# Coupled circularly polarized electromagnetic soliton states in magnetized
plasmas

G.P. Veldes  Department of Physics, University of Thessaly, Lamia 35100,
Greece  N. Lazarides  Department of Mathematics, Khalifa University, Abu
Dhabi, United Arab Emirates  D.J. Frantzeskakis  Department of Physics,
University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece  I.
Kourakis  Department of Mathematics, Khalifa University, Abu Dhabi, United
Arab Emirates  Space $\&$ Planetary Science Center, Khalifa University, Abu
Dhabi, United Arab Emirates  Visiting Researcher, Department of Physics,
University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
Adjunct Researcher, Hellenic Space Center, Leoforos Kifissias 178, Chalandri,
GR-15231 Athens, Greece

###### Abstract

The interaction between two co-propagating electromagnetic pulses in a
magnetized plasma is considered, from first principles, relying on a fluid-
Maxwell model. Two circularly polarized wavepackets by same group velocities
are considered, characterized by opposite circular polarization, to be
identified as left-hand- or right hand circularly polarized (i.e. LCP or RCP,
respectively). A multiscale perturbative technique is adopted, leading to a
pair of coupled nonlinear Schr¨odinger-type (NLS) equations for the modulated
amplitudes of the respective vector potentials associated with the two pulses.
Systematic analysis reveals the existence, in certain frequency bands, of
three different types of vector soliton modes: an LCP-bright/RCP-bright
coupled soliton pair state, an LCP-bright/RCP-dark soliton pair, and an LCP-
dark/RCP-bright soliton pair. The value of the magnetic field plays a critical
role since it determines the type of vector solitons that may occur in certain
frequency bands and, on the other hand, it affects the width of those
frequency bands that are characterized by a specific type of vector soliton
(type). The magnetic field (strength) thus arises as an order parameter,
affecting the existence conditions of each type of solution (in the form of an
envelope soliton pair). An exhaustive parametric investigation is presented in
terms of frequency bands and in a wide range of magnetic field (strength)
values, leading to results that may be applicable in beam-plasma interaction
scenarios as well as in space plasmas and in the ionosphere.

##  I Introduction

Electromagnetic (EM) pulse propagation in a plasma has been in the focus of
researchers for decades. From a basic theoretical point of view, it is
associated with fundamental physical processes involving various nonlinear
mechanisms and instabilities, which have been attracting increasing interest
since the early days of plasma research. Thanks to a series of seminal early
studies [1, 2, 3, 4] which set the foundations of contemporary laser-plasma
interaction science [5, 6], the study of electromagnetic pulse propagation has
been shown to be relevant with phenomena like parametric instabilities [7, 8],
harmonic generation [9], self-focusing [10], intense electric and magnetic
field [11, 12], and wakefield [13, 14] generation, among others. As laser
power and technological sophistication increased, in the last decades, new
regimes were attained where ultrashort ultrastrong electromagnetic (laser)
pulses could be realized in the laboratory, thus entering the realm of
relativistic dynamics (where the electron quiver velocity is comparable to the
speed of light) [15] and, more recently, even quantum-electrodynamics [16, 17,
18, 19]. EM beam propagation is nowadays a cutting edge topic in science, due
to its relevance with applications including laser-assisted (inertial
confinement fusion, ICF) schemes for energy production, to ion acceleration
[20] and to sophisticated experimental diagnostic techniques [21] and thus one
of the hottest topics in contemporary research.

The propagation of relativistic soliton pulses, which is our focus here, has
not only been predicted by analytical theory [2, 3, 4, 1, 22, 31] and computer
simulations [23, 24, 25, 26, 27, 28, 29, 30], but also observed in laser-
plasma interaction (LPI) experiments [32, 33, 34, 35, 36, 37, 38, 39]. These
are essentially electromagnetic pulses trapped in a plasma density cavitation
with over-dense boundaries [40, 41, 42]. As shown in Refs. 25 and 26, a big
portion of the laser pulse (electromagnetic) energy becomes localized into a
train of soliton-like structures, which may play a crucial role in laser-
matter and laser-plasma interaction processes.

The generation of relativistic electromagnetic solitons is governed by various
physical mechanisms, including dispersion due to the finite electron inertia
and nonlinearity due to a relativistic electron mass variation as well as to
ponderomotive forces. Various analytical studies have focused on elucidating
the properties of relativistic EM solitons, most of them focusing on a one-
dimensional (1D) relativistic fluid model comprising of coupled nonlinear
equations for the vector and scalar potentials, coupled to the plasma.
Circularly polarized (CP) EM solitons in a cold plasma were investigated early
on by Kozlov _et al_ [2], who relied on a quasineutrality hypothesis to
establish the occurrence of small-amplitude localized solutions (solitary
waves). Later, computer simulations have shed some light on large-amplitude
(relativistic) pulses, wherein charge separation in the plasma may be
significant [2]. Drifting envelope solitons of CP light were later modeled by
Kaw and coworkers [3]. A stationary (zero group velocity) relativistic EM
soliton solution with was shown to exist within a 1D cold plasma model in Ref.
4. Further remarkable works on 1D solitary waves in cold plasmas based on a
perturbation technique include the studies by Kuehl and Zhang [43] and by
Farina et al [42, 44], who incorporated the effect of ion motion in a fluid-
Maxwell description. Later, Poornakala et al [45] discussed the existence of
bright or dark type envelope solitons and identified associated regions in
parameter space. Thermal effects were taken into account in Refs. 47 and 48.
Single peak and multi-peak structures and their stability and mutual
interaction properties were studied by Saxena _et al_ , considering a non-
uniform plasma [46], and also later in Refs. 49 and 50. Note, for rigor, that
the aforementioned studies have focused on non-magnetized plasma.

Magnetization during laser plasma interactions has attracted significant
attention, as it impacts both the laser pulse dynamics and the substrate i.e.
the background plasma properties. Inertial confinement fusion (ICF) has
triggered interest in the study of magnetized plasma-laser interaction [51,
52]. Experimental observations have confirmed the generation of intense
magnetic fields (up to hundreds of MG) [53, 54, 12, 55] during LPI.

As regards EM pulse propagation in magnetized plasma, Shukla and Stenflo [56]
studied stationary solitary wave electromagnetic field structures adopting a
slowly varying envelope approximation, followed by Nagesha _et al_ [57], who
adopted a slowly varying envelop approximation and neglected perturbations in
the longitudinal component of the electron momentum, to find stationary
solutions for CP EM waves in cold magnetized plasma. Weakly super-acoustic
(supersonic) EM envelope waves were studied by Rao [58]. The occurrence of
standing 1D relativistic solitons in cold magnetized plasmas and the magnetic
field’s role on soliton stability was discussed by Farina _et al_ [59], who
found that the frequency interval for stable solitons depends on both the
magnitude and the orientation of the magnetic field in a significant manner.
Furthermore, the maximum field amplitude of the solitonic pulses essentially
depends on the ambient magnetic field (strength).

In the article at hand, we have undertaken an analytical and numerical study
of the interaction between left- and right-hand circularly polarized solitons
in magnetized plasmas. Based on a fluid-Maxwell model, as starting point, we
have established a system of scalar equations governing the propagation of
circularly polarized EM waves in magnetized plasma. First, analysis of the
linear regime has shown that, in certain frequency bands, left-hand circularly
polarized (LCP-) and right-hand circularly polarized (RCP-) modes can
propagate at the same group velocity. Next, we adopt a large-amplitude
(nonlinear) approach, treating the model equations within an asymptotic
multiscale expansion method. Assuming the existence of coupled localized
(soliton) states for the magnetic vector potential, the analysis leads to a
system of two coupled nonlinear Schrödinger (NLS) type equations for the
respective EM field envelopes, i.e. for the right-hand circularly polarized
(RCP-) and the left-hand circularly polarized (LCP-) mode respectively. The
analysis of the coupled NLS equations shows that there are three types coupled
(vector) solitons for certain frequency bands, namely,
(a) a RCP-propagating bright soliton coupled with a LCP-propagating bright
soliton,
(b) a RCP-propagating bright soliton coupled with a LCP-propagating dark
soliton, and
(c) a RCP-propagating dark soliton coupled with a LCP-propagating bright
soliton.

This paper is organized as follows. In Section II, we introduce the fluid-
Maxwell model for magnetized plasma and discuss the underlying physics. In
Section III, the system is linearized and the dispersive modes obtained are
identified. In Section IV, we assume a coupled-mode solution, and apply a
multiscale perturbative technique to derive a system of (two) coupled NLS
equations for the respective wavepacket amplitudes. In Section V, we present
analytical results for each type of vector soliton. Finally, in the concluding
Section VI, we summarize our results.

##  II Fluid-Maxwell model for EM waves

We consider an electron-ion plasma, permeated by a uniform magnetic field
along $\hat{x}$. The instantaneous state of the plasma is given by a number of
dynamical state variables, namely: the electron number density $n$, fluid
speed $\mathbf{u}=u\,\hat{x}$ and momentum (vector)
$\mathbf{p}=(p_{y},p_{z})$. These are subject to the influence of the
electrostatic (scalar) potential $\phi$ and the magnetic (vector) potential
$\mathbf{A}=(A_{y},A_{z})$. All of these variables are functions of space $x$
and time $t$, where one-dimensional (1D) propagation was considered, in the
direction parallel to the magnetic field, i.e. along $\hat{x}$. Given the
(high) frequency of interest, the ions will be assumed to be stationary.

Our analysis will be based on the following closed system of scalar equations
which govern the propagation of the circularly polarized EM waves in
magnetized plasma:

$\displaystyle\frac{\partial^{2}A_{y}}{\partial
x^{2}}-\frac{\partial^{2}A_{y}}{\partial t^{2}}=\frac{n}{\gamma}p_{y},$ (1)
$\displaystyle\frac{\partial^{2}A_{z}}{\partial
x^{2}}-\frac{\partial^{2}A_{z}}{\partial t^{2}}=\frac{n}{\gamma}p_{z},$ (2)
$\displaystyle\frac{\partial}{\partial
t}(p_{y}-A_{y})+u\frac{\partial}{\partial x}(p_{y}-A_{y})=-\frac{\Omega
p_{z}}{\gamma}$ (3) $\displaystyle\frac{\partial}{\partial
t}(p_{z}-A_{z})+u\frac{\partial}{\partial x}(p_{z}-A_{z})=\frac{\Omega
p_{y}}{\gamma}$ (4) $\displaystyle\frac{\partial n}{\partial
t}+\frac{\partial(nu)}{\partial x}=0,$ (5) $\displaystyle\frac{\partial(\gamma
u)}{\partial t}=\frac{\partial}{\partial x}(\phi-\gamma)$
$\displaystyle+\frac{1}{\gamma}\left[p_{y}\frac{\partial}{\partial
x}(p_{y}-A_{y})+p_{z}\frac{\partial}{\partial x}(p_{z}-A_{z})\right]$ (6)
$\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}=n-1.$ (7)

All physical quantities have been normalized by appropriate scales, namely:
the scalar and vector potentials are normalized by $mc^{2}/e$, the electric
field $\mathbf{E}$ by $mc\omega_{pe}/e$, the magnetic field by $\mathbf{B}$ by
$m\omega_{pe}/e$, the momentum by $mc$, the density by the $n_{0}$, the
electron velocity by the light velocity $c$. Thus, in this framework the
length is measured in units of the skin length $c/\omega_{p0}$, and time is
measured in units of the plasma period (inverse plasma frequency)
$(\omega_{pe}^{-1})$, where $\omega_{pe}=\sqrt{4\pi n_{0}e^{2}/m_{e}}$ (here
$n_{0}$ denotes the equilibrium electron density). Also, the electron momentum
can be expressed as ,

$\mathbf{P}=\mathbf{p}(x,t)+\gamma u(x,t)\hat{x}$ (8)

for a circularly polarized EM pulse, where $\gamma$ is the relativistic factor
and we have used the notation $\alpha=+1$ ($\alpha=-1$) for left- (right-)
hand circularly polarized electromagnetic waves.

##  III Multiple scale analysis for a single circularly polarized
monochromatic wavepacket

Let the state vector $\textbf{S}=(n,u,\phi;A_{y},A_{z};p_{y},p_{z})$ describe
the state of the system at a given position $x$ and time $t$. Assuming a
moderate (but finite) deviation from the equilibrium state
$S^{(0)}=(1,0,0;0,0;0,0)$, one may adopt a multiple scales perturbation
technique by expanding the system’s state as a series in terms of powers of a
small parameter $\epsilon\ll 1$ and subsequently considering multiharmonic
analysis in each order. The method was described exhaustively in the past,
e.g. in Ref. 60 in general, and also, for electrostatic and electromagnetic
waves in plasmas, in Ref. 61 and 62 respectively, hence lengthy details will
be omitted here.

The system’s state S is expanded as

S $\displaystyle=$
$\displaystyle\textbf{S}^{(0)}+\epsilon\textbf{S}^{(1)}+\epsilon^{2}\textbf{S}^{(2)}+...$
(9) $\displaystyle=$
$\displaystyle\textbf{S}^{(0)}+\sum_{n=1}^{\infty}\epsilon^{n}\textbf{S}^{(n)}$

where $S^{(0)}=(1,0,0;0,0;0,0)$ demonstrates the equilibrium state of the
system and $\epsilon\ll 1$ is a dimensionless, real and small parameter.

We now introduce new independent spatial and temporal variables,
$x_{m}=\epsilon^{m}x$, $t_{m}=\epsilon^{m}t$ ($m=0,1,2,\cdots$), and
accordingly expand the space and time derivative operators
$\partial_{x}=\partial_{x_{0}}+\epsilon\partial_{x_{1}}+\ldots$ and
$\partial_{t}=\partial_{t_{0}}+\epsilon\partial_{t_{1}}+\ldots$. Now, we use
the assumption that the wavenumber $k$ and the frequency $\omega$ (fast
variables) affect the perturbed state only via the phase $kx-\omega t$ of the
pulse, which are normalized by $\omega_{pe}$ and $k_{pe}=\omega/c$,
respectively. On the other hand, the amplitude of various frequency harmonics
is assumed to be a slowly varying function in space and time, and will thus
assumed to vary only on the slower scales ($x_{2},x_{3},...;t_{2},t_{3},...$).
The solutions to be sought of Eqs. (1-7) are therefore expressed in the form:

$\textbf{S}^{(n)}=\sum_{\ell=-\infty}^{\ell=\infty}\textbf{S}^{(n\ell)}(x_{m\geq
1},t_{m\geq 1})\,e^{i\ell(kx-\omega t)}.$ (10)

The methodology adopted in this work relies on a Newell type multiscale
technique, well known from nonlinear optics (where, interestingly space and
time are reversed, hence dispersion occurs in the frequency domain, rather
than the wavenumber space). The fundamental aim of the method it to allow for
a slow variation of the wavepacket’s amplitude (envelope) in space and time.
An excitation is considered around the equilibrium state (represented by the
zeroth order state in a smallness parameter ($\epsilon^{0}$), while the linear
(harmonic) carrier wave regime is obtained in first order ($\sim\epsilon^{1}$)
and the group velocity frame is defined in the second order
($\sim\epsilon^{2}$), upon imposing certain compatibility conditions (namely,
dictating the annihilation of secular terms occurring in that order). In each
order in $\epsilon$, a secondary expansion is considered in harmonics, from
the zeroth to the n-th harmonic. Within this (tedious but straightforward)
scheme, the first harmonic (carrier) amplitudes arise as solutions in the
first order, while the second harmonic amplitudes will arise via the second-
order second-harmonic amplitudes. Zeroth-order amplitudes will originate from
the 2nd order 2nd harmonic expressions, in combination with their 3rd order
counterparts. Finally, an evolution equation is obtained for the envelope, by
eliminating the secular terms occurring in 3rd order (1st harmonics). Linear
analysis is therefore embedded in this analysis, thus generalized to account
for the generation of higher harmonics.

Substituting Eqs. (9)-(10) into Eqs. (1)-(-7)) and collecting the terms
arising in each order in $\epsilon$, we obtain the amplitude evolution
equations at successive orders.

In order ${\cal O}(\epsilon$) (linear limit), we have the equations

$\displaystyle\partial_{t_{0}}n^{(1)}+\partial_{x_{0}}u^{(1)}=0,$ (11)
$\displaystyle\partial_{t_{0}}u^{(1)}-\partial_{x_{0}}\phi^{(1)}=0,$ (12)
$\displaystyle\partial_{x_{0}}^{2}\phi^{(1)}-n^{(1)}=0,$ (13)
$\displaystyle(\partial_{x_{0}}^{2}-\partial_{t_{0}}^{2})A_{y}^{(1)}-p_{y}^{(1)}=0,$
(14)
$\displaystyle(\partial_{x_{0}}^{2}-\partial_{t_{0}}^{2})A_{z}^{(1)}-p_{z}^{(1)}=0,$
(15) $\displaystyle\partial_{x_{0}}(p_{y}^{(1)}-A_{y}^{(1)})+\Omega
p_{z}^{(1)}=0,$ (16)
$\displaystyle\partial_{x_{0}}(p_{z}^{(1)}-A_{z}^{(1)})-\Omega p_{y}^{(1)}=0.$
(17)

  * •

for the zeroth harmonic ($\ell=0$)

$\displaystyle n^{(10)}=u^{(10)}=0,$ (18) $\displaystyle
p_{y}^{(10)}=p_{z}^{(10)}=0,$ (19) $\displaystyle
A_{y}^{(10)}=A_{z}^{(10)}=0.$ (20)

  * •

for the first harmonic ($\ell=1$)

$\displaystyle n^{(11)}=u^{(11)}=0,$ (21)
$\displaystyle(\omega^{2}-k^{2})A_{y}^{(11)}=p_{y}^{(11)},$ (22)
$\displaystyle(\omega^{2}-k^{2})A_{z}^{(11)}=p_{z}^{(11)},$ (23)
$\displaystyle-i\omega(p_{y}^{(11)}-A_{y}^{(11)})=-\Omega p_{z}^{(11)},$ (24)
$\displaystyle-i\omega(p_{z}^{(11)}-A_{z}^{(11)})=\Omega p_{y}^{(11)}.$ (25)

Considering the first harmonic amplitudes ($\ell=1$), Eqs. (22)-(25) require
an explicit compatibility condition to be adopted, in the form

$\omega^{2}-k^{2}=\frac{\omega}{\omega-\alpha\Omega}.$ (26)

One recognizes the dispersion relation for circularly-polarized waves [65].
The solution thus obtained for the first-harmonic amplitudes reads:

$\displaystyle p_{z}^{(11)}=i\alpha p^{(11)},\quad p^{(11)}\equiv
p_{y}^{(11)}.$ (27) $\displaystyle A_{z}^{(11)}=i\alpha A^{(11)},\quad
A^{(11)}\equiv A_{y}^{(11)}.$ (28)

In order ${\cal O}(\epsilon^{2}$), we have the equations

$\displaystyle\partial_{t_{0}}n^{(2)}+\partial_{x_{0}}u^{(2)}$
$\displaystyle=-\partial_{t_{1}}n^{(1)}-\partial_{x_{0}}(n^{(1)}u^{(1)})-\partial_{x_{1}}u^{(1)},$
(29) $\displaystyle\partial_{t_{0}}u^{(2)}-\partial_{x_{0}}\phi^{(2)}$
$\displaystyle=-\partial_{t_{1}}u^{(1)}+\partial_{x_{1}}(\phi^{(1)}+p_{y}^{(1)}\partial_{x_{0}}(p_{y}^{(1)}-A_{y}^{(1)})$
$\displaystyle+p_{z}^{(1)}\partial_{x_{0}}(p_{z}^{(1)}-A_{z}^{(1)}),$ (30)
$\displaystyle\partial_{x_{0}}^{2}\phi^{(2)}-n^{(2)}=-2\partial_{x_{0}}\partial_{x_{1}}\phi^{(1)},$
(31)
$\displaystyle(\partial_{x_{0}}^{2}-\partial_{t_{0}}^{2})A_{y}^{(2)}-p_{y}^{(2)}$
$\displaystyle=-2(\partial_{x_{0}}\partial_{x_{1}}-\partial_{t_{0}}\partial_{t_{1}})A_{y}^{(1)}+n^{(1)}p_{y}^{(1)},$
(32)
$\displaystyle(\partial_{x_{0}}^{2}-\partial_{t_{0}}^{2})A_{z}^{(2)}-p_{z}^{(2)}$
$\displaystyle=-2(\partial_{x_{0}}\partial_{x_{1}}-\partial_{t_{0}}\partial_{t_{1}})A_{z}^{(1)}+n^{(1)}p_{z}^{(1)},$
(33) $\displaystyle\partial_{t_{0}}(p_{y}^{(2)}-A_{y}^{(2)})+\Omega
p_{z}^{(2)}$
$\displaystyle=-(\partial_{t_{1}}+u^{(1)}\partial_{x_{0}})(p_{y}^{(1)}-A_{y}^{(1)}),$
(34) $\displaystyle\partial_{t_{0}}(p_{z}^{(2)}-A_{z}^{(2)})-\Omega
p_{z}^{(2)}$
$\displaystyle=-(\partial_{t_{1}}+u^{(1)}\partial_{x_{0}})(p_{z}^{(1)}-A_{z}^{(1)}).$
(35)

Same as in the Ref. [62], in the first harmonic (($\ell=1$), the condition for
annihilation of secular terms leads to $\partial\cdot/\partial
t_{1}+v_{g}\,\partial\cdot/\partial x_{1}=0$ (for $\cdot=p_{y/z}^{(1)}$ or
$A_{y/z}^{(1)}$), implying that the envelope moves at the group velocity
$v_{g}=\omega^{\prime}(k)$, given by:

$v_{g}=\frac{2k(\omega-\alpha\Omega)^{2}}{2\omega(\omega-\alpha\Omega)^{2}+\alpha\Omega}\,,$
(36)

Following the same procedure in order ${\cal O}(\epsilon^{3}$), and using the
variables $X=x_{1}-v_{g}t_{1}\equiv\epsilon(x-v_{g}t)$ and
$T=t_{2}\equiv\epsilon^{2}t$, we derive from the non-secularity condition at
$O(\epsilon^{3})$ the NLS Eq. (37)

$i\frac{\partial\psi}{\partial\tau}+P\frac{\partial^{2}\psi}{\partial\xi^{2}}+Q|\psi|^{2}=0,\\\
$ (37)

where $\psi$ denotes the amplitude $A_{y}^{(11)}$, the (slow) time and space
variables are $\tau=t_{2}$ and $\xi=x_{1}-v_{g}t_{1}$, and the dispersion
coefficient $P$ and the nonlinear self-phase modulation (SPM) coefficient $Q$
are respectively given by the (real) expressions:

$\displaystyle P\equiv\frac{1}{2}\frac{\partial^{2}\omega}{\partial
k^{2}}=\frac{v_{g}}{2k}+\frac{v_{g}^{2}}{\omega-\alpha\Omega}+\frac{v_{g}^{3}(3\omega-\alpha\Omega)}{2k(\omega-\alpha\Omega)}$
(38) $\displaystyle Q=\frac{v_{g}}{k}(\omega^{2}-k^{2})^{4}\,.$ (39)

Figure 1: The dispersion relation showing the frequency $\omega$ as a function
of the wave number $k$ (normalized values). The different values of $\Omega$,
i.e., $\Omega=2$, $\Omega=0.8$, and $\Omega=0.2$ are depicted by bold (green),
dashed (red) and thin solid (blue) lines, respectively. Top panel: The LCP EM
waves. Bottom panel: The RCP EM waves.

##  IV Multiple scale analysis for coupled modes

Now, we consider for a solution in the form

$S^{(11)}=\sum_{j=1}^{2}S_{j}^{(11)}(x_{1},x_{2},\ldots,t_{1},t_{2},\ldots)\exp(i\theta_{j})+{\rm
c.c.},$ (40)

where subscripts $j=1$ and $j=2$ correspond to the LCP and RCP waves,
$S_{j}^{11}$ is an unknown complex function,
$\theta_{j}=k_{j}x_{0}-\omega_{j}t_{0}$, while the wavenumbers $k_{j}$ and
frequencies $\omega_{j}$ satisfy the dispersion relation provided in Eq. (26).
In Fig. 1, we show the dispersion relation for the $\Omega=0.2$, $\Omega=0.8$,
and $\Omega=2$. It is clear, from Fig. 1, that for LCP EM wave (top panel)
there exist two frequency bands where the propagation is possible: the high-
frequency band and the low-frequency band. For $\Omega=0.2$ (depicted by solid
(blue line) the propagation is possible for $\omega>1.105$ (high-frequency
band) and for $\omega<0.2$ (low-frequency band) namely a gap appears between
$0.2<\omega<1.105$ where EM wave propagation is not possible. For $\Omega=0.8$
(depicted by dashed (red) line) the propagation is possible for $\omega>1.477$
(high-frequency band) and for $\omega<0.8$ (low-frequency band) where the gap
appears between $0.8<\omega<1.477$. Also, for $\Omega=2$ (depicted by thick
(green) line) the propagation is possible for $\omega>2.414$ (high-frequency
band) and for $\omega<2$ (low-frequency band). In this case the gap appears
between $2<\omega<2.414$. Moreover, for the RCP EM wave (bottom panel)
propagation is possible for frequencies $\omega>0.9047$ when $\Omega=0.2$
(depicted by solid (blue line). In case where $\Omega=0.8$(depicted by dashed
(red) line) and $\Omega=2$ (depicted by thick (green) line) the propagation is
possible for frequencies $\omega>0.6767$ and $\omega>0.414$,respectively.

Furthermore, the gap which appears for LCP EM waves increases (decreases) as
the $\Omega$ decreases (increases). It is important to notice that there is a
frequency region, which is independent from polarization of EM waves, where
the propagation cannot take place for values of $\Omega<0.707$.

Below we will demonstrate that in the nonlinear setting, coupling between LCP
and RCP EM waves with equal group velocities is possible, and we consider the
interaction between LCP and RCP waves. Observing Fig. 1 and knowing that the
group velocity at a given wavenumber represents a tangent (line) to the
dispersion curve, it is possible to identify domains, with equal group
velocities. Thus, using Eq. (26) we obtain the group velocity
$v_{g}=\partial\omega/\partial k$:

$v_{g}=\frac{2k(\omega-\alpha\Omega)^{2}}{2\omega(\omega-\alpha\Omega)^{2}+\alpha\Omega}\,,$

i.e. Eq. (36) above.

Figure 2: The group velocity $\upsilon_{g}$ as a function of the normalized
frequency $\omega$. The dashed (red) and solid (blue) dashed (red) lines show
the group velocity $\upsilon_{g}$ for RCP- and LCP- EM waves, respectively.

This result is presented in Fig. 2, where the group velocity $v_{g}$ is
plotted as a function of the normalized frequency $\omega$. Notice that the
dashed (red) solid (blue) and solid (blue) lines show the group velocity
$\upsilon_{g}$ for RCP- and LCP- EM waves, respectively.

---
Figure 3: The group velocity $\upsilon_{g}$ as a function of the normalized
frequency $\omega$ for different values of $\Omega$, i.e., $\Omega=0.2$ (top
panel), $\Omega=0.8$ (middle panel), and $\Omega=2$ (bottom panel). The dashed
(red) and solid (blue) dashed (red) lines show the group velocity
$\upsilon_{g}$ for RCP- and LCP- EM waves, respectively.

As seen in Fig. 2 using a horizontal cut we can to observe that there are a
RCP and a LCP electromagnetic wave which have the same group velocity (and are
coupled in the nonlinear regime).

The point A,B, C and D, in Fig. 2, have the same group velocity, namely,
$v_{g}=0.4$. Also, we see that there is a maximum possible common
$v_{g_{max}}^{\ell}$ in the low frequency band for LCP wave. Thus, one can
divide each of the group-velocity curves into three sub-regions, depending on
the sign of the group-velocity dispersion (GVD), $\partial
v_{g}/\partial\omega$, where the interaction with equal group velocities may
appear. As depicted in Fig. 2 these subregions are: (a) the sub-bands I and II
for the low frequency band of the LCP EM wave, characterized by positive and
negative GVD respectively, and the sub-bands III for the high frequency band
of the LCP EM wave, characterized by positive GVD, (b) the band of the RCP EM
wave, again characterized by positive GVD respectively. Thus, nonlinear LCP
and RCP modes of equal $v_{g}$ can feature the following three different
possible interactions:

  * •

Case 1:band of the RCP-mode and the LCP-mode in band I, both featuring
positive GVD for $v_{g}\leq v_{g_{max}}^{\ell}$.

  * •

Case 2: band of the RCP-mode and the LCP-mode in band II; here, the RCP (LCP)
mode features positive (negative) GVD for $v_{g}\leq v_{g_{max}}^{\ell}$.

  * •

Case 3: band of the RCP-mode and the LCP-mode in band III, both featuring
positive GVD for $0<v_{g}\leq 1$.

As shown in Fig. 3, upon inspection of the group-velocity curves for
$\Omega=0.8$ (middle panel) we see that the maximum possible common
$v_{g_{max}}^{\ell}=0.4661$ occurring at $\omega=0.198$, in the low frequency
band for LCP wave. Then, the subregions are: (a) the sub-bands I
($0<\omega<0.198$) and II ($0.198<\omega<0.8$) for the low frequency band and
the sub-bands III ($\omega>1.477$)for the high frequency band of the LCP EM
wave (b) the band of the RCP EM wave ($\omega>0.6767$).

Notice that, $v_{g}=1$ is the group velocity for unmagnetized plasma. It is
obvious that the above considerations is the result of the existence of the
frequency gap as shows in Fig. 3 for all values of $\Omega$, i.e, for
$\Omega=0.2$, $\Omega=0.8$and $\Omega=2$. Furthermore, we can observe that as
$\Omega$ increases the $v_{g}^{\ell}$ is raised and the gap is smaller. When
the $\Omega\gg 1$ then $v_{g}\rightarrow 1$ and the LCP and RCP wave are
degenerated.

###  IV.1 Nonlinear analysis: the coupled NLS equations

In our case, since we study the interaction between a LCP and a RCP nonlinear
mode with equal group velocities, we seek for a solution of Eqs. (1-7) in the
form:

$S^{(n)}=\sum_{j=1}^{2}S_{j}^{(n)}(X,T)\exp(i\theta_{j})+{\rm c.c.},$ (41)

where “c.c.” denotes complex conjugate. In Eq. (41), subscripts $j=1,2$
correspond to the LCP and RCP mode, $S_{j}^{(n)}(X,T)$ are unknown
(continuous) slowly-varying envelope functions depending on the slow scales
$X=\epsilon(x-v_{g}t)$ (where $v_{g}$ is the common group velocity) and
$T=\epsilon^{2}t$, while $\exp(i\theta_{j})$, with
$\theta_{j}=k_{j}x-\omega_{j}t$, are the carriers of frequencies $\omega_{j}$
and wavenumbers $k_{j}$. The $\epsilon$ is a formal small parameter related to
the soliton amplitude (see below). According to Eq. (41), the field
$S_{j}^{(n)}$ is the leading-order form of a more general ansatz employing
multiple time and space scales. In this context, use of a formal multi-scale
expansion method leads to a hierarchy of equations at various powers of
$\epsilon$, which are solved up to the third-order. Indeed, at orders
$\mathcal{O}(\epsilon)$ (linear limit) and $\mathcal{O}(\epsilon^{2})$, we
derive the dispersion relation, Eq. (26), and the group velocity, Eq. (36),
respectively. At the order, $\mathcal{O}(\epsilon^{3})$ we obtain the
nonlinear coupled NLS equations as

$\displaystyle
i\partial_{T}\Psi_{1}+D_{1}\partial_{X}^{2}\Psi_{1}+\left(g_{11}|\Psi_{1}|^{2}+g_{12}|\Psi_{2}|^{2}\right)\Psi_{1}=0,$
(42) $\displaystyle
i\partial_{T}\Psi_{2}+D_{2}\partial_{X}^{2}\Psi_{2}+\left(g_{21}|\Psi_{1}|^{2}+g_{22}|\Psi_{2}|^{2}\right)\Psi_{2}=0,$
(43)

where the amplitude $\Psi_{j}$, the normalized GVD coefficients $D_{j}$, the
self-phase modulation (SPM) coefficients $g_{jj}$, and the cross-phase
modulation (CPM) coefficients $g_{j,3-j}$ (with $j=1,2$) are respectively
given by:

$\displaystyle\Psi_{j}\equiv A_{j}^{(11)}$ (44) $\displaystyle
D_{j}\equiv\frac{1}{2}\frac{\partial^{2}\omega_{j}}{\partial
k_{j}^{2}}=\frac{v_{g}}{2k_{j}}+\frac{v_{g}^{2}}{\omega-\alpha\Omega}-\frac{v_{g}^{3}(3\omega-\alpha\Omega)}{2k_{j}(\omega-\alpha\Omega)}$
(45) $\displaystyle g_{jj}=\frac{v_{g}}{k_{j}}(\omega_{j}^{2}-k_{j}^{2})^{4},$
(46) $\displaystyle
g_{j,3-j}=\frac{2v_{g}}{k_{j}}(\omega_{j}^{2}-k_{j}^{2})^{2}(\omega_{3-j}^{2}-k_{3-j}^{2})^{2}.$
(47)

Note that, using the Eqs. (26) and (36) the above relations be cast in the
form:

$\displaystyle
g_{jj}=\frac{2\omega_{j}^{4}}{[2\omega_{j}(\omega_{j}-\alpha\Omega)^{2}+\alpha\Omega](\omega_{j}-\alpha\Omega)^{2}},$
(48) $\displaystyle
g_{j,3-j}=\frac{4\omega_{j}^{2}\omega_{j,3-j}^{2}}{[2\omega_{j}(\omega_{j}-\alpha\Omega)^{2}+\alpha\Omega](\omega_{j,3-j}-\alpha\Omega)^{2}}.$
(49)

As shown from Eqs. (46)-(47) the coefficients $g_{jj}$, and $g_{j,3-j}$ are
always positive. Next, using scale transformations, we measure normalized time
$T$ and densities $|\Psi_{j}|^{2}$ in units of $|D_{1}|^{-1}$ and
$|D_{1}/g_{jj}|$ respectively, and cast Eqs. (42)-(43) in the form:

$\displaystyle
i\partial_{T}\Psi_{1}+s~{}\partial_{X}^{2}\Psi_{1}+\left(\Psi_{1}|^{2}+\lambda_{1}|\Psi_{2}|^{2}\right)\Psi_{1}=0,$
(50) $\displaystyle
i\partial_{T}\Psi_{2}+d~{}\partial_{X}^{2}\Psi_{2}+\left(\lambda_{2}|\Psi_{1}|^{2}+\Psi_{2}|^{2}\right)\Psi_{2}=0,$
(51)

where

$\displaystyle s$ $\displaystyle=$ $\displaystyle{\rm sign}(D_{1}),\quad
d=\frac{D_{2}}{\left|D_{1}\right|},$ $\displaystyle\lambda_{1}$
$\displaystyle=$
$\displaystyle\frac{g_{12}}{|g_{22}|},\quad\lambda_{2}=\frac{g_{21}}{|g_{11}|}.$
(52)

where the coefficients $\lambda_{1,2}$ and $d$ are positive. As arises from
Eqs. (50) when the coupling coefficients ($\lambda_{j}=0$) are zero then the
evolution of either the LCP wave $\Psi_{1}$ or the RCP wave $\Psi_{2}$ is
described by a single NLS equation; the latter, supports soliton solutions of
the dark or the bright type, depending on the relative signs of dispersion and
nonlinearity coefficients (see, e.g., Ref. [70]). Particularly, the mode
$\Psi_{1}$ supports dark solitons for $s<0$ or bright solitons for $s>0$. The
mode $\Psi_{2}$ supports only bright solitons because the coefficient $d$ is
always positive. However, because the above conditions are modified when
$\lambda_{j}\neq 0$ various types of vector (coupled) solitons can be found as
seen from the Eqs. (50) and (51).

According to the Ref. [71], four types of vector solitons are possible:

  * •

bright-bright (BB) solitons, in the form:

$\displaystyle\Psi_{1}(X,T)$ $\displaystyle=$ $\displaystyle\Psi_{1,0}{\rm
sech}(bX)\exp(-i\eta_{1}T),$ (53) $\displaystyle\Psi_{2}(X,T)$
$\displaystyle=$ $\displaystyle\Psi_{2,0}{\rm sech}(bX)\exp(-i\eta_{2}T).$
(54)

  * •

bright-dark (BD) solitons, in the form:

$\displaystyle\Psi_{1}(X,T)$ $\displaystyle=$ $\displaystyle\Psi_{1,0}{\rm
sech}(bX)\exp(-i\eta_{1}T),$ (55) $\displaystyle\Psi_{2}(X,T)$
$\displaystyle=$ $\displaystyle\Psi_{2,0}{\rm tanh}(bX)\exp(-i\eta_{2}T),$
(56)

  * •

dark-bright (DB) solitons, in the form:

$\displaystyle\Psi_{1}(X,T)$ $\displaystyle=$ $\displaystyle\Psi_{1,0}{\rm
tanh}(bX)\exp(-i\eta_{1}T),$ (57) $\displaystyle\Psi_{2}(X,T)$
$\displaystyle=$ $\displaystyle\Psi_{2,0}{\rm sech}(bX)\exp(-i\eta_{2}T),$
(58)

  * •

dark-dark (DD) solitons, in the form:

$\displaystyle\Psi_{1}(X,T)$ $\displaystyle=$ $\displaystyle\Psi_{1,0}{\rm
tanh}(bX)\exp(-i\eta_{1}T),$ (59) $\displaystyle\Psi_{2}(X,T)$
$\displaystyle=$ $\displaystyle\Psi_{2,0}{\rm tanh}(bX)\exp(-i\eta_{2}T).$
(60)

In the above equations, $\eta_{j}$ (j=1,2) and $\Psi_{j,0}$ declare the
frequencies and amplitudes of each soliton, while $b$ is the (common) inverse
width of the solitons.

Now, each of the above ansatz is substituted in Eqs. (50) and (51), leading to
a set of equations connecting the soliton parameters. Particularly, the
equations connecting the amplitudes $\Psi_{j,0}$ and the inverse width $b$ are
of the form:

$\displaystyle(\Psi_{2,0}/\Psi_{1,0})^{2}$ $\displaystyle=$
$\displaystyle-\alpha_{1},\qquad(b/\Psi_{1,0})^{2}=-\alpha_{2},$ (61)
$\displaystyle(\Psi_{2,0}/\Psi_{1,0})^{2}$ $\displaystyle=$
$\displaystyle\alpha_{1},~{}~{}\qquad(b/\Psi_{1,0})^{2}=-\alpha_{2},$ (62)
$\displaystyle(\Psi_{2,0}/\Psi_{1,0})^{2}$ $\displaystyle=$
$\displaystyle\alpha_{1},~{}~{}\qquad(b/\Psi_{1,0})^{2}=\alpha_{2},$ (63)
$\displaystyle(\Psi_{2,0}/\Psi_{1,0})^{2}$ $\displaystyle=$
$\displaystyle-\alpha_{1},\qquad(b/\Psi_{1,0})^{2}=\alpha_{2},$ (64)

for the BB, BD, DB and DD solitons respectively, where parameters $\alpha_{j}$
($j=1,2$) are given by:

$\alpha_{1}=\frac{s\lambda_{2}-d}{s-d\lambda_{1}},~{}\alpha_{2}=\frac{\lambda_{1}\lambda_{2}-1}{2(s-d\lambda_{1})}.$
(65)

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|  |
Figure 4: Case 1: (Left panels) The dependence of the parameters,
$\lambda_{1}$ [thin solid (blue) line], $\lambda_{2}$ [dashed (blue) line] and
$d$ [bold solid (red) line] on the normalized frequency $\omega$, for
different values of $\Omega$, i.e., $\Omega=0.2$ (top panel), $\Omega=0.8$
(middle panel), and $\Omega=2$ (bottom panel). (Right panels) The same for the
coefficients $\alpha_{1}$ [solid (blue) line], $\alpha_{2}$ [dashed (red)
line].

It is worth noting that a similar set of coupled NLS equation has been derived
in the past for modulational wavepacket interaction in electron-ion plasmas,
considering either upper-hybrid [72] or, more recently, electrostatic [73, 74,
75] waves. Although the mathematical setting is analogous, and may therefore
interest the reader of this paper, the physical background in this paper is
distinct from those works; in those earlier studies, for instance, an
unmagnetized plasma was considered, in the electrostatic approximation (i.e.
neglecting magnetization, thus precluding electromagnetic excitations), with
the focus being on the nonthermal (kappa-distributed) background electron
statistics (a common occurrence in Space plasmas). Contrary to that picture,
our focus here is on electromagnetic solitary waves propagating in magnetized
plasma and, in particular, on the role of the ambient magnetic field on their
structural and propagation characteristics.

##  V Soliton interactions in different frequency bands

It is important to understand that the sign of the parameters $\alpha_{j}$
($j=1,2$) defined above determines the type of vector solitons that may occur
in the plasma, as dictated by Eqs. (61)-(64). Also, the parameters
$\alpha_{j}$ depend on the frequency and the magnetic field according to Eq.
(65), through coefficients $\lambda_{1,2}$, $s$ and $d$. Consequently, we
expect that the precise solitons solutions will arise from the investigation
of the sign of the parameters $\alpha_{j}$ and the value of magnetic field,
for all cases (Case 1, 2 and 3) as have been defined in Section V. On the
other hand, the Eqs. (50)-(51) are no longer of the Manakov type and, thus,
generally, they are not completely integrable. Nevertheless, standing wave
soliton solutions whose exact analytical form depend on by sign of the
parameters $\alpha_{j}$ can still be found. That is valid for all cases of
interactions (Case 1, 2, and 3).

###  V.1 Case 1: Solitons in bands I and RCP.

Initially, we study the coupling between a propagating soliton, with a
frequency that is in band I, and a propagating soliton, with a frequency that
is in RCP band. In Fig. 4 we depict the dependence of the parameters
$\lambda_{1}$, $\lambda_{2}$ and $d$, as a function of the normalized
frequency $\omega$ (for $\Omega=0.2$, $\Omega=0.8$ and $\Omega=2$). Now, we
have $s=+1$ (cf. Fig. 3) and $\lambda_{1,2}>0$ and $d>0$, as can be seen in
the Fig. 4. Also, the parameters $\alpha_{j}$ ($j=1,2$) are given by:

$\displaystyle\alpha_{1}=\frac{\lambda_{2}-d}{1-d\lambda_{1}},$ (66)
$\displaystyle\alpha_{2}=\frac{\lambda_{1}\lambda_{2}-1}{2(1-d\lambda_{1})}.$
(67)

In this case, the Eqs. (50)-(51) are no longer of the Manakov type and, thus,
generally, they are not completely integrable. Nevertheless, standing wave
solitons solutions can still be found whose exact analytical form depend on by
sign of the parameters $\alpha_{j}$. 1. When $\alpha_{1}>0$ and $\alpha_{2}>0$
then we have a case of coupled solitons, a dark soliton and a bright soliton,
whose exact analytical form are given by Eqs. (57)-(58), where the soliton
amplitude parameters $\eta_{1,2}$ and the inverse width $b$ are connected via
the following equations:

$\displaystyle\eta_{1}=-\Psi_{1,0}^{2},$ (68)
$\displaystyle\eta_{2}=-(d\alpha_{2}+\lambda_{2})\Psi_{1,0}^{2},$ (69)
$\displaystyle b^{2}=\alpha_{2}\Psi_{1,0}^{2},$ (70)

It is thus clear that the above solutions have one free parameter. Employing
the solutions (57)-(58), we can approximate the vector potential $A_{y}(x,t)$,
in terms of the original coordinates $x$ and $t$, as follows:

$\displaystyle A_{y}(x,t)$ $\displaystyle\approx$ $\displaystyle
A_{y0}[\Psi_{1}(x,t)\cos(k_{1}x-\omega_{01}t)$ (71) $\displaystyle+$
$\displaystyle\Psi_{2}(x,t)\cos(k_{2}x-\omega_{02}t)],$

The amplitude $A_{0}$ and the frequencies $\omega_{0j}$ ($j=1,2$) of the
soliton are given by:

$\displaystyle A_{y0}$ $\displaystyle=$ $\displaystyle
2\epsilon\sqrt{\left|\frac{D_{1}}{g_{11}}\right|},$ (72)
$\displaystyle\omega_{0j}$ $\displaystyle=$
$\displaystyle\omega_{j}+\epsilon^{2}\eta_{j}|D_{1}|.$ (73)

In this case,

$\displaystyle\Psi_{1}=\Psi_{1,0}\tanh[\epsilon b(x-v_{g}t)],$ (74)
$\displaystyle\Psi_{2}=\sqrt{\left|\frac{g_{11}}{g_{22}}\right|\alpha_{1}}\Psi_{1,0}\operatorname{sech}[\epsilon
b(x-v_{g}t)].$ (75)

---
Figure 5: Case 1: The profiles (at t=0) of the dark and bright solitons (top
panel) and dark-bright soliton (bottom panel) for $\Omega=0.8$ . The
parameters used are $v_{g}=0.2965$, $\omega_{1}=0.0335$ ($k_{1}=0.2119$) and
$\omega_{2}=0.7007$ ($k_{2}=0.1551$). Also, we use $\epsilon=0.1$ and
$\Psi_{1,0}=0.05$.

In Fig. 5 we show the profiles (at t=0) of the dark (LCP mode) and bright (RCP
mode) solitons in the absence of coupling (top panel) as well as the dark-
bright soliton (bottom panel), for $\Omega=0.8$. Now, the group velocity of
the DB soliton (common for both components) is $v_{g}=0.2965$, which occurs
when the angular frequencies for the modes $\Psi_{1}$ and $\Psi_{2}$ take,
respectively, the values $\omega_{1}=0.0335$ ($k_{1}=0.2119$) and
$\omega_{2}=0.7007$ ($k_{2}=0.1551$). 2. When $\alpha_{1}<0$ and
$\alpha_{2}<0$ then we have a case of coupled bright solitons, whose exact
analytical form are given by Eqs. (53)-(54). Now, the parameters $\eta_{1,2}$
and $b$ are connected via the following equations:

$\displaystyle\eta_{1}=\alpha_{2}\Psi_{1,0}^{2},$ (76)
$\displaystyle\eta_{2}=d\alpha_{2}\Psi_{1,0}^{2},$ (77) $\displaystyle
b^{2}=-\alpha_{2}\Psi_{1,0}^{2}.$ (78)

The vector potential $A_{y}(x,t)$, in terms of the original coordinates $x$
and $t$ and soliton parameters, are given by the Eq. (71) and Eqs. (72)-(73)
respectively, where, now

$\displaystyle\Psi_{1}=\Psi_{1,0}\operatorname{sech}[\epsilon b(x-v_{g}t)],$
(79)
$\displaystyle\Psi_{2}=\sqrt{\left|\frac{g_{11}}{g_{22}}\alpha_{1}\right|}\Psi_{1,0}\operatorname{sech}[\epsilon
b(x-v_{g}t)].$ (80)

---
Figure 6: Case 1: The profiles (at t=0) of the bright solitons (top panel) and
dark-bright soliton (bottom panel) for $\Omega=0.8$ . The parameters used are
$v_{g}=0.4572$, $\omega_{1}=0.1454$ ($k_{1}=0.4932$) and $\omega_{2}=0.77414$
($k_{2}=0.2621$). Also, we use $\epsilon=0.1$ and $\Psi_{1,0}=0.05$.

In Fig. 6 we show the profiles (at t=0) of the bright (LCP mode) and bright
(RCP mode) solitons in the absence of coupling (top panel) as well as the
bright-bright soliton (bottom panel), for $\Omega=0.8$. Now, the group
velocity of the BB soliton (common for both components) is $v_{g}=0.4572$,
which occurs when the angular frequencies for the modes $\Psi_{1}$ and
$\Psi_{2}$ take, respectively, the values $\omega_{1}=0.1454$ ($k_{1}=0.4932$)
and $\omega_{2}=0.77414$ ($k_{2}=0.2621$).

---
Figure 7: Case 1: The profiles (at t=0) of the bright and dark solitons (top
panel) and bright-dark soliton (bottom panel) for $\Omega=0.8$ . The
parameters used are $v_{g}=0.4647$, $\omega_{1}=0.1743$ ($k_{1}=0.5558$) and
$\omega_{2}=0.744$ ($k_{2}=0.2677$). Also, we use $\epsilon=0.1$ and
$\Psi_{1,0}=0.05$.

3. When $\alpha_{1}>0$ and $\alpha_{2}<0$ then we have a case of coupled solitons, a bright soliton and a dark soliton, whose exact analytical form are given by Eqs. (55)-(56), where the soliton amplitude parameters $\eta_{1,2}$ and the inverse width $b$ are connected via the following equations:

$\displaystyle\eta_{1}=(\alpha_{2}-\alpha_{1}\lambda_{1})\Psi_{1,0}^{2},$ (81)
$\displaystyle\eta_{2}=-\alpha_{1}\Psi_{1,0}^{2},$ (82) $\displaystyle
b^{2}=-\alpha_{2}\Psi_{1,0}^{2}.$ (83)

Now, we can approximate the vector potential $A_{y}(x,t)$, in terms of the
original coordinates $x$ and $t$, as in the Eq. (71) where, in this case,

$\displaystyle\Psi_{1}=\Psi_{1,0}\operatorname{sech}[\epsilon b(x-v_{g}t)],$
(84)
$\displaystyle\Psi_{2}=\sqrt{\left|\frac{g_{11}}{g_{22}}\right|\alpha_{1}}\Psi_{1,0}\tanh[\epsilon
b(x-v_{g}t)].$ (85)

In this case, the solution amplitude $A_{0}$ and the frequencies $\omega_{0j}$
($j=1,2$) are given by Eqs. (72)-(73).

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Figure 8: Same as Fig. 4, but for Case 2. |  |
---|---|---
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|  |
Figure 9: Same as Fig. 4, but for Case 3.

In Fig. 7 we show the profiles (at t=0) of the bright (LCP mode) and dark (RCP
mode) solitons in the absence of coupling (top panel) as well as the bright-
dark soliton (bottom panel), for $\Omega=0.8$. Now, the group velocity of the
BD soliton (common for both components) is $v_{g}=0.4647$, which occurs when
the angular frequencies for the modes $\Psi_{1}$ and $\Psi_{2}$ take,
respectively, the values $\omega_{1}=0.1743$ ($k_{1}=0.5558$) and
$\omega_{2}=0.744$ ($k_{2}=0.2677$).

TABLE 1

$\alpha_{1}$
---
$\alpha_{2}$ | - | +
- |  BB |  BD
+ |  |  DB

The above table shows the existence of vector solitons according to the sign
of the coefficients $\alpha_{1}$ and $\alpha_{2}$, for the case 1. Moreover,
as are depicted in Fig. 4, for $\Omega=0.8$, the above solutions are defined
only for a narrow frequency band $(0-0.129)$, $(0.136-0.166)$ and
$(0.17-0.198)$ for dark-bright (DB), bright-bright (BB) and bright-dark (BD)
interactions , respectively. Also, as is observed in the Fig. 4, as $\Omega$
is decreased (increased) lead to an increase (decrease) of the width of the DB
band where DB solitons can be formed, while the width of BB and BD band is
decreased (increased). As seen in this figure, for $\Omega=0.2$ the DB, BB and
BD band possess, approximately, the $80\%$, $10\%$ and $6\%$ of I-band,
respectively. Notice that for $\Omega=0.8$ ($\Omega=2$) the DB, BB and BD band
possess, the $65\%$ ($47\%$), $15\%$ ($21\%$) and $14\%$ ($30\%$) of I-band,
respectively.

###  V.2 Case 2: Bright-dark solitons in bands II and RCP.

In this case $s=-1$, we study the coupling between a propagating soliton, with
a frequency that is in band II, and a propagating soliton, with a frequency
that is in RCP band. In Fig. 8 we depict the dependence of the parameters
$\lambda_{1}$, $\lambda_{2}$ and $d$, as a function of the normalized
frequency $\omega$ (for $\Omega=0.2$, $\Omega=0.8$ and $\Omega=2$). Now, the
coefficients $\alpha_{1}$ and $\alpha_{2}$ take the following form:

$\displaystyle\alpha_{1}=\frac{\lambda_{2}+d}{1+d\lambda_{1}},$ (86)
$\displaystyle\alpha_{2}=-\frac{\lambda_{1}\lambda_{2}-1}{2(1+d\lambda_{1})}.$
(87)

Following our previous considerations, here we have a case of coupled bright-
dark solitons where the exact analytical form is given by Eqs. (55)-(56). The
soliton amplitude parameters $\eta_{1,2}$ and the inverse width $b$ are
connected via the following equations:

$\displaystyle\eta_{1}=-(\alpha_{2}+\alpha_{1}\lambda_{1})\Psi_{1,0}^{2},$
(88) $\displaystyle\eta_{2}=-\alpha_{1}\Psi_{1,0}^{2},$ (89) $\displaystyle
b^{2}=-\alpha_{2}\Psi_{1,0}^{2}.$ (90)

The analytical form for vector potential $A_{y}(x,t)$ and the solitons
parameters are same with the previous case, namely, in the interaction in
bands I and RCP [c.f Eqs. (84)-(85)].

Figure 10: Different types of solitons as a function of the angular
(normalized) frequency $\omega$ are depicted in the table, for the different
values of $\Omega$.

###  V.3 Case 3: Solitons in bands III and RCP.

Finally, we study the coupling between a propagating soliton, with a frequency
that is in band III, and a propagating soliton, with a frequency that is in
RCP band. Now, $s=+1$ (cf. Fig. 3) while the other dispersion and nonlinearity
coefficients are shown in Fig. 9 as functions of the normalized frequency
$\omega$ for $\Omega=0.2$, $\Omega=0.8$ and $\Omega=2$. Also, as is observed
in the Fig. 9, as $\Omega$ is increased lead to an increase of the width of
the BD band where BD solitons can be formed, while the width of BB is
decreased.

###  V.4 Discussions

The analysis of previous section end in some important results which confirm
our expectations. First of all in the Case 1 (Solitons in bands I and RCP) we
obtain three types of vector solitons, namely bright-bright, bright-dark and
dark-bright. Also, we observe that for low frequencies and $\Omega$ values the
width of the dark-bright band where dark-bright solitons can be formed
possesses the larger percentage of the band. However as $\Omega$ is increased
the occurrence of the dark-bright type is decreased while the width of the
bright-bright and mainly also that of the bright-dark category are increased.
In Case 2 (Case 2: Bright-dark solitons in bands II and RCP) for all the
frequencies and $\Omega$ values, we have only the existence of bright-dark
solitons. Finally, in Case 3 (Solitons in bands III and RCP) we have only
bright-bright solitons in all the frequencies for low $\Omega$ values. As
$\Omega$ is increased bright-dark solitons appeared which dominate in all
frequency band for the large values of $\Omega$.

According to the table in Fig. 10, we found that bright-dark solitons exist in
all frequency bands for all values of $\Omega$. Furthermore, we observe
bright-bright solitons in bands (I-RCP) and (III-RCP) for the small values of
$\Omega$. Also, we found that the dark-bright solitons exist only in I-RCP
frequency band for small values of $\omega$ and $\Omega$.

Generally our detailed analysis can support various coupled NLS systems in
magnetized plasmas with the equal group velocities between the propagating
waves. Our analytical investigation reveals the existence of various type
vector solitons taking into account the certain frequency bands and the
strength of the magnetic field. It is very important to note that from our
suggested method the capability control of a solitonic mode e.g. LCP (RCP)
from the change the amplitude or width of other mode RCP (LCP) emerges.

Although our present work has an analogous mathematical settings with the
works of Refs. [62], [71], [73], and [75] it has serious differences as: a)
The physical background in this paper is different compared to Refs. [71],
[73], and [75] . In Refs. [73] and [75] the study was carried out in the
framework of an unmagnetized plasma while in the Ref. [71] the physical
background was completely different where a nonlinear composite right left
handed transmission line was studied, b) With regards to Ref. [62], where the
physical background is same, namely the magnetized plasma, in this paper we
study the interaction between left hand circularly polarized (LCP) and right
hand circularly polarized (RCP) solitons while in Ref. [62], we investigated
only the beam-plasma interaction, c) As previously mentioned, in this paper,
we investigate the interaction between of a LCP and RCP electromagnetic wave
which have the same group velocity in different frequency bands. In Refs [73]
and [75] the electrostatic wave packets are studied in a regime with different
group velocities, d) In this work, we study the vector soliton solutions in
different frequency bands and for different magnetic fields. The analysis in
Ref. [71] was carried out only for different frequency bands, in Ref. [73],
the instability growth rate was investigated while in the Ref. [75], the
analysis was focused on the values of the $\kappa$-index of k-distribution
function.

##  VI Conclusion

In conclusion, we have performed analytical techniques to study the existence
of coupled left- and right-circularly polarized solitons in magnetized
plasmas. Our analysis started with the derivation of a closed system of scalar
equations which the govern of the propagation of LCP and RCP EM waves in
magnetized plasmas. Upon the linear limit, the dispersion relation has been
derived and we have shown that there are LCP- and RCP modes, with the same
group velocity, which can be propagate for specific frequency bands. Next, in
the nonlinear regime we used the multiple scale perturbation method in order
to investigated the coupling between LCP- and RCP modes. In this framework, we
obtain, a system of two coupled nonlinear Schrödinger (NLS) equations for the
unknown vector potential envelope functions. The above system of equations was
used in order to predict the existence of coupled left- and right-circularly
polarized solitons in magnetized plasmas, of the bright-bright, bright-dark
and dark-bright type.

Thus, we found vector solitons of the same type (namely, bright-bright) as
well as ones of the mixed type (namely, bright-dark and dark-bright) in
specific frequency bands. Also, the values of $\Omega$ change the width of
frequency band where the vector solitons can be formed. Thus, we found vector
solitons of the type dark-bright, bright-bright and bright-dark in the bands I
and RCP where the increase of the $\Omega$ lead to a decrease of the width of
the band of dark-bright solitons with simultaneous increase of the width of
the band of the bright-bright and bright-dark solitons. In the bands II and
RCP only bright-dark solitons were found. Finally, we found bright-dark and
bright-bright solitons in the bands III and RCP where the increase of the
$\Omega$ lead to a decrease of the width of the band of bright-bright solitons
with simultaneous increase of the width of the band of the bright-dark
solitons.

Our results should be of interest to the plasma science community, where beam-
plasma interactions are a hot topic of research nowadays, both theoretically
and in the laboratory (laser plasma interaction experiments), and also in
nonlinear optics, where coupled beam propagation is an ubiquitous point of
focus. Beyond these physical settings, coupled NLS equations such as our Eqs.
(24)-(25) arise in various contexts, ranging from hydrodynamics to
supraconductivity (Bose-Einstein condensates) and left-hand materials, to
mention a few. The mathematical setting and the methodology of our study is
thus expected to apply in a wider context.

## Acknowledgements

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

This work was completed during a 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, represented by Professor
D.J. Frantzeskakis and (HSC Director) Professor I. Daglis, respectively, is
warmly acknowledged.

Author GV acknowledges support from ADEK (Abu Dhabi Education and Knowledge
Council, currently ASPIRE-UAE) AARE (Abu Dhabi Award of Research Excellence)
grant (AARE18-179) during a research visit in 2022. Hospitality from the host
(Khalifa University) during that visit is gratefully acknowledged.

## References

  * [1] Akhiezer, A., Polovin, R.V.: Theory of wave motion of an electron plasma. Zh. Eksp. Teor. Fiz. 30, 915 (1956) [Sov. Phys. JETP 3, 696 (1956)]
  * [2] Kozlov, V.A., Litvak, A.G., Suvorov, E.V.: Envelope solitons of relativistic strong electromagnetic waves. Soviet Phys. JETP 49, 75–80 (1979)
  * [3] Kaw, P.K., Sen, A., Katsouleas, T.: Nonlinear 1D laser pulse solitons in a plasma. Phys. Rev. Lett. 68, 3172 (1992); https://doi.org/10.1103/PhysRevLett.68.3172
  * [4] Esirkepov, T.Zh., Kamenets, F.F., Bulanov, S.V., Naumova, N.M.: Low-frequency relativistic electromagnetic solitons in collisionless plasmas. JETP Lett. 68, 36 (1998); https://doi.org/10.1134/1.567817
  * [5] Kruer, W.: The Physics Of Laser Plasma Interactions (1st Edition). CRC Press (2003); https://doi.org/10.1201/9781003003243.
  * [6] Eliezer, S., Mima, K.: Applications of Laser-Plasma Interactions, Series in Plasma Physics (Taylor $\&$ Francis, 2009).
  * [7] Quesnel, B., Mora, P., Adam, J.C., Héron, A., Laval, G.: Electron parametric instabilities of ultraintense laser pulses propagating in plasmas of arbitrary density. Phys. Plasmas 4, 3358 (1997); https://doi.org/10.1063/1.872494 ; Barr, H.C., Mason, P., Parr, D.M.: Electron parametric instabilities driven by relativistically intense laser light in plasma. Phys. Rev. Lett. 83, 1606 (1999); https://doi.org/10.1103/PhysRevLett.83.1606
  * [8] Gleixner, F., and Kumar, N.: Electronic parametric instabilities of an ultrarelativistic laser pulse in a plasma. Phys. Rev. E 101, 033201 (2020); doi:10.1103/physreve.101.033201
  * [9] Mori, W.B., Decker, C.D., Leemans, W.P.: Relativistic harmonic content of nonlinear electromagnetic waves in underdense plasmas. IEEE Trans. Plasma Sci. 21 (1), 110–119 (1993); DOI: 10.1109/27.221109 Esarey, E., Ting, A., Sprangle, P., Umstadter, D., Liu, X.: Nonlinear analysis of relativistic harmonic generation by intense lasers in plasmas. IEEE Trans. Plasma Sci. 21 (1), 95–104 (1993); DOI: 10.1109/27.221107
  * [10] Esarey, E., Sprangle, P., Krall, J., Ting, A.: Self-focusing and guiding of short laser pulses in ionizing gases and plasmas. IEEE J. Quantum Electron. 33 (11), 1879–1914 (1997); DOI: 10.1109/3.641305
  * [11] Borghesi, M., Campbell, D.H., Schiavi, A. Haines, M.G., Willi, O., MacKinnon, A.J., Patel, P., Gizzi, L.A., Galimberti, M., Clarke, R.J., Pegoraro, F., Ruhl, H., Bulanov, S.: Electric field detection in laser-plasma interaction experiments via the proton imaging technique. Phys. Plasmas 9, 2214–2220 (2002); https://doi.org/10.1063/1.1459457
  * [12] Wagner, U., Tatarakis, M., Gopal, A., Beg, F.N., Clark, E.L., Dangor, A.E., Evans, R.G., Haines, M.G., Mangles, S.P.D., Norreys, P.A., Wei, M.-S., Zepf, M., Krushelnick, K.: Laboratory measurements of $0.7~{}GG$ magnetic fields generated during high-intensity laser interactions with dense plasmas. Phys. Rev. E 70, 026401 (2004); https://doi.org/10.1103/PhysRevE.70.026401
  * [13] Chen, S.-Y., Krishnan, M., Maksimchuk, A., Umstadter, D.: Excitation and damping of a self-modulated laser wakefield. Phys. Plasmas 7, 403–413 (2000); https://doi.org/10.1063/1.873809
  * [14] Roy, S., Chatterjee, D., and Misra, A. P.: Generation of wakefields and electromagnetic solitons in relativistic degenerate plasmas. Phys. Scr. 95, 015603. (2019); doi:10.1088/1402-4896/ab447d
  * [15] Melrose, D.B., McPhedran, R.C.: Electromagnetic Processes in Dispersive Media. Cambridge University Press (1991); https://doi.org/10.1017/CBO9780511600036 .
  * [16] Di Piazza, A., Müller, C., Hatsagortsyan, K.Z., Keitel, C.H.: Extremely high-intensity laser interactions with fundamental quantum systems. Rev. Mod. Phys. 84, 1177 (2012); https://doi.org/10.1103/RevModPhys.84.1177
  * [17] Brodin, G., Marklund, M., Stenflo, L.: Proposal for detection of QED vacuum nonlinearities in Maxwell’s equations by the use of waveguides. Phys. Rev. Lett. 87 (17), 171801 (2001); https://doi.org/10.1103/PhysRevLett.87.171801
  * [18] Nargesi, N., Miraboutalebi, S., Rajaei, L., Samavati, K.: The quantum exchange effects on the electromagnetic solitonic excitations in warm plasma. Results in Physics 52 106877 (2023); https://doi.org/10.1016/j.rinp.2023.106877
  * [19] Bulanov, S. V., Sasorov, P. V., Pegoraro, F., Kadlecov, H., Bulanov, S. S, Esirkepov, T. Zh., Rosanov, N. N., Korn, G.: Electromagnetic Solitons in Quantum Vacuum. Phys. Rev. D 101, 016016 (2020); https://doi.org/10.1103/PhysRevD.101.016016
  * [20] Macchi, A., Borghesi, M., Passoni, M.: Ion acceleration by superintense laser-plasma interaction. Rev. Mod. Phys. 85, 751 (2013); https://doi.org/10.1103/RevModPhys.85.751
  * [21] Borghesi, M., Audebert, P., Bulanov, S.V., Cowan, T., Fuchs, J., Gauthier, J.C., Mackinnon, A.J., Patel, P.K., Pretzler, G., Romagnani, L., Schiavi, A., Toncian, T., Willi, O.: High-intensity laser-plasma interaction studies employing laser-driven proton probes. Laser Part. Beams 23, 291 (2005); https://doi.org/10.1017/S0263034605050408
  * [22] Marburger, J.H., Tooper, R.F.: Nonlinear optical standing waves in overdense plasmas. Phys. Rev. Lett. 35, 1001–1004 (1975); https://doi.org/10.1103/PhysRevLett.35.1001
  * [23] Bulanov, S.V., Inovenkov, I.N., Kirsanov, V.I., Naumova, N.M., Sakharov, A.S.: Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma. Phys. Fluids B 4, 1935–1942 (1992); https://doi.org/10.1063/1.860046
  * [24] Bulanov, S.V., Esirkepov, T.Z., Naumova, N.M., Pegoraro, F., Vshivkov, V.A.: Solitonlike electromagnetic waves behind a superintense laser pulse in a plasma. Phys. Rev. Lett. 82, 3440–3443 (1999); https://doi.org/10.1103/PhysRevLett.82.3440
  * [25] Sentoku, Y., Esirkepov, T.Zh., Mima, K., Nishihara, K., Califano, F., Pegoraro, F., Sakagami, H., Kitagawa, Y., Naumova, N.M. Bulanov, S.V.: Bursts of superreflected laser light from inhomogeneous plasmas due to the generation of relativistic solitary waves. Phys. Rev. Lett. 83, 3434 (1999); https://doi.org/10.1103/PhysRevLett.83.3434
  * [26] Bulanov, S.V., Califano, F., Esirkepov, T.Zh., Mima, K., Naumova, N.M., Nishihara K., Pegoraro, F., Sentoku, Y., Vshivkov, V.A.: Generation of subcycle relativistic solitons by super intense laser pulses in plasmas. Physica D 152-153, 682 (2001); https://doi.org/10.1016/S0167-2789(01)00201-9
  * [27] Naumova, N.M., Bulanov, S.V., Esirkepov, T.Z., Farina, D., Nishihara, K., Pegoraro, F., Ruhl, H., Sakharov, A.S.: Formation of electromagnetic postsolitons in plasmas. Phys. Rev. Lett. 87, 185004 (2001); https://doi.org/10.1103/PhysRevLett.87.185004
  * [28] Esirkepov, T., Nishihara, K., Bulanov, S.V., Pegoraro, F.: Three-dimensional relativistic electromagnetic subcycle solitons. Phys. Rev. Lett. 89, 275002 (2002); https://doi.org/10.1103/PhysRevLett.89.275002
  * [29] Bulanov, S.S., Esirkepov, T.Z., Kamenets, F.f., Pegoraro, F.: Single-cycle high-intensity electromagnetic pulse generation in the interaction of a plasma wakefield with regular nonlinear structures. Phys. Rev. E 73, 036408 (2006); https://doi.org/10.1103/PhysRevE.73.036408
  * [30] Wu, D., Zheng, C.Y., Yan, X.Q., Yu, M.Y., He, X.T.: Breather-like penetration of ultrashort linearly polarized laser into over-dense plasmas. Phys. Plasmas 20, 033101 (2013); https://doi.org/10.1063/1.4794197
  * [31] Feng, W., Li, J. Q, Kishimoto, Y.: Laser propagation and soliton generation in strongly magnetized plasmas. Phys. Plasmas 23, 032102 (2016); http://dx.doi.org/10.1063/1.4942789
  * [32] Borghesi, M., Bulanov, S., Campbell, D.H., Clarke, R.J., Esirkepov, T.J., Galimberti, M., Gizzi, L.A., MacKinnon, A.J., Naumova, N.M., Pegoraro, F., Ruhl, H., Schiavi, A., Willi, O.: Macroscopic evidence of soliton formation in multiterawatt laser-plasma interaction. Phys. Rev. Lett. 88, 135002 (2002); https://doi.org/10.1103/PhysRevLett.88.135002
  * [33] Romagnani, L., Bigongiari, A., Kar, S., Bulanov, S.V., Cecchetti, C.A., Esirkepov, T.Z., Galimberti, M., Jung, R., Liseykina, T.V., Macchi, A., Osterholz, J., Pegoraro, F., Willi, O., Borghesi, M.: Observation of magnetized soliton remnants in the wake of intense laser pulse propagation through plasmas. Phys. Rev. Lett. 105, 175002 (2010); https://doi.org/10.1103/PhysRevLett.105.175002
  * [34] Sarri, G., Singh, D.K., Davies, J.R., Fiuza, F., Lancaster, K.L., Clark, E.L., Hassan, S., Jiang, J., Kageiwa, N., Lopes, N., Rehman, A., Russo, C., Scott, R.H.H., Tanimoto, T., Najmudin, Z., Tanaka, K.A., Tatarakis, M., Borghesi, M., Norreys, P.A.: Observation of postsoliton expansion following laser propagation through an underdense plasma. Phys. Rev. Lett. 105, 175007 (2010); https://doi.org/10.1103/PhysRevLett.105.175007
  * [35] Sarri, G., Kar, S., Romagnani, L., Bulanov, S.V., Cecchetti, C.A., Galimberti, M., Gizzi, L.A., Heathcote, R., Jung, R., Kourakis, I., Osterholz, J., Schiavi, A., Willi, O., Borghesi, M.: Observation of plasma density dependence of electromagnetic soliton excitation by an intense laser pulse. Phys. Plasmas 18, 080704 (2011); https://doi.org/10.1063/1.3625261
  * [36] Chen, L.M., Kotaki, H., Nakajima, K., Koga, J., Bulanov, S.V., Tajima, T., Gu, Y.Q., Peng, H.S., Wang, X.X., Wen, T.S., Liu, H.J., Jiao, C.Y., Zhang, C.G., Huang, X.J., Guo, Y., Zhou, K.N., Hua, J.F., An, W.M., Tang, C.X., Lin, Y.Z.: Self-guiding of $100~{}TW$ femtosecond laser pulses in centimeter-scale underdense plasma. Phys. Plasmas 14, 040703 (2007); https://doi.org/10.1063/1.2720374
  * [37] Pirozhkov, A.S., Ma, J., Kando, M., Esirkepov, T.Z., Fukuda, Y., Chen, L.-M., Daito, I., Ogura, K., Homma, T., Hayashi, Y., Kotaki, H., Sagisaka, A., Mori, M., Koga, J.K., Kawachi, T., Daido, H., Bulanov, S.V., Kimura, T., Kato, Y., Tajima, T.: Frequency multiplication of light back-reflected from a relativistic wake wave. Phys. Plasmas 14, 123106 (2007); https://doi.org/10.1063/1.2816443
  * [38] Sylla, F., Flacco, A., Kahaly, S., Veltcheva, M., Lifschitz, A., Sanchez-Arriaga, G., Lefebvre, E., Malka, V.: Anticorrelation between ion acceleration and nonlinear coherent structures from laser-underdense plasma interaction. Phys. Rev. Lett. 108, 115003 (2012); https://doi.org/10.1103/PhysRevLett.108.115003
  * [39] Trines, R. M. G. M., Robinson, A. P. L., Nazarov, W., Kumar, G. R., Pasley, J.: Formation and evolution of post-solitons following a high intensity laser-plasma interaction with a low-density foam target. Plasma Phys. Control. Fusion 63 074001 (2021); https://doi.org/10.1088/1361-6587/abf85c
  * [40] Sen, A., Kaw, P.K.: Nonlinear 1D laser pulse solitons for particle acceleration. Phys. Scr. T50, 47 (1994); https://dx.doi.org/10.1088/0031-8949/1994/T50/007
  * [41] Mima, K., Ohsuga, T., Takabe, H., Nishihara, K., Tajima, T., Zaidman, E., Horton, W.: Wakeless triple-soliton accelerator. Phys. Rev. Lett. 57, 1421 (1986); https://doi.org/10.1103/PhysRevLett.57.1421
  * [42] Farina, D., Bulanov, S.V.: Relativistic electromagnetic solitons in the electron-ion plasma. Phys. Rev. Lett. 86, 5289–5292 (2001); https://doi.org/10.1103/PhysRevLett.86.5289
  * [43] Kuehl, H.H., Zhang, C.Y.: One-dimensional, weakly nonlinear electromagnetic solitary waves in a plasma. Phys. Rev. E 48, 1316 (1993); https://doi.org/10.1103/PhysRevE.48.1316
  * [44] Farina, D., Bulanov, S.V.: Dynamics of relativistic solitons. Plasma Phys. Control. Fusion 47, A73 (2005); https://dx.doi.org/10.1088/0741-3335/47/5A/007 ; Farina, D., Bulanov, S.V.: Slow electromagnetic solitons in electron-ion plasmas. Plasma Phys. Rep. 27, 641–651 (2001); https://doi.org/10.1134/1.1390536
  * [45] Poornakala, S., Das, A., Kaw, P.K., Sen, A., Sheng, Z.M., Sentoku, Y., Mima, K., Nishikawa, K.: Weakly relativistic one-dimensional laser pulse envelope solitons in a warm plasma. Phys. Plasmas 9, 3802–3810 (2002); https://doi.org/10.1063/1.1496085
  * [46] Saxena, A., Das, A., Sen, A., Kaw, P., Fluid simulation studies of the dynamical behavior of one-dimensional relativistic electromagnetic solitons. Phys. Plasmas 13, 032309 (2006); https://doi.org/10.1063/1.2187447
  * [47] Lontano, M., Passoni, M., Bulanov, S.V.: Relativistic electromagnetic solitons in a warm quasineutral electron-ion plasma. Phys. Plasmas 10, 639 (2003); https://doi.org/10.1063/1.1544666 Lontano, M., Bulanov, S.V., Koga, J., Passoni, M., Tajima, T.: A kinetic model for the one-dimensional electromagnetic solitons in an isothermal plasma Phys. Plasmas 9, 2562–2568 (2002); https://doi.org/10.1063/1.1476307
  * [48] Eliasson, B., Shukla, P.K.: Kinetic effects on relativistic solitons in plasmas. Phys. Lett. A 354 (5-6), 453–456 (2006); https://doi.org/10.1016/j.physleta.2006.01.083
  * [49] Saxena, V., Das, A., Sengupta, S., Kaw, P., Sen, A.: Stability of nonlinear one-dimensional laser pulse solitons in a plasma. Phys. Plasmas 14, 072307 (2007); https://doi.org/10.1063/1.2749227 Saxena, V., Das, A., Sengupta, S., Kaw, P., Sen, A.: Stability of one-dimensional relativistic laser plasma envelope solitons. J. Phys.: Conf. Ser. 112, 022110 (2008); https://dx.doi.org/10.1088/1742-6596/112/2/022110
  * [50] Lehmann, G., Laedke, E.W., Spatschek, K.H.: Stability and evolution of one-dimensional relativistic solitons on the ion time scale. Phys. Plasmas 13, 092302 (2006); https://doi.org/10.1063/1.2338820
  * [51] Pukhov, A., Meyer-ter-Vehn, J.: Relativistic magnetic self-channeling of light in near-critical plasma: Three-dimensional particle-in-cell simulation. Phys. Rev. Lett. 76, 3975–3978 (1996); https://doi.org/10.1103/PhysRevLett.76.3975
  * [52] Hurricane, O.A., Callahan, D.A., Springer, P.T., Edwards, M.J., Patel, P., Baker, K., Casey, D.T., Divol, L., Döppner, T., Hinkel, D.E., et al. Beyond alpha-heating: Driving inertially confined fusion implosions toward a burning-plasma state on the National Ignition Facility. Plasma Phys. Control. Fusion, 61, 014033. (2018); https://doi.org/10.1088/1361-6587/aaed71
  * [53] Borghesi, M., MacKinnon, A.L., Bell, A.R., Gaillard, R., Willi, O.: Megagauss magnetic field generation and plasma jet formation on solid targets irradiated by an ultraintense picosecond laser pulse. Phys. Rev. Lett. 81, 112–115 (1998); https://doi.org/10.1103/PhysRevLett.81.112
  * [54] Briand, J., Adrian, V., Tamer, M.El., Gomes, A., Quemener, Y., Dinguirard, J.P., Kiefferet, J.C.: Axial magnetic fields in laser-produced plasmas. Phys. Rev. Lett. 54, 38 (1985); https://doi.org/10.1103/PhysRevLett.54.38
  * [55] Tatarakis, M., Gopal, A., Watts, I., Beg, F.N., Dangor, A.E., Krushelnick, K., Wagner, U., Norreys, P.A., Clark, E.L., Zepf, M., Evans, R.G.: Measurements of ultrastrong magnetic fields during relativistic laser–plasma interactions. Phys. Plasmas 9, 2244 (2002); https://doi.org/10.1063/1.1469027
  * [56] Shukla, P.K., Stenflo, L.: Nonlinear propagation of electromagnetic waves in magnetized plasmas. Phys. Rev. A 30, 2110 (1984); https://doi.org/10.1103/PhysRevA.30.2110
  * [57] Nagesha Rao, N., Shukla, P.K., Yu, M.Y.: Strong electromagnetic pulses in magnetized plasmas. Phys. Fluids 27, 2664–2668 (1984); https://doi.org/10.1063/1.864568
  * [58] Rao, N.N.: Theory of near-sonic envelope electromagnetic waves in magnetized plasmas. Phys. Rev. A 37, 4846–4853 (1988); https://doi.org/10.1103/PhysRevA.37.4846
  * [59] Farina, D., Lontano, M., Bulanov, S.: Relativistic solitons in magnetized plasmas. Phys. Rev. E 62, 4146– (2000); https://doi.org/10.1103/PhysRevE.62.4146
  * [60] Jeffrey, A., Kawahara, T.: Asymptotic Methods in Nonlinear Wave Theory. Pitman, Boston (1982)
  * [61] Kourakis, I., Shukla, P.K.: Exact theory for localized envelope modulated electrostatic wavepackets in space and dusty plasmas. Nonlinear Process. Geophys. 12, 407–423 (2005); https://doi.org/10.5194/npg-12-407-2005
  * [62] Veldes, G.P., Borhanian, J., McKerr, M., Saxena, V., Frantzeskakis, D.J., Kourakis, I.: Electromagnetic rogue waves in beam-plasma interactions. J. Opt. 15 (6), 064003 (2013); https://dx.doi.org/10.1088/2040-8978/15/6/064003
  * [63] Segre, S.E.: A review of plasma polarimetry - theory and methods. Plasma Phys. Control. Fusion 41, R57 (1999); https://dx.doi.org/10.1088/0741-3335/41/2/001
  * [64] Taniuti, T., Yajima, Y.: Perturbation Method for a Nonlinear Wave Modulation. I. J. Math. Phys. 10, 1369–1372 (1969); https://doi.org/10.1063/1.1664975 ; Asano, N., Taniuti, T., Yajima, N.: Perturbation Method for a Nonlinear Wave Modulation. II. _ibid._ 10, 2020–2024 (1969); https://doi.org/10.1063/1.1664797
  * [65] Swanson, D.G.: Plasma Waves, Academic Press, INC, San Diego (1989)
  * [66] Remoissenet, M.: Waves Called Solitons. Springer-Verlag, Berlin, 2nd Ed. (1996)
  * [67] Dauxois, T., Peyrard, M.: Physics of Solitons. Cambridge University Press, Cambridge (2006)
  * [68] Fedele, R., Schamel, H.: Solitary waves in the Madelung’s fluid: Connection between the nonlinear Schrödinger equation and the Korteweg-de Vries equation. Eur. Phys. J. B 27, 313–320 (2002); https://doi.org/10.1140/epjb/e2002-00160-7 ; Fedele, R.: Envelope solitons versus solitons. Phys. Scr. 65, 502 (2002); https://dx.doi.org/10.1238/Physica.Regular.065a00502
  * [69] Borhanian, J., Sobhanian, S., Kourakis, I., Esfandyari-Kalejahi, A.: Evolution of linearly polarized electromagnetic pulses in laser plasmas. Phys. Plasmas 15, 093108 (2008); https://doi.org/10.1063/1.2990023
  * [70] Kivshar, Yu.S., Agrawal, G.P.: Optical solitons: from fibers to photonic crystals. Academic Press, San Diego (2003)
  * [71] Veldes, G.P., Cuevas, J., Kevrekidis, P.G., Frantzeskakis, D.J.: Coupled backward- and forward-propagating solitons in a composite right- and left-handed transmission line. Phys. Rev. E 88, 013203 (2013); https://doi.org/10.1103/PhysRevE.88.013203
  * [72] Kourakis, I., Shukla, P.K., Morfill, G.: Modulational instability and localized excitations involving two nonlinearly coupled upper-hybrid waves in plasmas. New J. Phys. 7, 153 (2005); https://dx.doi.org/10.1088/1367-2630/7/1/153
  * [73] Lazarides, N., Veldes, G.P., Javed, A., Kourakis, I.: Modulational electrostatic wave-wave interactions in plasma fluids modeled by asymmetric coupled nonlinear Schrödinger (CNLS) equations. Chaos, Solitons and Fractals 175, 113974 (2023); https://doi.org/10.1016/j.chaos.2023.113974
  * [74] Lazarides, N., Kourakis, I.: Coupled nonlinear Schrödinger (CNLS) equations for two interacting electrostatic wavepackets in a non-Maxwellian fluid plasma model. Nonlinear Dyn. 112, 2795–2819 (2024). https://doi.org/10.1007/s11071-023-09165-4
  * [75] Lazarides, N., Veldes, G.P., Frantzeskakis, D.J., Kourakis, I.: Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas. Sci. Rep. 14, 2150 (2024). https://doi.org/10.1038/s41598-024-52431-7

##  VII Declarations

###  VII.1 Funding

Authors IK and NL gratefully acknowledge financial support from Khalifa
University of Science and Technology, Abu Dhabi, United Arab Emirates, via the
project CIRA-2021-064 (8474000412) (PI Ioannis Kourakis). IK also acknowledges
financial support from KU via the project FSU-2021-012 (8474000352) (PI
Ioannis Kourakis) as well as from KU Space and Planetary Science Center, via
grant No. KU-SPSC-8474000336 (PI Mohamed Ramy Mohamed Elmaarry).

###  VII.2 Competing Interests

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

###  VII.3 Author Contributions

G. P. Veldes carried out the algebraic work, contributed to the methodology,
software development, and numerical analysis. N. Lazarides contributed to the
methodology and reviewed the manuscript. D. J. Frantzeskakis contributed to
the concept and design, and reviewed the manuscript. I. Kourakis contributed
to the problem conception, project design, launch and methodology, reviewed
the manuscript, and managed project funding. All authors read and approved the
final manuscript.

###  VII.4 Data Availability

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