton (which exists for $\alpha>0$
and $\beta<0$) to a DD vector soliton (which exists for $\alpha<0$ and
$\beta>0$) takes place. (Note that the denominators of $\alpha$ and $\beta$
differ by an overall sign only.) Very close to the transition point, where the
amplitude of the second component may be much larger than unity, the CNLS
equations (8) and (9) are not expected to provide a valid description of
vector solitons in our plasma fluid model.

###  5.1 Extremely asymmetric waves emerging as vector soliton components

Note that, with reference to Figs. 6 and 7, there may exist boundaries between
existence areas of different type of vector solitons in which the amplitude
$\Psi_{2}$ of the envelope soliton component acquires large values, while its
width $b$ acquires very low values simultaneously. Such a case can be observed
in Fig. 7(e), for a value of $k_{1}$ slightly above $1.25$ (green curves in
Figs. 6(e) and 7(e)), for which DD vector solitons exist. For that
$k_{1}\simeq 1.3$, the width $b$ acquires very low values while at the same
time the amplitude $A_{2}$ of the second vector soliton component acquires
values around $A_{2}\simeq 0.4$, which are four (4) times larger that the
corresponding amplitude $A_{1}=0.1$ (fixed) of the first vector soliton
component. Such a highly localized and high-amplitude envelope soliton can be
characterized as “extremely asymmetric wave”, i.e. a large amplitude breather-
like structure which co-exists (co-propagates) with an ordinary sister
envelope soliton (pulse). Of course, the specific choice of values makes the
components of the (DD in the described case) vector soliton highly
a-symmetric. This possibility, as well as the stability and geometry of
extremely asymmetric vector soliton (pairs) will be analyzed in detail in a
future a work.

In Figs. 8-11, several illustrative examples of all the four types of vector
solitons which may exist in certain areas on the $k_{1}-k_{2}$ plane, are
provided for several values of the spectral index $\kappa$ and the wavenumbers
$k_{1}$ and $k_{2}$. The first two of these figures, in particular, i.e., Fig.
8 and Fig. 9, provide illustrative examples of DB and DD vector solitons,
respectively, for $\kappa=2$, $3$, $5$, and $100$. E.g., in Fig. 8, the values
for the two wavenumbers were chosen to be $k_{1}=0.25$ and $k_{2}=1.5$ (for
all subfigures), while $\kappa$ varies as shown on the figure. In this
particular case, the values of $k_{1}$ and $k_{2}$ favor the existence of DB
vector solitons for any $\kappa>2$. Note that the amplitude of the envelop
soliton component (black curve), $\Psi_{1}$ , $A_{1}$, is fixed to $0.1$, and
thus it is the same in Figs. 8(a) through (d), even though its width $b$ does
change slightly its value. The second vector soliton component (red curve),
$\Psi_{2}$, has a varying amplitude which increases considerably with
increasing $\kappa$, as can be observed in Fig. 8 (e.g., from $A_{2}\simeq
0.06$ in Fig. 8(a) to $A_{2}\simeq 0.29$ in Fig. 8(d).

Similarly, in Fig. 9, the same plots of vector solitons as those presented
before in Fig. 8 are shown but for a different $k_{1}$ and $k_{2}$ pair of
values on the $k_{1}-k_{2}$ plane, i.e., $k_{1}=0.9$ and $k_{2}=1$. for this
choice of the $k_{1}$ and $k_{2}$ pair, we get DD vector solitons for all the
values of the spectral index $\kappa$ considered. Of course the width of the
envelopes $b$ and the amplitude of the vector soliton component $\Psi_{2}$,
$A_{2}$, change considerably with $\kappa$. Interestingly, the width $b$ of
both soliton components is rather large for an extreme (strongly non-
Mawxellian) value $\kappa=2$ (Fig. 9(a)). Also, the amplitude $A_{2}$ is very
low in this case, i.e., $A_{2}=0.05$. In Fig. 9(b), however, for $\kappa=3$,
the width $b$ decreases considerably, so that the soliton components become
narrower and thus highly localized, while at the same time the amplitude
$A_{2}$ increases to more than $0.22$. With further increasing $\kappa$, both
$b$ and $A_{2}$ gradually and slowly decrease, while they tend to become equal
to each other for very large values of $\kappa$, e.g., for $\kappa=100$ in
Fig. 9(d).

Figure 9:  Same as Fig. 8 but for $k_{1}=0.9$ and $k_{2}=1$.  Figure 10:
Vector solitons along a “cut” of the $k_{1}-k_{2}$ plane in Fig. 3(a) at
$k_{2}=0.95$ for $\kappa=2$ and (a) $k_{1}=0.1$; (b) $k_{1}=0.3$; (c)
$k_{1}=1$; (d) $k_{1}=1.8$. We observe a DB vector soliton in (a), a DD vector
soliton in (b), and a BB vector solitons in (c)-(d).  Figure 11:  Vector
solitons along a “cut” of the $k_{1}-k_{2}$ plane in Fig. 3(b) at $k_{2}=0.95$
for $\kappa=3$ and (a) $k_{1}=0.1$; (b) $k_{1}=0.3$; (c) $k_{1}=1$; (d)
$k_{1}=1.8$. We observe a DB vector soliton in (a)-(b), a DD vector soliton in
(c), and a BD vector soliton in (d).

The vector solitons shown in Figs. 10 and 11 are obtained for spectral index
$\kappa=2$ and $3$, respectively, and different pairs of wavenumbers $k_{1}$
and $k_{2}$, along $k_{2}=0.95$ on the $k_{1}-k_{2}$ plane. In Fig. 10(a), a
DB vector soliton is shown for $k_{1}=0.1$ and $k_{2}=0.95$, which are both
very narrow (small width $b$) and have roughly the same amplitude, i.e.,
$A_{1}\simeq A_{2}\simeq 0.1$. A DD vector soliton is shown in Fig. 10(b) for
$k_{1}=0.3$ and $k_{2}=0.95$, where the amplitude $A_{1}$ is twice the
amplitude $A_{2}$. In Figs. 10(c) and (d), two BB vector solitons are shown
for $k_{1}=1.0,k_{2}=0.95$, and $k_{1}=1.8,k_{2}=0.95$, respectively. The two
components of these vector solitons do not differ very much and, in one case
(Fig. 10(d)), the two amplitudes $A_{1}$ and $A_{2}$ as well as their width
$b$ are practically the same. This seems reasonable since the values of
$k_{1}$ and $k_{2}$ in this case are very close together and thus close to the
curve $k_{1}=k_{2}$ on the $k_{1}-k_{2}$ plane, where the CNLS system becomes
nearly symmetric.

In Fig. 11, several vector solitons are shown for $\kappa=3$ and the same
pairs of values of the wavenumbers $k_{1}$ and $k_{2}$ as those used in Fig.
10. Here, a narrow (small $b$) highly localized DB vector soliton is shown in
Fig. 11(a), whose $\Psi_{2}$ component has a rather high amplitude
$A_{2}\simeq 0.3$, about three times the amplitude $A_{1}$. In Fig. 11(b) we
have another DB vector solitons whose components are however much wider than
the previous one (large $b$) and $A_{2}$ is roughly $1.5A_{1}$. In Fig. 11(c),
we have a DD vector soliton whose components have similar amplitudes but they
are rather wide (large $b$), Next, in Fig. 11(d), we have two BD vector
solitons, whose components exhibit comparable amplitude.

Figure 12:  Extreme amplitude (asymmetric) vector solitons obtained around the
boundary of the area of existence of dark-bright (DB, blue color) and dark-
dark (DD, green color) in Fig. 3(b) for spectral index $\kappa=3$,
$k_{2}=1.5$, and (a) $k_{1}=1.26$; (b) $k_{1}=1.30$. In both plots, the
amplitude of the second component of the vector soliton (bright and dark,
respectively) is significantly larger than that of the first one.

An illustrative example of vector solitons with an extreme amplitude component
is shown in Fig. 12. In that figure, two such vector solitons obtained for the
same spectral index $\kappa=3$ and wavenumber of the second carrier wave
$k_{2}=1.5$ are shown for slightly different values of $k_{1}$. These
parameter values correspond to points very close to the boundary of the area
of existence of dark-bright (DB, blue color) and dark-dark (DD, green color)
of Fig. 3(b) for $\kappa=3$. At these points, the second components of the
existing vector solitons are expected from Fig. 6(e) to have a high amplitude
as compared with the first one. In Fig. 12(a), for $k_{1}=1.26$, in which the
vector soliton is of the DB type, the amplitude of the second (bright)
component which is shown in red color is about four times larger than that of
the first one. The same remark holds for the amplitude of the second (dark)
component shown in red color in Fig. 12(b), for $k_{1}=1.30$, is again about
four times larger than the first one. Note the both in Figs. 12(a) and 12(b)
the first component of the corresponding vector soliton is shown in black
color and it is of the dark type. Thus, in this particular case one may switch
from a certain type of vector soliton with a large component to another type
(i.e., bright to dark, in this case) by a small shift in one of the
wavenumbers, thus crossing a boundary between adjacent areas in the existence
diagram. Note that the extreme amplitude component is the one that changes
type in this case, i.e. from bright in Fig. 12(a) to dark in Fig. 12(b).
(Recall that the amplitude of the first component in all vector solitons
presented here is numerically fixed to $A_{1}=0.1$, for comparison and
reference.)

Figure 13:  Comparison among representative vector soliton pairs obtained for
small and large values of the spectral index $\kappa$, i.e., for $\kappa=3$
(left panels; strongly non-thermal distribution): (a) $k_{1}=0.86$, $k_{2}=1$;
(c) $k_{1}=1,49$, $k_{2}=1.75$; (e) $k_{1}=1.01$, $k_{2}=1.6$, and
$\kappa=100$ (right panels; quasi-Maxwellian): (b) $k_{1}=0.86$, $k_{2}=$; (d)
$k_{1}=1.49$, $k_{2}=1.75$; (f) $k_{1}=1.01$, $k_{2}=1.6$. The vertical scales
are the same for all panels in each row, to facilitate comparison. The
horizontal scales are the same for all panels.

Further examples of vector solitons with an extreme amplitude component are
shown in Fig. 13, in which the effect of extreme variation of the spectral
index $\kappa$ is illustrated. The two panels in each row are obtained for the
same pairs of the wavenumbers $k_{1}$ and $k_{2}$ but for different $\kappa$.
In all left panels, the value of spectral index $\kappa=3$ has been used,
while in all right panels a large value of $\kappa$ is used ($\kappa=100$) for
which the electron distribution practically coincides with a Maxwell-Boltzmann
one. In Fig. 13(b), for $\kappa=100$, the selected values of $k_{1}$ and
$k_{2}$ are such that the vector soliton is of the dark-dark (DD) type whose
components have almost the same (relatively low) amplitude. However, as the
spectral index is decreased to the relatively small value $\kappa=3$ in Fig.
13(a), the amplitude of the second component increases significantly with
respect to that of the first one, although the the vector soliton type (DD)
remains the same. Thus, by decreasing $\kappa$, the resulting increase in the
suprathermal electron population provides the necessary energy for the
emergence of a dark type extreme amplitude excitation shown in red color.

In the second row, i.e. in Figs. 13(c) and 13(d), we observe the emergence of
an extreme amplitude _bright_ type component upon decreasing the value of
$\kappa$, respectively. It appears that moving far from Maxwellian equilibrium
results in energizing the electrons, that supply the energy required to excite
a large amplitude “breather” type envelope structure seven (!) times higher
than the sister component. A different trend is witnessed in Figs. 13(e) and
13(f). In this case, somehow counter-intuitively, a small-amplitude dark (wave
1) / large amplitude bright (wave 2) vector soliton excited for $\kappa=100$ –
see 13(f) – is in fact suppressed in amplitude, upon reducing the spectral
index to $\kappa=3$ – see 13(e) – however without changing structural type
(DB).

Figure 14:  Temporal evolution of bright-bright vector solitons profiles
obtained from numerical simulations using Eqs. (8) and (9) with initial
condition from Eq. (13) to which small random noise is added. The profiles in
(a) and (b) (resp. (c) and (d)) for $\bar{\Psi}_{1}$ and $\bar{\Psi}_{2}$ are
obtained for $k_{1}=1.5$ and $k_{2}=1$ (resp. $k_{1}=1.673$ and $k_{2}=1.95$).
The parameters of the vector soliton in (a) and (b) (resp. in (c) and (d)) are
$A_{1}=0.1$, $A_{2}\simeq 0.129$, $b\simeq 0.759$ (resp. $A_{1}=0.1$,
$A_{2}\simeq 0.964$, $b\simeq 2.13$).

In order to address the stability of the obtained vector solitons, linear
stability analysis corroborated by direct numerical simulations and/or
(numerical) spectral stability analysis should be used to obtain the parameter
regimes which can support stable vector solitons. Further analytical
approaches may involve methods applied, e.g., to birefringent optical fibers
[62] or the application of the Vakitov-Kolokolov criterion [63]. The stability
analysis of the various types vector solitons in plasmas is certainly
important and worth studying on its own right and it is a matter of future
work. Here, stability is demonstrated using direct numerical simulations for
two bright-bright (BB) vector solitons; the parameters are chosen so that the
components of one of them have similar amplitude while the components of the
other are highly asymmetric (e.g. the amplitude of the second component is
much higher than that of the first).

These two cases are illustrated in Fig. 14 in which (a) and (b) show the
temporal evolution of the $|\bar{\Psi}_{1}|$ and $|\bar{\Psi}_{2}|$ profiles
for the first set of parameters, while (c) and (d) those for the second set of
parameters (see caption in Fig. 14). These simulations are initialized with
the corresponding analytical solutions given in Eqs. (13), to which small
random noise was incorporated. In Figs. 14(a) and (b) that random noise is
relatively large so that it is clearly visible for small $\tau$. At both ends
of of the $\xi$ interval, dissipation has been added by hand to remove the
excess energy introduced by the noise. In this way, the excess energy leaves
the system and at large $\tau$ the profiles practically coincide with the
analytical ones. We have also checked that at large $\tau$ the norms of the
two components $|\bar{\Psi}_{1}|^{2}$ and $|\bar{\Psi}_{2}|^{2}$ saturate to a
constant value.

##  6 Discussion and Conclusions

We have considered the simultaneous propagation of a pair of (nonlinearly
interacting) electrostatic wavepackets in a collisionless unmagnetized
electron-ion plasma, from first principles. The wavepackets are not identical,
in the sense that both their amplitudes and (carrier) wavenumbers $k_{1}$ and
$k_{2}$ are allowed to differ. Adopting a Newell type multiple (time and
space) scales technique, a pair of CNLS equations was derived. A standard non-
magnetized plasma fluid model was adopted, for simplicity, comprising cold
inertial ions evolving against an inertialess electron background. The
electron population was assumed to obey a kappa-type distribution, which is
characterized by the spectral index parameter $\kappa$, a situation often
occurring in space plasmas (with typical values of $\kappa$ usually ranging
from $2$ to $6$). As the kappa distribution diverges significantly from a
Maxwell-Boltzmann distribution, the highly energetic (suprathermal) electron
component results in significant modification of the modulated wavepackets
characteristics and interactions thereof. We have investigated the
modulational (in)stability profile of the coupled wavepacket pair, focusing on
its dependence on the electron spectral index ($\kappa$). We have shown that
various types of vector solitons may exist in different areas on the
$k_{1}-k_{2}$ plane, while their shape depends on (the value of) $\kappa$. The
strongest variation is observed in the interval of $\kappa$ from $2$ to $3$.

The six coefficients of the CNLS equations, i.e., the dispersion coefficients
$P_{j}$, the nonlinearity coefficients $Q_{jj}$, and the nonlinear coupling
coefficients $Q_{ij}$ (with $i\neq j$), are given by complicated algebraic
expressions (see Supplementary Information) as functions of the wavenumbers
$k_{1}$ and $k_{2}$ and the spectral index $\kappa$. For arbitrary values of
$k_{1}$ and $k_{2}$ ($\neq k_{1}$), these coefficients do not possess any
particular symmetry, hence the generalized system of CNLS equations thus
obtained is most likely non-integrable, in the general case. (Obviously, the
integrable Manakov case is recovered if $k_{1}=k_{2}$.) The asymmetry of the
CNLS equations does not prevent one from obtaining vector soliton solutions,
as combinations of bright (B) and dark (D) envelopes structures, i.e. of BB,
BD, DB or DD type, each of which will occur in particular areas on the
$k_{1}-k_{2}$ parameter plane.

The “area of existence” of these vector solitons exhibits strong variation
with respect to $\kappa$, in particular for values of $\kappa$ between $2$ and
$3$. For $\kappa$ close to $2$, BB vector solitons exist in the largest part
of the plane, while BD and DB vector solitons exist only for low $k_{1}$ and
$k_{2}$, respectively. Also, DD vector solitons exist in narrow areas between
BB-BD and BB-DB vector solitons. However, for increasing $\kappa$, the area of
existence of BD and DB vector solitons increases at the expense of the area of
BB vector solitons. At the same time, the area of existence of DD vector
solitons also increases. For $\kappa$ greater than $3$, the pattern of the
areas of existence of the four vector solitons change only slightly with
$\kappa$.

The vector soliton parameters, i.e., their amplitude and width, both of which
have been calculated analytically, also vary significantly with varying
$k_{1}$, $k_{2}$ and $\kappa$. It is interesting to see how the transition
between different types of vector solitons occurs, upon varying one of these
parameters, keeping the remaining two parameters fixed. As illustrated in the
figures above, this transition between different types of vector solitons can
be either smooth, or associated with a divergence of, say, the amplitude of
one wave at the transition point, i.e. at the boundary separating areas of
existence of different types of vector solitons.

Of particular interest is the situation in which, close to a transition point
where a soliton parameter diverges, the amplitude of one of the components may
acquire extreme values, i.e. far exceeding its sister wave’s amplitude, thus
forming what could be characterized as an extreme amplitude wave (component)
pair. These highly asymmetric vector solitons are a peculiarity which is
attributed to the general asymmetry of the CNLS equations. The investigation
of the stability of these vector solitons using semi-analytic and numerical
methods will be a subject of future work.

Focusing on the role of suprathermal electrons, our investigation has shown
that the spectral index $\kappa$ affects significantly the modulational
(in)stability profile of the CNLS system as well as the characteristic of
vector soliton types that may be sustained in the plasma. For smaller
$\kappa$, i.e., as the electron distribution deviates from the Maxwellian one,
the areas on the parameter planes in which modulational instability appears
become larger, while at the same time the growth rate becomes higher in those
areas (i.e., enhancing the instability). Concerning vector solitons, a
variation in the value of $\kappa$ modifies the existence diagram in parameter
space, in which different types of vector solitons may occur. The most
prominent variation occurs below $\kappa=3$, where DB, BD, and DD vector
solitons exist in substantial areas of the $k_{1}-k_{2}$ plane, down to
$\kappa=2$, where BB vector solitons become dominant.

The existence of all four types of vector solitons on a parameter plane
involving the spectral index $\kappa$ and the walk-off parameter $\delta$ was
also illustrated and discussed. Notably, for the particular plasma model
considered here, the parameter $\delta$ takes low values (in fact, the lower
$\kappa$, the smaller the range of $\delta$), which are thus not expected to
affect the formation or the stability of vector solitons.

The formation of solitary waves/solitons is a phenomenon that commonly occurs
in space plasmas, e.g., in the solar wind and in planetary magnetospheres; cf.
observations of electrostatic solitary waves by the Cluster satellites [64,
65] (and references therein). As one example, electrostatic solitary waves
have been observed in the Earth’s magnetopause [66, 67], and their theoretical
interpretation requires resorting to multicomponent plasma fluid models [68].
Remarkably, envelope structures (breathers, rogue waves) modeled by the NLS
equation have been realized in laboratory plasmas, little more than a decade
ago [69].

Based on earlier considerations, where modulational instability has been
proposed as an intermediate stage between amplitude modulation of a Stokes
wave and higher-order effects leading to rogue wave formation [70], we
anticipate that the creation of extreme amplitude soliton-pair structures
predicted by our model may provide an effective framework as a precursor
towards freak wave occurrence in relation with electtostatic plasma modes.

Our work aims at providing a platform for modeling solitons/solitary waves in
space plasmas, where modulated envelope pairs may emerge from two or more
interacting nonlinear waves. In a wider context, our results will be valuable
in other disciplines where wavepackets may propagate in nonlinear dispersive
media, including – but not being limited to – hydrodynamics, nonlinear (fiber)
optics and telecommunications (signal transmission via optical pulses), to
mention a few.

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## Acknowledgements

Authors IK and NL gratefully acknowledge financial support from Khalifa
University (United Arab Emirates) via the project CIRA-2021-064 (8474000412).
IK gratefully acknowledges financial support from KU via the project
FSU-2021-012 (8474000352) as well as from KU Space and Planetary Science
Center, via grant No. KU-SPSC-8474000336.

This work was completed during a two-semester long research visit by author IK
to the Physics Department, National and Kapodistrian University of Athens,
Greece. During the same period, author IK also held an Adjunct Researcher
status at the Hellenic Space Center, Greece. The hospitality of both hosts is
warmly acknowledged.

## Author contributions statement

N.L. carried out the algebraic work, contributed to the methodology, software
development, and numerical analysis. G.P.V. contributed to the methodology and
reviewed the manuscript. D.J.F. contributed to the concept and design, and
reviewed the manuscript. I.K. contributed to the problem conception, project
design, launch and methodology, and reviewed the manuscript.

## Competing Interests

The authors declare no competing interests.

## Data Availability

The datasets generated during and/or analysed during the current study are
available from the corresponding author upon reasonable request.


# 3D solar coronal loop reconstructions with machine learning

Iulia Chifu Max-Planck-Institut für Sonnensystemforschung,
Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany  Astronomical Institute of
Romanian Academy,
Cutitul de Argint 5, Bucharest, Romania  Ricardo Gafeira Univ Coimbra, IA,
CITEUC, OGAUC, Coimbra, Portugal
Rua do Observatório s/n,3040-004 Coimbra, Portugal  Instituto de Astrofísica
de Andalucía (CSIC),
Apartado de Correos 3004, E-18080 Granada, Spain<EMAIL_ADDRESS>
###### Abstract

The magnetic field plays an essential role in the initiation and evolution of
different solar phenomena in the corona. The structure and evolution of the 3D
coronal magnetic field are still not very well known. A way to get the 3D
structure of the coronal magnetic field is by performing magnetic field
extrapolations from the photosphere to the corona. In previous work, it was
shown that by prescribing the 3D reconstructed loops’ geometry, the magnetic
field extrapolation finds a solution with a better agreement between the
modeled field and the reconstructed loops. Also, it improves the quality of
the field extrapolation. Stereoscopy represents the classical method for
performing 3D coronal loop reconstruction. It uses at least two view
directions. When only one vantage point of the coronal loops is available,
other 3D reconstruction methods must be applied. Within this work, we present
a method for the 3D loop reconstruction based on machine learning. Our purpose
for developing this method is to use as many observed coronal loops in space
and time for the modeling of the coronal magnetic field. Our results show that
we can build machine learning models that can retrieve 3D loops based only on
their projection information. In the end, the neural network model will be
able to use only 2D information of the coronal loops, identified, traced and
extracted from the EUV images, for the calculation of their 3D geometry.

3D reconstruction, coronal loops

##  1 Introduction

Coronal loops are approximated to a 3D arch-like structure that has their
footpoints located in the opposite polarity of the magnetic field. In the
solar corona, the plasma pressure is dominated by the magnetic pressure (Gary,
2001) which makes the coronal loop plasma confined in the 3D flux tubes. The
coronal loops are mainly visible on the solar disk in the EUV images which
shows only the projection of the 3D structures. The reconstruction of the
three-dimensional shape of coronal loops is important for the implication on
the magnetic field and topology (Reale, 2010).

The magnetic field plays an important role in the initiation and evolution of
different solar phenomena in the corona. While at the photospheric level full-
disk vector magnetic field measurements are provided by several observatories,
in the chromosphere and corona the vector magnetic field is only sometimes
provided and only for small fields of views. Different approaches are used to
obtain the coronal magnetic field. One of the approaches is based on magnetic
field extrapolations into the corona having as bottom boundary photospheric
magnetic field data (see Wiegelmann & Sakurai (2012); Wiegelmann et al.
(2015), for reviews). Another approach is a stereoscopic reconstruction based
on simultaneously recorded images from at least two view directions in the EUV
wavelengths (chromosphere and low corona) and white light (high corona) (see
Aschwanden (2011) for a review). When applied for the 3D reconstruction of the
coronal magnetic loops, the output of the two methods does not match (De Rosa
et al., 2009). Chifu et al. (2017) showed that by constraining the nonlinear
force-free field (NLFFF) extrapolations with the 3D stereoscopically
reconstructed loops, one can obtain a better field model.

The 3D stereoscopic reconstruction method requires images of the same object
recorded from at least two view directions. This requirement limits the time
range for finding the 3D information on solar coronal loops. One of the steps
for stereoscopy is identification and matching of the same object in the
available images (Inhester, 2006). This condition limits the possibility of
selecting large number of coronal loops.

With the method laid out in this work, we can make use of all of the epochs at
which exists simultaneously EUV images and vector magnetograms. The 2D loops
which one would be able to identify, visually or automatically, in an EUV
image, will be 3D reconstructed by the neural network method. The resulted 3D
loops can be therefore used as a constrain for the magnetic field
extrapolation as shown by Chifu et al. (2017). This method allows us to use
many more loops as a constraint for extrapolation than was possible till now.
The method we present here for the 3D coronal loop reconstruction will be able
to constrain the coronal magnetic field models to fit the observations. These
coronal magnetic field models can be further used by the MHD models in the
attempt to investigate the heating mechanisms of the coronal loops.

##  2 Methods

###  2.1 Convolution Neural Network

Neural networks (NN) are designed to work like the processes inside a human
brain. In this context, what we call a neuron is a mathematical function that
infers quantities based on the input data according to the NN architecture.

Among the various different NN architectures, there is a subset called
convolutional neural network (CNN). These networks use convolutional layers
where one or two-dimensional filters are applied to the input data. The
unidimensional (1D) CNN basic structure used in this work can be then resumed
as a combination of the following layers: 1.) layers that apply a convolution
between a kernel and each set of data points (in our case loop position), with
the goal of enhancing particular features in the data; 2.) pooling layer that
downsamples the data by reducing the data information but keeping the more
prominent features; 3.) Finally, the so-called fully connected layers which
are taking the results of the previous operations and multiply them by a set
of weights to estimate the final output. Most of the operations performed in
these layers have an associated activation function, which are mathematical
equations that determines the importance of a certain variation.

One key element of any NN is the training process in which an input data set
and its respective output data are used. This allows the model to determine
the weights that the fully connected layers will apply. The network uses an
optimization procedure to minimize the differences, using a loss function,
between the training set and the estimated output. It is common practice to
use a fraction of the input data not used for the training and compare it to
the model output. That is called validation loss. It allows us to evaluate the
evolution of the training and identify problems like overfitting. To optimize
the minimization process, we can divide the data into small batches. During
the training, the weights will be updated at the end of each batch, leading to
a better and faster convergence. The overfitting can be avoided by introducing
dropout elements that randomly remove a fraction of the points from the
training process introducing a level of randomness to the system and helping
it to avoid false local minima. We call epoch each full iteration of the
training process described before.

In this work, we follow the CNN architecture model presented on Gafeira et al.
(2021). We optimized the hyperparameters by trial-and-error until we found a
CNN model satisfying a loss function being at least 10-3 of the normalization
values to all considered cases. We trained the model using the optimizer
AdaMax witch is a first-order gradient-based optimization of stochastic
objective functions (Kingma & Ba, 2014). We used a batch size of 64 and a
validation split of 0.3. The convolutional layers and the first dense layer
use the nonlinear function called the rectified linear activation unit (ReLU)
(Nair & Hinton, 2010) and the last dense layer a linear function. The global
structure of the CNN and the respective hyperparameters used in this work is
summarised in Table 1.

The input parameters are the 2D position of the loops (X, Y), the length of
the loop, the distance between the footpoints of the loops, and the angle
defined by the two footpoints with the top of the loops. The 2D positions X
and Y are interpolated to all have a fixed size of 1500 points.

The output and training set of the CNN is the Z component of the corresponding
2D loop with a fixed size of 1500 points.

We applied a normalization factor that corresponds to the loop average height
used in the training of CNN. In this work the normalization used is 8.75 ADU
for Data type 1, and for Data type 2 Case 1 is 10.1 Mm, Case 2 is 77.54 Mm and
Case 3 is 73.15 Mm.

Layer | Size | Activation
---|---|---
1D convolutional | 150 | ReLU
1D MaxPooling | 2 |
1D convolutional | 90 | ReLU
1D MaxPooling | 2 |
1D convolutional | 45 | ReLU
Dropout | factor of 0.25 |
Fully connected | 4 times the loop |
dense layer | length | ReLU
Fully connected | loop length |
dense layer | length | linear
Table 1: CNN model architecture used in this work.

###  2.2 Nonlinear Force-Free Field extrapolation

To produce a big enough data set for training the CNN we used the nonlinear
force-free field extrapolation method, proposed initially by Wheatland et al.
(2000) and implemented by Wiegelmann (2004). The NLFFF optimization method is
based on the minimization of a functional L${}_{\text{tot}}$ as a sum of the
following terms

$\displaystyle\text{L}_{1}=\int_{V}w_{f}\frac{|(\nabla\times\mathbf{B})\times\mathbf{B}|^{2}}{B^{2}}\;d^{3}r,$
(1)
$\displaystyle\text{L}_{2}=\int_{V}w_{f}|\nabla\cdot\mathbf{B}|^{2}\;dr^{3},$
(2)
$\displaystyle\text{L}_{3}=\int_{S}(\mathbf{B}-\mathbf{B}_{obs})\cdot\mathrm{diag(\sigma^{-2}_{\alpha})}\cdot(\mathbf{B}-\mathbf{B}_{obs})\;d^{2}r,$
(3)

where the minimization of $L_{1}$ satisfies the force-free condition and the
$L_{2}$ satisfies the Gauss Law. The minimization of the $L_{3}$ ensures that
at the photospheric level the modeled field ($\mathbf{B}$) is as close as
possible to the observed one ($\mathbf{B_{obs}}$). $w_{f}$ is the boundary
weight function (more details in Wiegelmann, 2004),
$\mathrm{diag(\sigma^{-2}_{\alpha})}$ represents the estimated measurement
errors for the three field component (see Tadesse et al., 2011, for more
details). The method was implemented in the Cartesian (Wiegelmann, 2004) and
spherical (Tadesse et al., 2011) coordinate system. Both of the
implementations are using as a bottom boundary the vector magnetic field. The
lateral and top boundaries are fixed from an initial calculated potential
field.

##  3 Data preparation

We tested the CNN method in four different cases by using input loops obtained
from different solutions of the NLFFF extrapolation. The first case uses
synthetic data and the last three are based on observational data:

  1. 1.

Data type 1. The NLFFF extrapolation uses as bottom boundary a semi-analytical
force-free field solution proposed by Low & Lou (1990). The solution is
obtained by solving the Grad-Shafranov equation for an axisymmetric nonlinear
force-free field. The results obtained based on the synthetic data are defined
as Case 0.

  2. 2.

Data type 2. In this category are the data obtained by the Heliospheric
Magnetic Imager (HMI) instrument (Couvidat et al., 2012) onboard Solar Dynamic
Observatory (SDO) spacecraft (Pesnell et al., 2012). We defined three Cases:

     * •

Case 1. We use as input for the NLFFF a field of view of 1280$\times$400 from
a full disk HMI vector magnetogram.

     * •

Case 2. For the NLFFF extrapolation method in spherical coordinates, we used
as bottom boundary a full disk vector magnetogram recorded on 02 August 2010.

     * •

Case 3. For the last case, we use synoptic vector magnetograms which are HMI
data products suitable for the full Sun extrapolation (Liu et al., 2017).
These maps are constructed from the daily HMI vector magnetograms. The size of
the HMI synoptic magnetograms is 3600$\times$1440 pixels and the pixel size is
0.1 deg in longitude and 0.001 in sin latitude.

From the solutions of the extrapolations, using a fourth-order Runge-Kutta
method, we traced 3D magnetic field lines and we projected them on the solar
surface.

##  4 Results

###  4.1 Data type 1

Case 0. We tested the CNN method described in Section 2.1 with 3D and 2D loops
obtained from an NLFFF extrapolation which used as bottom boundary a Low & Lou
semi-analytical force-free field solution. We used the Cartesian coordinate
system for a computational box in the X, Y, and Z-direction of
192$\times$192$\times$96 grid points. For the training of the CNN, we used
21704 loops traced from the NLFFF solution.

(a) Display of the training (blue solid line) and validation (orange dashed
line) loss function during the CNN training.

(b) Distribution of the root mean square for Case 0 loops.

(c) 3D CNN reconstructed loops with solid orange lines and original 3D loops
with black dashed lines overplotted on the Low & Lou bottom boundary.

Figure 1: CNN results for Data type 1, Case 0.

We trained the CNN initially using 250 epochs. After the training, we evaluate
the result of the training and validation loss function. Since the model has
converged after 250 epochs, we stopped the training process. In the presented
model we did not find any indication of overfitting. That would be visible as
a sudden increase of the validation loss function. In Fig.1(a) we display the
validation and training loss evolution obtained from the CNN model for Case 0.

For the evaluation of the CNN solution, we calculated the root mean square

$RMS=\sqrt{\frac{\sum\mathfrak{R^{2}}}{N}},$ (4)

where $\mathfrak{R}$ is the ratio between the Z coordinate of the
reconstructed and the original loops and N is the number of elements in a
loop.

In Fig.1(b) we show the RMS distribution for Case 0. The RMS values spanning
0.9 to 1.1 represent 99.96% from the entire data set of loops. An RMS value of
unity means the CNN reconstructed loop and the original ones are perfectly
matched. The CNN could obtain almost perfectly the 3D position for 99.96% of
loops. In Fig. 1(c) we display the Low & Lou magnetic field solution used for
the NLFFF extrapolation overplotted with a randomly selected subset (for
better visibility) of the original traced and the reconstructed 3D loops.

###  4.2 Data type 2

Case 1. We applied the NLFFF extrapolation in the Cartesian geometry to an HMI
vector magnetic field recorded on 15 July 2010. From the full disk
4096$\times$4096 pixels we selected a 1280$\times$400 field of view. The
height of the box is 320 pixels. From the solution of the extrapolation, we
traced 10127 loops. The CNN network reached a reasonable solution after 250
epochs similar to Case 0. In Fig. 2(a) we plot the evolution of the training
and validation losses with the epochs. Fig. 2(b) shows the distribution of the
RMS (Eq. 4). The RMS values between 0.9 and 1.1 represent a percentage of
99.06 of all loops. In Fig.2(c) we present the HMI vector magnetogram used as
input for extrapolation together with few randomly selected 3D original and
reconstructed loops (orange and blue solid line). The two different categories
of reconstructed loops are made based on the RMS values. The orange loops are
the ones with RMS smaller than 1.1 and the blue loops with RMS larger than
1.1.

(a) Display of the training (blue solid line) and validation (orange dashed
line) loss function during the CNN training.

(b) Distribution of the root mean square for Case 1 loops.

(c) Randomly selected 3D loops overplotted on the HMI phototspheric radial
magnetic field. The black dashed lines shows the original 3d loops. The CNN
reconstructed 3D loops with an RMS smaller than 1.1 are displayed with solid
orange lines. In blue solid line are shown the 3D loops with an RMS larger
than 1.1.

Figure 2: CNN results for Data type 2, Case 1.

Case 2. For the spherical NLFFF optimization code we use full disk HMI vector
magnetic field recorded on 02 August 2010. The computational domain of
256$\times$372$\times$512 grid points extends over r = [1, 2.5] R⊙, $\theta$ =
[-70∘, 70∘] in latitude and $\phi$ = [90∘, 270∘] in longitude. The latitudinal
boundaries exclude the polar areas because the surface data in polar latitudes
have poor quality and the numerical finite-difference representation used for
the $L_{tot}$ functional expressed in the spherical coordinates becomes
singular at the poles. From the solution of the extrapolation, we traced and
used for the CNN 10368 3D loops. 98.31% of the loops had an RMS (Eq. 4)
between 0.5 and 1.5.

Case 3. We used as boundary data full sun synoptic vector magnetogram.

(a) Display of the training (blue solid line) and validation (orange dashed
line) loss function during the CNN training.

(b) Distribution of the root mean square for Case 3 loops.

(c) Randomly selected 3D loops overplotted on the HMI phototspheric radial
magnetic field. The black dashed lines shows the original 3D loops. The CNN
reconstructed 3D loops with an RMS (Eq. 4) between 0.5 an 1.5 are displayed
with solid orange lines. In blue solid line are shown the 3D loops with an RMS
between 0. and 0.5 and between 1.5 and 2. In red solid lines are presented the
3D loops with an RMS larger than 2.

Figure 3: CNN results for Data type 2, Case 3.

For the NLFFF extrapolation we used the computational domain of
180$\times$280$\times$720 grid points extending over r = [1, 2.5] R⊙, $\theta$
= [-70∘, 70∘] in latitude and $\phi$ = [0∘, 360∘] in longitude. From the
solution of the extrapolation, we traced loops higher than r = 1.05 Rs. For
the training, we used a total of 18555 loops. We applied to the loops
reversible transformations that are useful for the CNN method and which are:
coordinate transformations, rotation to ensure z is positive, transformations
that x and y components are in the positive domain. We also filtered the loops
based on their inflection points to exclude the over wiggling loops which in
general are not visible in the EUV images. Compared with the other cases, the
Case 3 loops have a wider range in the values of length, distance between
footpoints and heights so the CNN method did not perform as well as in the
previous cases. In Fig. 3(a) we show the evolution of the training and
validation loss function. For this case, the CNN model converged after 400
epochs. Fig. 3(b) displays the distribution of the RMS ratios of the loops.
From all of the loops, 94.5% is in the RMS range of 0.5 to 1.5. The
performance is not as good as in the previous cases. In Fig. 3(c) we show the
radial magnetic field of the synoptic map at the photospheric level with the
disk center corresponding to 180∘ longitude.

##  5 Discussions and conclusions

Stereoscopy is a suitable method for deriving the 3D geometry of solar
phenomena. The study, evolution, and kinematics of these phenomena are
essential for understanding their physical processes and the space weather
forecasting models. This analysis is constrained by the data availability of
at least two view directions of the same object, reducing the reconstruction
possibilities. On the other hand, solar observations from one view direction
are more abundant than two simultaneous views. When only one vantage point is
available, the CNN method can be used and applied to coronal loops and other
curve-like structures.

In conclusion, the new 3D loop reconstruction method will allow the user to
employ only one image from a single vantage point. One would select as many
loops as possible from the EUV image by visual inspection or automatic
identification. The 2D curve-like structure parameters mentioned in the
Section 2.1 can be used as input to the CNN models already trained (see Case 0
…Case 3 from Section 4). The output of the used CNN model will be the third
component of the 2D curve-like structures. In the end, one will have all the
three components of the curve-like structures.

The neural network model presented in this paper can only perform the 3D
geometrical reconstruction of loop types used for the training. It does not
have any direct contribution to the magnetic field modeling. Indirectly, the
3D coordinates of the CNN reconstructed loops can be, for example, used as
constraint for the magnetic field modeling as shown by Chifu et al. (2017).

##  6 Acknowledgements

Data are courtesy of NASA/SDO and the HMI science teams. IC acknowledge DFG-
grant WI 3211/5-1. The HMI data are provided courtesy of NASA/SDO and the HMI
science team. I.C. would like to thank M. Madjarska for helpful discussions.
R.G acknowledge financial support by the Portuguese Government throughthe
Foundation for Science and Technology - FCT FEDER - European Regional
Development Fund through COMPETE 2020 - Operational Programme Competitiveness
and Internationalization.

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