# Effect of weak nonlocal nonlinearity on generalized sixth-order dispersion
modulational instability in optical media

Conrad B. Tabi<EMAIL_ADDRESS>Department of Physics and Astronomy,
Botswana International University of Science and Technology, Private Mail Bag
16 Palapye, Botswana  Camus G. Latchio Tiofack<EMAIL_ADDRESS>Faculty of
Sciences, University of Maroua, P.O. Box 814, Maroua, Cameroon  Hippolyte
Tagwo  University Institute of Wood Technology, P.O. Box 306, Mbalmayo,
Cameroon  Laboratory of Mechanics, Department of Physics, Faculty of Science,
University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon<EMAIL_ADDRESS>Timoléon C. Kofané<EMAIL_ADDRESS>Department of Physics and Astronomy,
Botswana International University of Science and Technology, Private Mail Bag
16 Palapye, Botswana  Laboratory of Mechanics, Department of Physics, Faculty
of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

###### Abstract

This paper analyzes the behaviors of solitons in even higher-order dispersive
media and explores the modulational instability phenomenon in optical media.
The analysis considers quadratic, quartic, and sextic dispersions with weakly
nonlocal Kerr nonlinearity. The results show that nonlocality enhances the MI
gain and leads to rogue waves in response to different combinations of even
dispersions and nonlocal nonlinearity. The study suggests that Kerr
nonlocality can enhance the excitation of extreme events in even higher-order
dispersive nonlinear media with potential applications in optical fibers and
fiber lasers.

##  I Introduction

Nonlinear dynamics of electromagnetic waves inside a medium is one of the
sources of integrable, near-integrable, and non-integrable nonlinear evolution
equations of mathematical physics having solutions of solitonic and solitonic
dissipative types, and which is connected with plenty of interesting and
practically important phenomena. It is well-known that the self-phase
modulation of waves propagating in nonlinear optical fiber is governed by the
one-dimensional cubic nonlinear Schrödinger (NLS) equation which is an
integrable partial differential equation possessing an infinite number of
symmetries corresponding to an infinite number of conserved quantities. In the
case of temporal solitons, the balance between the self-focusing nonlinearity
(negative Kerr-nonlinearity) and the anomalous dispersion regime (positive
group velocity) leads to the bright solitons, while the interplay between the
self-defocusing nonlinearity (positive Kerr-nonlinearity) and the normal
dispersion regime (negative group velocity), leads to the dark solitons. In
the case of spatial solitons, the balance between Kerr-nonlinearity and
diffraction can exhibit bright or dark solitons for the self-focusing and
self-defocusing nonlinear media, respectively. Beside the temporal and spatial
optical solitons, there exist optical structures which are fully confined in
space and time and are referred to as spatiotemporal optical solitons, also
known as ”light bullets” LambSoliton1980 ; AblowitzInverse1981 ;
ZakharovInverse1984 ; DoddSolitons1982 ; KivsharPhotonics2003 ;
SilberbergOPlet1990 . The term ”optical bullet” was proposed by Silberberg
SilberbergOPlet1990  to indicate a complete spatiotemporal soliton, for which
confinement in the three spatial dimensions and localization in the temporal
domain are achieved by the balance between a focusing nonlinearity and the
spreading due to chromatic dispersion and angular diffraction.

Inverse scattering theory has been applied with great success to the NLS
equation, and solutions are dark and bright solitons, with sech and tanh
profiles, respectively. Under some physical conditions, the self-modulation of
the carrier wave leads to a spontaneous energy localization via the generation
of rational solitons which are localized in space and time, having envelope
structures such as Akhmediev-Peregrine waves, Akhmediev wave trains, and
Kuznetsov-Ma waves, respectively KivsharPhotonics2003 ; KharifRW2009 ;
SolliNature2007 ; PeregrineJAust1983 ; KuznetsovSovPhys1977 ; Ma1979 ;
AkhmedievTMP1987 .

The possibility of using optical solitons not only as carriers of information
in high-speed and long-haul communication systems but also in ultrashort
lasers as well has been the focus of intense research activity for almost
three decades, leading to various scenarios of the pulse evolution, which are
resulted in particular changes of the pulse shape, spectrum, and chirp.
Indeed, Optical solitons are very stable against changes in the properties of
the medium, provided that these changes occur over distances that are long
compared with a soliton period, and optical solitons can adapt their shape to
slowly varying parameters of the medium. However, such optical solitons are
perturbed by several higher-order dispersive and nonlinear effects, mainly
when ultrashort optical pulses are used to excite them AgrawalNFO1989 .

So said, for the specific case in which a fundamental soliton is perturbed by
third-order dispersion (TOD) only WaiPRA1990 ; ElginPRA1993 ; KarpmanPRE1993 ;
KodamaOLett1994 ; AkhmedievPRA1995 ; ElginOpCom1995 , there is the emission of
nonsolitonic radiation at a frequency different from those associated with the
soliton WaiPRA1990 , and analytic expressions for the frequency and amplitude
of the nonsolitonic radiation peak that is generated in the output spectrum
when the TOD perturbs a higher-order soliton have been found RoyPRA2009 .
Several studies suggest that effects of group-velocity dispersion (GVD) and
fourth-order dispersion (FOD) can be minimized under some specific physical
conditions, and new types of solitary wave solutions for second and FOD NLS
equation only have been reported since the1990s HookOpLett1993 ;
AkhmedievOpCom1994 ; ChristovOpLett1994 ; KarlssonOptCom1994 ;
AbdullaevOpCom1994 ; AkhmedievOpCom1995 ; PicheOpLett1996 ; ZakharovJETP1998 ;
KalithasanJOpt2010 ; ArmaroliOpLett2014 ; RedondoNatCom2016 ; LoOpExp2018 ;
TaheriOpLett2019 . In such cases, propagation of high-intensity ultrashort
pulse can be accurately described by considering higher-order dispersion
effects like FOD. In particular, FOD effects lead the ultrashort solitons to
decay due to dispersive radiation on optical-fiber-guided HookOpLett1993 . In
the mode-locked laser with the near-zero second as well as near-zero TOD,
pulses much shorter than 10fs can be generated if FOD is further reduced
ChristovOpLett1994 . An exact localized analytical solution has been obtained
in the case where negative FOD is added to anomalous second-order dispersion
KarlssonOptCom1994 . In the case of an optical fiber near the zero dispersion
point, a new region with modulational instability (MI) process of
electromagnetic waves has been found, and a recurrence phenomenon similar to
the Fermi-Pasta-Ulam problem has been observed for both positive and negative
FOD AbdullaevOpCom1994 . Moreover, it has been shown that the bright optical
soliton’s temporal shape and peak power can be stable when a weak TOD is
introduced in the presence of FOD PicheOpLett1996 . Indeed, higher-order
dispersion effects in the anomalous dispersion regime can suppress the MI gain
spectra of an electromagnetic wave in a resonant optical fiber with a two-
level system KalithasanJOpt2010 . Remarkably, the possibility of multiple
instability peaks originating from the parametric resonance, coexisting with
the conventional MI because of FOD has been thoroughly analyzed, where
periodic variations of GVD in the proximity of the zero dispersion point have
been considered ArmaroliOpLett2014 .

The recently discovered pure-quartic solitons RedondoNatCom2016 ; LoOpExp2018
; TaheriOpLett2019 ; KruglovPRA2018 ; TamOpLett2019 ; GaoPRE2020 ; TamPRA2020
; ParkerPhysD2021 ; BandaraPRA2021 ; ZhaoOPLett2021 ; ZhangOpLett2022 ;
QianOpLett2022 ; YaoPRR2022 ; AlexanderOpLett2022 ; LiOpCom2022 ; QiangPRA2022
; MartijnAPLPhot2021  arising from the interaction of quartic dispersion and
Kerr nonlinearity, formed a framework for subsequent research in the field of
communications, slow-light devices, and ultrafast laser science and open also
the door to unexplored soliton regimes. A class of bright solitons arising
purely from negative FOD and self-phase modulation interaction, which can
occur even for normal group-velocity dispersion, has been demonstrated
experimentally RedondoNatCom2016 . Importantly, a general analysis of the
dispersion and nonlinear properties necessary to observe pure-quartic solitons
in optical platforms has been reported LoOpExp2018 . Moreover, the Lagrangian
variational method has been used to show that continuous wave coherent light
of a Kerr resonator featuring quartic GVD forms a pure-quartic soliton with a
Gaussian envelope TaheriOpLett2019 . An exact solitary wave solution that
applies for all orders of dispersion up to the fourth order has been proposed
KruglovPRA2018 . A family of pure-quartic solitons, existing through a balance
of positive Kerr nonlinearity and negative quartic dispersion, has been found
TamOpLett2019 . The propagation velocities, periodicity, and localization of a
Gaussian-type initial perturbation with cosine-type modulation on a plane wave
inside a nonlinear fiber model purely with FOD have been predicted by the
modified linear stability analysis GaoPRE2020 . A combination of negative
quartic and some quadratic dispersions leading to high soliton energies has
been reported TamPRA2020 . Multi-pulse solitary wave solutions’ existence and
spectral stability to an NLS with both second-order dispersion and FOD have
been developed ParkerPhysD2021 . A dynamical system approach has been
successfully applied to the NLS equation in the presence of both quartic and
quadratic dispersion terms, and families of solitons with different symmetry
properties have been characterized BandaraPRA2021 . The vector properties of
quartic solitons in a pure FOD birefringent fiber and a quartic dispersion
dominant mode-locked fiber laser have been investigated ZhaoOPLett2021 .
Indeed, the bifurcation diagrams in a passively mode-locked fiber laser have
revealed that the pure-quartic solitons can alternate between the stable
single soliton and pulsating regimes multiple times before transiting into the
chaotic state ZhangOpLett2022 . Moreover, it has been demonstrated that
dissipative pure-quartic solitons in a fiber laser possesses a higher energy
scaling ability compared to conventional dissipative solitons QianOpLett2022 .
The dynamic orbits outside the heteroclinic separatrix in the phase plane,
corresponding to the Fermi-Pasta-Ulam recurrences and Akhmediev breathers
(ABs), have been discussed in the pure quartic systems YaoPRR2022 . Stable
dark solitons in the presence of pure-quartic dispersion and both quadratic
and quartic dispersion effects have been theoretically predicted
AlexanderOpLett2022 . The standard Gaussian trial solution has been modified
to reflect the radiation-induced oscillation damping of the breathing pure-
quartic solitons LiOpCom2022 . Subsequently, bright solitons can exist, for
the particular choice of the dispersion parameters, in the NLE equation with
quadratic, quartic, and sextic dispersion effects QiangPRA2022 .

Our primary objective in this paper is to investigate the MI process in a
fiber system taking into account quadratic, quartic, and sextic dispersion
terms as well as the inherently nonlocal character of nonlinearity
SalernoPRE2018 ; ZangaCNSNS2020 . Nonlocal nonlinearity means that the light-
induced refractive index change of a material at a particular location is
determined by the light intensity in a certain neighborhood of this location.
MI induced by the interplay between nonlinearity and dispersion (in the
temporal domain) or diffraction (in the spatial domain) is a generic
phenomenon in nonlinear physics. In particular, MI has been found to be
relevant in understanding the formation and propagation of solitonic waves
KevrekidisPRA2003 , explaining the domain wall formation ZhangPRL2005 , and
quantum phase transition SmerziPRL2002 . Our results suggest nonlocality to
enhance MI even in some nonintegrable cases and additionally confirm MI to be
a key mechanism for generating rogue waves (RW) under different combinations
of even dispersions and nonlinear terms.

The remaining part of the paper is structured as follows. Section II presents
the governing weakly nonlocal model under study. Then, in Section III, we
investigate, through analytical methods, the MI excitation in view of the
generation of bright solitary waves in the optical fiber. Section IV discusses
the numerical experiment of the model equation. Section V summarizes our
investigations and concludes the work.

##  II Model

The model proposed in the present work combines the effect of even-order
dispersions up to the sixth order that interacts with nonlocal nonlinear
effects for optical solitons to take place. This is modeled by the following
modified NLS equation:

$\begin{split}i\frac{\partial\psi}{\partial
z}&-\frac{\beta_{2}}{2}\frac{\partial^{2}\psi}{\partial
t^{2}}+\frac{\beta_{4}}{24}\frac{\partial^{4}\psi}{\partial t^{4}}\\\
&-\frac{\beta_{6}}{720}\frac{\partial^{6}\psi}{\partial t^{6}}+\Delta
n\psi(z,t)=0,\end{split}$ (1)

where $\psi(z,t)$ is the pulse complex envelope, $z$ is the propagation
coordinate, and $t$ is the local time. The $\beta_{n}$ represent the
$n^{th}-$order dispersion coefficients, with

$\beta_{n}=\frac{\partial^{n}\beta}{\partial\omega^{n}}=\frac{\partial^{n-1}v_{g}^{-1}}{\partial\omega^{n-1}},$
(2)

where $\beta$ is the mode propagation constant and $v_{g}$ is the group
velocity.

It is well-known in nonlinear Kerr media that the intensity-dependent
refractive index change is given by $\Delta n(t,I)=I$, with the intensity of
the beam $I=|\psi(z,t)|^{2}$. In the case of nonlinear media with a nonlocal
nonlinearity, the nonlinear refractive index change of the medium can be
represented by the following phenomenological model

$\begin{split}\Delta
n(t,I)&=\alpha\int^{+\infty}_{-\infty}R(t-t^{{}^{\prime}})I(t^{\prime},z)dt^{{}^{\prime}},\end{split}$
(3)

where $\alpha$ is the strength of the nonlocal Kerr nonlinearity. Under weak
nonlocality, which is the main concern of this work, the above nonlinear term
is expanded as

$\Delta
n\left(t,I\right)=\alpha\left|\psi\left(t,z\right)\right|^{2}+\alpha\gamma\frac{\partial^{2}\left(\left|\psi\left(t,z\right)\right|^{2}\right)}{\partial
t^{2}},$ (4)

where $\gamma=\frac{1}{2}\int_{-\infty}^{+\infty}t^{2}\\!R\left(t\right){dt}$
is the nonlocal parameter. Inserting Eq.(4) into Eq.(1), one gets the
following modified NLS equation:

$\begin{split}i\frac{\partial\psi}{\partial
z}&-\frac{\beta_{2}}{2}\frac{\partial^{2}\psi}{\partial
t^{2}}+\frac{\beta_{4}}{24}\frac{\partial^{4}\psi}{\partial
t^{4}}-\frac{\beta_{6}}{720}\frac{\partial^{6}\psi}{\partial t^{6}}\\\
&+\alpha\left|\psi\right|^{2}\psi+\chi\frac{\partial^{2}\left(\left|\psi\right|^{2}\right)}{\partial
t^{2}}\psi=0,\end{split}$ (5)

where $\chi=\alpha\gamma$. Compared to the model used in Ref. QiangPRA2022 ,
the nonlocality brings about the additional term
$\chi\frac{\partial^{2}\left(\left|\psi\right|^{2}\right)}{\partial
t^{2}}\psi$ to the equation, which can be switched off by imposing $\gamma=0$.
In such a context, solitons with oscillating tails were numerically
characterized along with their dynamical response to the interplay between
higher-order dispersions and Kerr nonlinearity. Other studies have explored
the cases $\gamma=0$, $\beta_{2}=\beta_{6}=0$, where the interplay between the
negative quartic and Kerr nonlinearity was found to sustain the stability of
pure-quartic solitons of the NLS equation RungePRR2021 . More recently, a
modified linear stability analysis on the latter was proposed, where Gao et
al. GaoPRE2020  further proposed a modified gain of MI from a localized
perturbation. Soliton properties can be modified due to the presence of
nonlocalities such as strong modification of modulational instability
KrolikowskiJOptB2004 ; TagwoOptik2017 ; ZangaCNSNS2020 ; TabiOpLett2022 ;
TabiPRE2022 , suppression of beam collapse BangPRE2002 , dramatic change of
the soliton interaction PecciantiOpLett2002 ; NikolovOpLett2004 , formation of
multi-soliton bound states XuOpLett2005 , stabilization of spatially localized
vortex solitons DesyatnikovPOpt2005 , symmetry breaking azimuthal instability
YakimenkoPRE2005 ; BriedisOpExp2005  as well as stabilization of different
nonlinear structures such as ring-like clusters of many solitons
DesyatnikovPRL2002  and modulated localized vortex beams or azimuthons
DesyatnikovPRL2005 . A simple procedure for estimating the strength of a weak
nonlocality in the propagation of a Gaussian beam has been formulated, and the
conditions for breathing soliton formation in both one and two transverse
dimensions have been derived BezuhanovPRA2008 . Moreover, the high limit of
the nonlocal 2D NLS equation has been derived SkupinPRE2006 . Indeed, the
stabilization of 2D ring dark solitons and ring anti-dark solitons has been
demonstrated in nonlocal media ChenJPB2017 .

##  III Linear stability analysis and gain spectrum

###  III.1 Linear stability analysis

To exclusively address the analytical study of MI in the above-introduced
model, we assume it admits continuous wave (CW) solutions that propagates
inside the fiber with the initial input power $P_{0}$ so that

$\psi(z,t)=\sqrt{P_{0}}e^{i\phi_{NL}z},$ (6)

where $\phi_{NL}=\alpha P_{0}$ is the nonlinear phase shift due to the
nonlinearity. To further proceed, a slight perturbation is introduced into
solution (6) to study its stability,

$\psi(z,t)=[\sqrt{P_{0}}+\zeta(z,t)]e^{i\phi_{NL}z},$ (7)

where $\zeta(z,t)$ is a small perturbation, with
$\left|\zeta(z,t)\right|\ll\sqrt{P_{0}}$ being the complex field. Substituting
the perturbed solution (7) into Eq.(5) and linearizing around the unperturbed
solution, the perturbed field is found to be governed by the following
equations:

$\begin{split}i\frac{\partial\zeta}{\partial
z}&-\frac{\beta_{2}}{2}\frac{\partial^{2}\zeta}{\partial
t^{2}}+\frac{\beta_{4}}{24}\frac{\partial^{4}\zeta}{\partial
t^{4}}-\frac{\beta_{6}}{720}\frac{\partial^{6}\zeta}{\partial t^{6}}\\\
&+\alpha P_{0}(\zeta+\zeta^{*})+\chi
P_{0}\left(\frac{\partial^{2}\zeta}{\partial
t^{2}}+\frac{\partial^{2}\zeta^{*}}{\partial t^{2}}\right)=0,\end{split}$ (8)

where $\zeta^{\ast}$ is the complex conjugate of the perturbed field. Further,
the solution for Eq.(7) is taken in the form

$\displaystyle\zeta(z,t)=\zeta_{1}(z)e^{-i\Omega t}+\zeta_{2}(z)e^{i\Omega
t},$ (9)

where $\zeta_{1}(z)$ and $\zeta_{2}(z)$ are the complex perturbation fields
and $\Omega$ is the complex modulation frequency. Applying the above ansatz to
the perturbed field Eq. (7) leads to a $2\times 2$ system of the following
form for the perturbation fields:

$i\frac{\partial}{\partial s}\left[\begin{array}[]{c}\zeta_{1}(z)\\\
\zeta_{2}^{\ast}(z)\end{array}\right]=\left[\begin{array}[]{cc}a_{11}(\Omega)&a_{12}(\Omega)\\\
a_{21}(\Omega)&a_{22}(\Omega)\end{array}\right]\left[\begin{array}[]{c}\zeta_{1}(z)\\\
\zeta_{2}^{\ast}(z)\end{array}\right],$ (10)

where $\zeta_{j}^{\ast}(z)$($j=1,2$) is the complex conjugate of the field
$\zeta_{j}(z)$($j=1,2$), with the elements of the system’s matrix $M$ being
given by

$\begin{split}a_{11}=&\frac{\beta_{2}}{2}\Omega^{2}+\frac{\beta_{4}}{24}\Omega^{4}+\frac{\beta_{6}}{720}\Omega^{6}+\alpha
P_{0}-\chi P_{0}\Omega^{2},\\\ a_{12}=&\alpha P_{0}-\chi
P_{0}\Omega^{2},\;\;a_{21}=-\left(\alpha P_{0}-\chi P_{0}\Omega^{2}\right),\\\
a_{22}=&-\left(\frac{\beta_{2}}{2}\Omega^{2}+\frac{\beta_{4}}{24}\Omega^{4}+\frac{\beta_{6}}{720}\Omega^{6}+\alpha
P_{0}-\chi P_{0}\Omega^{2}\right).\end{split}$ (11)

Figure 1: Plots of the OMF versus the pump power $P_{0}$, computed from
solution (15). The panel (a) shows results for $\beta_{2}=-0.001$ps2mm-1 and
panel (b) displays the OMF for $\beta_{2}=-0.8$ps2mm-1, with $|\beta_{4}|$
taking increasing values in each of the cases. The other parameter values are:
$\alpha=0.5$W-1.mm-1 and $\beta_{6}=0$.

The MI gain is investigated from the imaginary part of the wavenumber $k$ of
the perturbation that can be obtained using the characteristic equation ${\rm
Det}\left|M-kI\right|=0$, with $I$ being a $2\times 2$ identity matrix. One
therefore obtains the dispersion relation

$\begin{split}k=\sqrt{\left(\alpha
P_{0}+m\Omega^{2}+\frac{\beta_{4}}{24}\Omega^{4}+\frac{\beta_{6}}{720}\Omega^{6}\right)^{2}-\left(\alpha
P_{0}-\chi P_{0}\Omega^{2}\right)^{2}},\end{split}$ (12)

where $m=({\beta_{2}}/{2}-\chi P_{0})$. The MI will take place under the
condition where the wavenumber $k$ has a nonzero imaginary part. The latter
will be responsible for an amplitude exponential growth. The power gain,
through which the MI is predicted is such that $\Lambda(\Omega)=2{\rm Im}(k)$,
which is specifically obtained in the form

$\Lambda(\Omega)=2\sqrt{\left(\alpha P_{0}-\chi
P_{0}\Omega^{2}\right)^{2}-\left(\alpha
P_{0}+m\Omega^{2}+\frac{\beta_{4}}{24}\Omega^{4}+\frac{\beta_{6}}{720}\Omega^{6}\right)^{2}}.$
(13)

Interestingly, it is obvious that the above MI gain depends on all the
involved terms, including the nonlocal one. To carry out our analysis, the MI
gain will be parametrically studied using different combinations of
parameters, especially the higher-order dispersions and the nonlocal
contribution to the nonlinearity. In the meantime, the Optimum modulation
frequency (OMF) can be obtained from the above expression of the MI gain by
solving the equation $d\Lambda/d\Omega=0$. This leads to the equation:

$\begin{split}&\frac{\beta_{6}^{2}}{43200}\Omega^{10}+\frac{\beta_{4}\beta_{6}}{864}\Omega^{8}+\left(\frac{\beta_{4}^{2}}{72}+\frac{\beta_{6}m}{45}\right)\Omega^{6}\\\
&+\left(\frac{\beta_{4}m}{2}+\frac{\alpha\beta_{6}P_{0}}{60}\right)\Omega^{4}-\left(4P_{0}^{2}\chi^{2}-4m^{2}-\frac{\alpha\beta_{4}P_{0}}{3}\right)\Omega^{2}\\\
&+4\alpha P_{0}\left(m+P_{0}\chi\right)=0.\end{split}$ (14)

Eq.(14) constitutes a higher-order polynomial equation whose solutions cannot
easily be found analytically. Even with softwares like Mathematica or Matlab,
only solutions related to specific choices of parameters can be obtained using
symbolic computations. In what follows, we address some of such accessible
cases that will allow us to qualitatively study the impact of the different
constituents on the MI gain.

Figure 2: Plots of the OMF as a function of the pump power $P_{0}$ for
$\alpha=-0.5$W-1.mm-1, obtained from Eq.(15). In panel (a), the second-order
dispersion is switched off, i.e., $\beta_{2}=0$, while in panel (b)
$\beta_{2}$ should take weak values for the OMF to exist. The values chosen in
this case is $\beta_{2}=-0.8$ps2mm-1, with $|\beta_{4}|$ taking increasing
values in each of the cases, the rest of the parameters are such that
$\gamma=0$, and $\beta_{6}=0$.

###  III.2 Optimum modulation frequency and MI gain spectrum

####  III.2.1 The optimum modulation frequency

On solving Eq.(14), it is possible to find frequency modes from which all the
physical processes underlying the MI mechanism can be obtained. This will be
possible only if the different frequencies are real and positive. One should
also be aware that depending on the parameter choices, mainly the dispersion
coefficients $\beta_{2}$, $\beta_{4}$, and $\beta_{6}$, one might get
different results as is clearly addressed here, through the following cases.

(i) Case $\beta_{6}=0$, $\beta_{4}\neq 0$, $\beta_{2}\neq 0$, and $\chi=0$

This particular case addresses the situation where the nonlocal term and the
sextic dispersion term are switched off. Eq.(14), therefore, admits six
solutions, with only the following being likely to be real and positive,
depending on the parameter choices:

$\begin{split}\Omega_{\rm
Opt}=\sqrt{-\frac{12m}{\beta_{4}}\pm\frac{2\sqrt{6}\sqrt{6m^{2}-\alpha\beta_{4}P_{0}}}{\beta_{4}}}.\end{split}$
(15)

The corresponding features are displayed in Fig. 1, where the OMF is plotted
versus the pump power $P_{0}$ with $|\beta_{2}|$ taking increasing values. In
Fig. 1(a), where we have fixed $|\beta_{2}|=0.001$ps2mm-1, the action of the
quartic dispersion is perceptible, due to the weak value of the quadratic
dispersion. Moreover, the OMF is a decreasing function of $|\beta_{4}|$, a
behavior that is also perceptible in Fig. 1(b), where
$|\beta_{2}|=0.8$ps2mm-1. However, the magnitude of the OMF drastically drops,
while it is not dramatically sensitive to the effect of increasing
$|\beta_{4}|$. In the particular case where the quadratic dispersion and the
nonlocality are completely switched off, the OMF reduces to $\Omega_{\rm
Opt}=(-72\alpha P_{0}/\beta_{4})^{1/4}$. Such a critical case, for
$\alpha=-0.5$W-1.mm-1, is shown in Fig. 2(a), where the OMF is extracted as a
function of the pump power. When $|\beta_{4}|$ increases as previously, the
OMF drops. Under the same conditions but with $|\beta_{2}|\neq 0$, its value
should be strong enough for the function of the pump to be effective in MI
onset. This is shown in Fig. 2(b), where $|\beta_{4}|=0.0004$ps4mm-1 excludes
some interval of $P_{0}$ after which the OMF exists. For the remaining values
of $\beta_{4}$, the OMF drops as previously, while the instability gets
preserved.

Figure 3: Plots of the OMF versus the pump power $P_{0}$ in the absence [panel
(a)] and presence [panel (b)] of the nonlocal term. Computed from the OMF
formula (15), each of the cases is studied for the fourth-order dispersion
coefficient taking the respective values $\beta_{4}=-0.003$ps4mm-1 (solid blue
line), $\beta_{4}=-0.02$ps4mm-1 (dashed red line) and $\beta_{4}=-0.8$ ps4mm-1
(dotted yellow line). The other parameter values are: $\alpha=0.5$W-1.mm-1,
$\beta_{6}=0$, and $\beta_{2}=0$. Figure 4: The OMF of Eq.(15) is plotted
versus the pump power for (a): the nonlinearity coefficient changing and (b)
the nonlocality parameter changing. In panel (a), the nonlocality parameter is
taken to zero and $\alpha$ takes the respective values $\alpha=0.01$ W-1.mm-1
(solid blue line), $\alpha=0.1$ W.mm-2 (dashed red line) and $\alpha=0.5$
W-1.mm-1 (dotted yellow line). To plot panel (b), we have fixed $\alpha=0.5$
W-1.mm-1 and the nonlocality parameter has been given the values
$\gamma=0.001$ (solid blue line), $\gamma=0.05$ (dashed red line) and
$\gamma=0.1$ (yellow dotted line). The other parameter values are:
$\beta_{6}=0$, $\beta_{4}=-0.8$ps4mm-1, and $\beta_{2}=0$. Figure 5: The OMF
is numerically extracted and plotted versus the pump power in the context
where the $\beta_{4}=0$, with $\beta_{2}=-0.8$ps2mm-1 and the sextic
dispersion taking the values $\beta_{6}=-0.001$ps6mm-1 (solid blue line),
$\beta_{6}=-0.004$ps6mm-1 (dashed red line) and $\beta_{6}=-0.004$ps6mm-1
(dotted yellow line) . In panel(a), the nonlocality parameter is given the
value $\gamma=0.01$, while in panel (b) it takes the value $\gamma=-0.01$,
with $\alpha=0.1$ W-1.mm-1. Figure 6: The OMF is numerically extracted and
plotted versus the pump power for (a): the nonlinearity coefficient changing
and (b) the nonlocality parameter changing. In panel (a), the nonlocality
parameter is taken to zero and $\alpha$ takes the respective values
$\alpha=0.01$ W.mm-2 (solid blue line), $\alpha=0.1$ W-1.mm-1 (dashed red
line) and $\alpha=0.5$ W-1.mm-1 (dotted yellow line). To plot panel (b), we
have fixed $\alpha=0.5$ W-1.mm-1 and the nonlocality parameter has been given
the values $\gamma=0.001$ (solid blue line), $\gamma=0.05$ (dashed red line)
and $\gamma=0.1$ (yellow dotted line). The other parameter values are:
$\beta_{6}=0$, $\beta_{4}=-0.8$ps4mm-1, and $\beta_{2}=0$.

(ii) Case $\beta_{6}=0$, $\beta_{4}\neq 0$, $\beta_{2}=0$, and $\chi\neq 0$

In this case, the sextic and quadratic dispersions are canceled, while the
nonlocality is existent, leading to the solution

$\Omega_{\rm
Opt}=\sqrt{\frac{18P_{0}\chi}{\beta_{4}}\pm\frac{2\sqrt{3}\sqrt{P_{0}\left(27P_{0}\chi^{2}-2\alpha\beta_{4}\right)}}{\beta_{4}}}$
(16)

for Eq.(14). The results related to this case are delivered by the plots of
Fig. 3(a), where the nonlocality is switched off, with $\beta_{2}=0$, which
shows the possibility of MI. In presence of the nonlocality, the decreasing
values of $|\beta_{4}|$ show a saturation behavior where after a transient
increase, the values of the pump power that support the OMF seem to be uniform
[see Fig. 3(a)]. On the other hand, increasing the Kerr nonlinear coefficient
enhances the OMF [see Fig. 4(a)], and a saturation effect takes place when the
nonlocality parameter increases in a context where the OMF drops with
increasing $\gamma$ [see Fig. 4(b)].

From the above two cases, it is obvious that the FOD, second-order dispersion
and the nonlocality parameter importantly affect the occurrence of MI. As
mentioned earlier, one of the problems faced here is solving cases involving
the SOD coefficient, which may be addressed numerically. In what follows, we
discuss some of such few cases.

(iii) Case $\beta_{6}\neq 0$, $\beta_{4}=0$, $\beta_{2}\neq 0$, and
$\gamma\neq 0$

This case engages the sextic dispersion, the second-order dispersion, the Kerr
nonlinearity, and the weak nonlocality. The corresponding results are
displayed in Fig. 5, where panel (a) is dedicated to the positive nonlocality
parameter $\gamma=0.01$ under the competition between negative second-order
and sextic dispersion coefficients. In general, there is a high sensitivity of
the OMF versus the magnitude of $|\beta_{6}|$. MI is characterized by a
perpetual frequency shift with the value of the pump power increasing [see
Fig. 5(a)]. Such a shift, when $\gamma<0$, is still perceptible, except that
the OMF is a decreasing function of $|\beta_{6}|$. This clearly shows that the
two cases of $\gamma$ (positive or negative) may support MI but with different
solitonic features.

(iv) Case $\beta_{6}\neq 0$, $\beta_{4}\neq 0$, $\beta_{2}=0$, and $\gamma\neq
0$

In the absence of nonlocality, panel (a) of Fig. 6 testifies to the
possibility of MI, especially when the quartic and sextic dispersions coexist
in the absence of the second-order dispersion. The OMF shows high sensitivity
against the magnitude of $|\beta_{6}|$, which, however, decreases with the
pump power $P_{0}$ decreasing. The same scenario is ostensible in Fig. 6(b),
where the nonlocality is taken into consideration, with the other conditions
remaining the same. This shows a competition between the dispersive
nonlinearity and Kerr one and shows that when the sextic and quartic
dispersions are well-chosen, the effect of the nonlocality is diluted by the
presence of the Kerr nonlinearity, probably due to the weak factor
$\chi=\alpha\gamma$ of the nonlocality.

Except for the few cases regarded above, other features may be explored using
more combinations of the involved parameters. The study cannot, therefore, be
exhaustive in that direction. However, the few studied cases can give the
reader a flavor of what can be brought by including the nonlocality in such
models, opening a new route for more comprehensive theoretical and
experimental investigations. Nevertheless, to get more insight into the
problem under study, the next step is devoted to studying the full MI gain
using the above predictions, and we go beyond them to show the impact of
combining system parameters on the MI gain. To that effect, the MI gain
proposed in Eq.(13) will be used, and calculations will be conducted in the
case where $\beta_{2}$ is negative when considered.

Figure 7: The panels show the variations of the MI gain in the
$(\Omega,\beta_{4})-$plane in the absence of the second-order dispersion.
Panels (aj)j=1,2 are plotted in absence of nonlocality with $\alpha$ taking
the values -0.5 W-1.mm-1 [panel (a1)] and 0.5 W.mm-2 [panel (a2)]. In presence
of nonlocality ($\gamma=0.01$), one obtains the features of panels (bj)j=1,2,
with the nonlinearity coefficient taking the respective values -0.5 W-1.mm-1
and 0.5 W-1.mm-1 for panels (b1) and (b2). The rest of the parameters are such
that $\beta_{6}=0$, and $P_{0}=1$W. Figure 8: The MI gain is represented in
the $(\Omega,\beta_{4})-$plane in presence of the in presence of the second-
order dispersion $\beta_{2}=-1.5$ps2mm-1. Panels (aj)j=1,2 correspond to the
case where the nonlocality is switched off, with the nonlinear coefficient
taking the respective values $\alpha=-0.5$ W.mm-2 and $\alpha=0.5$ W-1.mm-1
for panels (a1) and (a2). In panels (bj)j=1,2, similar calculations are
repeated but in presence of the nonlocal parameter that takes the value
$\gamma=0.01$, with the other parameters being $\beta_{6}=0$, and $P_{0}=1$ W.
Figure 9: The panels show the variations of the MI gain in the
$(\Omega,\beta_{4})-$plane in absence of the second-order dispersion
($\beta_{2}=0$), a non-nil nonlocality, $\gamma=0.01$. The effect of the
sixth-order dispersion coefficient $\beta_{6}$ is confronted to the effect of
the cubic nonlinearity for the emergence of the gain spectrum. Panels (aj)
show $\Lambda(\Omega)$ for $\alpha=0.5$ W-1.mm-1, with $\beta_{6}=0.2$ ps6mm-1
[panel (a1)] and $\beta_{6}=-0.2$ ps6mm-1 [panel (a2)]. Panels (bj) display
$\Lambda(\Omega)$ for $\alpha=-0.5$ W-1.mm-1, where $\beta_{6}=0.2$
ps6mm-1[panel (b1)] and $\beta_{6}=-0.2$ ps6mm-1 [panel (b2)], with the input
power being $P_{0}=1$ W.

####  III.2.2 The MI gain spectrum

In order to conduct exhaustively our MI analysis and predict regions of
parameter for the occurrence of MI, the MI gain will be, in general, plotted
versus the perturbation frequency $\Omega$ and the quartic dispersion
coefficient $\beta_{4}$.

The panels of Fig. 7 show the MI gain versus the perturbation frequency and
the quartic dispersion coefficient $\beta_{4}$ for negative and positive cubic
nonlinearity coefficient $\alpha$. This is carried out in the absence of
second-order and sextic dispersions. In general, the instability is manifested
by two frequency sidebands that vanish at $\beta_{4}=0$. On the one hand, when
$\alpha<0$, the instability takes part in the area $\beta_{4}>0$, while the MI
emerges for $\beta_{4}<0$ when $\alpha>0$. We should stress that in this
particular case, the nonlocality is switched off, and the windows of
instability are symmetrically large, compared to the case where $\gamma\neq
0$, as shown in Fig. 7(bj)j=1,2, with $\gamma=0.01$. Obviously, the MI gain
displays the same features, except that including the nonlocality shrinks the
instability sidebands that, nevertheless, remain symmetric. Precision should,
however, be made that the symmetric character of the MI gain is inherent to
optical fibers relying on the NLS equation, whose MI gain is usually a
quadratic function of the perturbation frequency $\Omega$. This has been
reported even in the presence of other physical effects such as higher-order
dispersion, higher-order nonlinearities, and cross-phase modulation
TabiOpLett2022 , to cite a few. In the meantime, recent works on materials
have demonstrated that effects related to spin-orbit (SO) coupling, for
example, can affect the symmetric features of the MI gain. This has been the
case for Bose-Einstein condensates with helicoidal SO coupling in one- and
two-space dimensions li2019modulational ; TabiPLA2022a , as well as Bose-Bose
mixtures with helicoidal SO coupling and, more recently, with Rashba and
Dresselhaus SO couplings TabiPRA2021 ; TabiPLA2022 . Interestingly, the three
types of SO coupling commonly include coupling through the first-order
derivatives with respect to the space coordinate of the BEC components.
Referring to the optical fiber, the appearance of symmetric sidebands is also
referred to as Stokes, and anti-Stokes sidebands, which for some input powers,
can become asymmetric DindaOpCom2006 . For now, the nonlocality contributes to
reducing the frequency band gap, while the Stokes and anti-Stokes sidebands
get enhanced. An important remark to be made here is that under the conditions
where the nonlocality is off, the system, whose MI gain is shown in Fig. 7, is
nonintegrable. However, the stability and existence of some pure-quartic
solitons with temporal profiles characterized by exponentially decaying tails,
with additional oscillations, were comprehensively discussed in Ref.
TamPRA2020  for $\beta_{4}<0$. Nevertheless, a conclusion was reached that
such entities were not really solitons and needed to be properly
characterized.

Beyond the above-discussed scenarios, one of the systems that were recently
given attention is the modified NLS equation in the presence of quadratic and
quartic dispersions. In presence of the nonlocality contribution, the figures
of instability are displayed in Fig. 8, where panels (aj)j=1,2 show
$\Lambda(\Omega)$ for $\gamma=0$ and $\beta_{2}=-1.5$ ps2mm-1. For
$\alpha=-0.5$ W-1.mm-1 [panel(a1)], the instability is supported by two
symmetrical lobes showing the MI gain exclusively in the area $\beta_{4}>0$,
while the maximum MI gain takes place in the area $\beta_{4}<0$ for
$\alpha=0.5$ W-1.mm-1 and decreases when $\beta_{4}$ increases from 0 to
infinity. In presence of the nonlocality, i.e., $\gamma=0.01$, the features
delivered by Fig. 8(bj)j=1,2 are similar to those of panels (aj)j=1,2, except
that the sidebands get shrunk, and the instability tends to be restrained to
the allocated frequency bands. Also, for $\alpha<0$, the band gap gets
extended while it gets reduced under negative nonlinear effects. The solution
of such a system is well-known, especially when the system is clear of the
nonlocality effect, as initially proposed by Karlsson and Höök
KarlssonOptCom1994  (when $\beta_{2}<0$ and $\beta_{4}<0$) and used by several
authors under different contexts.

Figure 10: Propagation of the perturbed plane wave intensity and wave pattern
formation in presence of only the quadratic dispersion. Panels (aj)j=1,2
correspond to the case where the nonlocality is switched off ($\gamma=0$),
while panels (bj)j=1,2 are obtained in presence of the nonlocality
($\gamma=0.01$). The rest of the parameters are such that: $\beta_{6}=0$,
$\beta_{4}=-0.24$ ps4mm-1 , $\beta_{2}=0$, $\beta_{6}=0$, $\Omega_{m}=0.75$
THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1. The upper line shows the evolution
of the perturbed plane wave, while the lower line displays the spectral
amplitude versus the propagation distance $z$ and the frequency shift. Figure
11: Panel (a) shows a section of Fig. 10(a) confirming the disintegration of
the plane wave solution into trains of solitonic objects while panel (a)
displays the maximum of the wave amplitude versus the propagation distance
$z$. This happens in presence of the nonlocality which shows to support
instability in pure-quartic NLS equation with parameters values being:
$\gamma=0.01$, $\beta_{6}=0$, $\beta_{4}=-0.2$ ps4mm-1, $\beta_{2}=0$,
$\Omega_{m}=0.75$ THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1. Figure 12:
Propagation of the perturbed plane wave intensity and wave pattern formation
in absence of the nonlocality [panels (aj)j=1,2] and in presence of the
nonlocality [panel (bj)j=1,2]. Panels (b1) and (b2) show the disintegration of
the CW in the frequency-space for $\gamma=0$ and $\gamma=0.01$, respectively,
with $\beta_{6}=0$, $\beta_{4}=-0.2$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1,
$\Omega=0.75$ THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1. The upper line shows
the evolution of the perturbed plane wave, while the lower line displays the
spectral amplitude versus the propagation distance $z$ and the frequency
shift.

A special case of the MI gain is displayed in Fig. 9, where the sextic
dispersion, materialized by $\beta_{6}$, is confronted with the other
dispersion orders, in presence of the nonlocality. We should stress that
recent investigations have addressed the type of bright soliton that may exist
in such a system, free from the nonlocality effect QiangPRA2022 . In the
meantime, the features of Fig. 9 reveal the existence of MI in such a model up
to the extent the nonlocality is accommodated by the process of wave pattern
generation. In this particular case, we have switched off the effect of the
quadratic dispersion $\beta_{2}$, and panels (aj)j=1,2 have been obtained for
$\alpha=0.5$ W-1.mm-1, with $\beta_{6}$ taking the respective values $0.2$
ps6mm-1 and $-0.2$ ps6mm-1. Here, the MI gain displays a set of two
symmetrical sidebands, but the effect of the negative sextic dispersion brings
about re-entrant instability tails, with the whole gain pattern being
restrained to the $\beta_{4}>0$ area [see Fig. 9(a1)]. In Fig. 9(a2), the
previous conditions are kept, except that $\beta_{6}=-0.2$ ps6mm-1 shifts the
major lobes of instability to the area $\beta_{4}<0$ along with extended
symmetrical tails of instability in the area $\beta_{4}>0$. A reversed
instability scenario appears for $\alpha=-0.5$W-1.mm-1, where curved
instability tails are present for $\beta_{6}=-0.2$ ps6mm-1 [see Fig. 9(b2)]
and extended tails for $\beta_{6}=0.2$ ps6mm-1 [see Fig. 9(b1)].

Figure 13: Propagation of the perturbed plane wave intensity and wave pattern
formation in presence of nonlocality for (aj)j=1,2: $\beta_{4}=-0.01$ ps4mm-1,
$\beta_{2}=-0.8$ ps2mm-1 and (bj)j=1,2: $\beta_{4}=-0.2$ ps4mm-1,
$\beta_{2}=-0.8$ ps2mm-1, with $\gamma=0.01$, $\beta_{6}=0$, $\Omega=0.75$
THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1. The upper line shows the evolution
of the perturbed plane wave, while the lower line displays the spectral
amplitude versus the propagation distance $z$ and the frequency shift. Figure
14: Propagation of the perturbed plane wave intensity and wave pattern
formation in absence of nonlocality for (aj)j=1,2: $\beta_{4}=-0.01$ ps4mm-1,
$\beta_{2}=-0.8$ ps2mm-1 and (bj)j=1,2: $\beta_{4}=-0.2$ ps4mm-1,
$\beta_{2}=-0.8$ ps2mm-1, with $\gamma=0.01$, $\beta_{6}=0$, $\Omega=0.75$
THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1. The upper line shows the evolution
of the perturbed plane wave, while the lower line displays the spectral
amplitude versus the propagation distance $z$ and the frequency shift.

##  IV Numerical experiment

The linear stability analysis has given us information about the onset of the
instability using various combinations of system parameters. Although the
number of parameters is not high compared to some recent NLS models, the
information gathered from the previous section are a clear evidence of the
possibility of MI in the studied model. However, the main drawback of the
linear stability analysis is its limitation in giving reliable information
about the long-distance propagation of the input CW signal, the reason being
that it results from the linearization around the unperturbed CW. Therefore,
to confirm the accuracy of our analytical predictions, Eq.(5) has been
integrated using the split-step Fourier method. An initial signal input of the
form

$\begin{split}\psi(t,z=0)=\sqrt{P_{0}}+\varepsilon\cos(\Omega_{m}t),\end{split}$
(17)

has been injected, with $\Omega_{m}=0.75$ THz being the modulation frequency,
$\varepsilon=10^{-3}$ and $P_{0}=1$ W. Although the quadratic dispersion
coefficient can take both positive and negative values, the whole study will
be conducted using $\beta_{2}<0$, while the other parameters will be varied
according to predictions from the linear stability analysis. The results will
be generally extracted in terms of the signal intensity $I=|\psi(t,z)|^{2}$.
Additionally, simulations will be performed using a propagation distance
$z=30L_{NL}$, where $L_{NL}$ represents a nonlinear length defined as
$L_{NL}=(\alpha P_{0})^{-1}$, with $P_{0}$ being the input power defined
above.

###  IV.1 MI under pure quartic dispersion: $\beta_{2}=0$, $\beta_{4}\neq 0$
and $\beta_{6}=0$

Figure 15: Propagation of the perturbed plane wave intensity and wave pattern
formation in absence of nonlocality for (a): $\beta_{4}=-0.01$ ps4mm-1,
$\beta_{2}=-0.8$ ps2mm-1 and (b): $\beta_{4}=-0.2$ ps4mm-1, $\beta_{2}=-0.8$
ps2mm-1, with $\gamma=0.01$, $\beta_{6}=0$, $\Omega_{m}=0.75$ THz, $P_{0}=1$W,
and $\alpha=0.5$ W-1.mm-1.

Considerable efforts have been made both analytically and numerically to
understand the dynamics of pure-quartic solitons. In this context, from a
theoretical point of view, the evolution of pure-quartic soliton has been
studied using a Gaussian-like shape ansatz based on a variational approach. In
the meantime, from the numerical point of view, it has been demonstrated that
the dynamics of the pure-quartic soliton behaves like a temporal profile
characterized by exponentially decaying tails with additional oscillations.
Only recently, close forms of pure-quartic solitons have been proposed in the
forms of $sech$ and $tanh$ functions, which do not possess oscillating tails,
markedly different from the conventional pure-quartic solitons in the absence
of additional effects such as weak nonlocality TrikiPLA2023 . This is also
true in the present work when the nonlocality is not considered, as shown in
Fig. 10(aj)j=1,2, where the CW under modulation remains stable, with
$\beta_{2}=-1.5$ ps2mm-1. In the presence of nonlocality, i.e., $\gamma\neq
0$, the instability develops, which shows that the non-integrability of the
purely quartic NLS equation can be violated under additional input from other
nonlinear terms, as depicted in Fig. 10(bj)j=1,2. Obviously, the MI is
manifested by solitonic objects that form trains and have pulse-like shapes
[see Fig. 11]. Indeed, this confirms our analytical predictions and gives more
credit to MI as a universal mechanism capable of leading to soliton formation
in nonlinear systems when the balance between nonlinearity and dispersion is
appropriate. Particularly, the behaviors of Fig. 10(b2) show some frequency
shift, which confirms the emergence of sidebands. We should, however, stress
that even though no exact solution can be found under the pure quartic
dispersion, it was experimentally revealed by Blanco-Redondo et al.
RedondoNatCom2016  that it may contribute to preserving the shape of any
pulsating signal. Moreover, they proposed an approximate shape of the
fundamental pure-quartic solution in the form of Gaussian and found agreement
with their experimental observations. Other contributions have revealed that
coupling negative quadric and quartic dispersions gives a possibility to solve
the NLS equations, with a solution that was proposed by Karlsson and Höök
KarlssonOptCom1994 . The MI in such a situation is explored next.

###  IV.2 MI under quadratic and quartic dispersions: $\beta_{2}\neq 0$,
$\beta_{4}\neq 0$ and $\beta_{6}=0$

The results related to this case are shown in Figs. 12 and 13. The first case
that is addressed here compares the situations where the nonlocality is absent
and when it is taken into account. In the absence of the nonlocality, the
competition between nonlinear and the available dispersion effects gives rise
to the patterns of Fig. 12(aj)j=1,2. The instability takes place around the
distance $z=15L_{NL}$ and manifests itself via the emergence of RW-like
patterns. When the nonlocality is involved, the initial CW breaks into the
patterns of Fig. 12(bj)j=1,2, where sneaky breathing patterns appear, with a
reduced wave intensity. However, the frequency shift takes place over a longer
distance compared to the events of Fig. 12(aj)j=1,2. This equally confirms our
analytical predictions from which, when parameters are well-chosen inside the
instability zone, the perturbation will grow exponentially and give rise to
wave patterns. Of course, we have restricted ourselves to the case of negative
quadratic dispersion. The reader should, however, be reminded that our
analytical calculations also predicted the existence of MI in the case where
$\beta_{2}$ is positive. In that direction, Tam and Alexander TamPRA2020
recently proposed a generalized solution that they named meta-soliton that was
found for $\beta_{2}>0$ and $\beta_{4}<0$, with specific characteristics
mostly applicable to laser systems. Moreover, single and multi-pulse solitary
wave solutions to a generalized NLS equation, in presence of both quadratic
and quartic dispersion terms were addressed by Parker and Aceves
ParkerPhysD2021 . The criteria for their existence and stability were also
studied and it was revealed that multi-soliton structures were fundamentally
unstable. In our context, another scenario relying on coupled quartic and
quadratic dispersions is shown in Fig. 13, where $\beta_{2}$ takes the value
$\beta_{2}=-0.8$ ps2mm-1, with $|\beta_{4}|$ increasing. In Fig. 13(aj)j=1,2,
where we have fixed $\beta_{4}=-0.01$ ps4mm-1, the instability develops
through the disintegration of the CW under a broad frequency shift as
testified by Fig. 13(a2). However, when $|\beta_{4}|$ is increased to 0.2
ps4mm-1, the CW disintegrates through a broad range of frequency sidebands,
which identifies erratic dynamics of the instability. This is supported by
highly localized structures as per the features of Fig. 13(b2). The patterns
are also found to be very sensitive to the inclusion of the nonlocality, which
further affects their dynamics when increased. For the patterns of Fig. 14,
the nonlocality parameter has been increased to $\gamma=0.08$, under the
conditions where $\beta_{2}$ remains fixed and $|\beta_{4}|$ takes values
$0.01$ ps4mm-1 [see Fig. 14(aj)j=1,2] and $0.2$ ps4mm-1 [see Fig.
14(bj)j=1,2]. In the first case, after initiation, many frequency shifts take
place as the patterns propagate, showing the long-distance robustness of MI
[see Fig. 14(a2)]. On the contrary, for $\beta_{4}=-0.2$ ps4mm-1, the
appearance of frequency shifts is reduced, although the initial step of MI is
supported by the emergence of RWs. Moreover, the final stage of MI here is
pictured in Fig. 15, where panels (a), related to Fig 14(aj)j=1,2, shows a
train of pulse and panel (b), extracted from Fig 14(bj)j=1,2, shows the
disintegration of the CW into two-humped solitons.

Figure 16: Propagation of the perturbed plane wave intensity and wave pattern
formation in absence of nonlocality for (aj)j=1,2: $\beta_{6}=-0.1$ ps6mm-1,
$\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1 and (bj)j=1,2:
$\beta_{6}=-0.5$ ps6mm-1, $\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1,
with $\gamma=0$, $\Omega_{m}=0.75$ THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1.
The upper line shows the evolution of the perturbed plane wave, while the
lower line displays the spectral amplitude versus the propagation distance $z$
and the frequency shift. Figure 17: Propagation of the perturbed plane wave
intensity and wave pattern formation in absence of nonlocality for (a):
$\beta_{6}=-0.1$ ps6mm-1, $\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1
and (b): $\beta_{6}=-0.5$ ps6mm-1, $\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$
ps2mm-1, with $\gamma=0$, $\Omega=0.75$ THz, $P_{0}=1$W, and $\alpha=0.5$
W-1.mm-1.

###  IV.3 MI under quadratic, quartic and sextic dispersions: $\beta_{2}\neq
0$, $\beta_{4}\neq 0$ and $\beta_{6}\neq 0$

The results of numerical simulations on the MI in the presence of the
quadratic, quartic, and septic dispersions are presented in Figs. 16-18 in the
absence and presence of the nonlocality. To remind, the effect of the sextic
dispersion is governed by the coefficient $\beta_{6}$ that can be positive or
negative. In absence of the nonlocality, the MI is manifested by the patterns
of Fig. 16, where we have fixed $\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$
ps2mm-1, while $|\beta_{6}|$ is attributed increasing values. The frequency
domain shows some frequency shifts from $z=15$$L_{NL}$ that evolve as distance
increases, showing a maintained instability of the CW for $\beta_{6}=-0.1$
ps6mm-1. In the same context, when $|\beta_{6}|$ is increased to $0.5$
ps6mm-1, a recombination phenomenon affects the temporal frequency of the RWs.
In the same direction, the dynamics of MI shows multiple frequency shifts that
testify to an early instability occurrence under sustained CW disintegration,
as shown in Fig. 16(b2). Fig. 17(a) shows how the AB breather evolves during
propagation, which illustrates the strong impact of propagation distance on
the development of MI through which the multi-alternation behavior can be
attributed to energy redistribution over the propagation distance. In the same
context and based on the AB patterns of Fig. 16(bj)j=1,2, the impact of
reducing the magnitude of $|\beta_{2}|$ is shown in Fig. 17(b) and supports
the interplay between the quadratic and sextic dispersion in maintaining
highly localized structures. It is obvious that decreasing $|\beta_{2}|$
contributes to reducing the amplitude of the rogue waves, which are
indubitably of type A according to the available nomenclature. In the presence
of the nonlocality, the features of instability given by Fig. 16 completely
change as shown in Fig. 18. In general, the inclusion of the nonlocality in
the presence of the sextic dispersion affects the propagation of the
instability. The nonlinear manifestation of MI is driven by AB lattice of type
A for $\beta_{6}=-0.1$ ps6mm-1, while the type-B breather patterns are
obtained for $\beta_{6}=-0.5$ ps6mm-1. The propagation features are also
different in terms of frequency shifts, where the distances between successive
shifts get reduced with $|\beta_{6}|$ increasing. This seems to support
efficient soliton fiber communication as it enhances regular bits of
information release [see Fig. 18(b2)]. Interestingly, each of the individual
elements of the detected bit displays a Peregrine soliton-like shape [see Fig.
18(c)] that is a straightforward signature of MI. Therefore, the MI approach
is a way to make use of the quartic and sextic dispersions to balance higher-
order nonlinear effects. This results in the shape of the obtained quartic-
sextic optical pulses being preserved while propagating along the optical
fiber.

Figure 18: Propagation of the perturbed plane wave intensity and wave pattern
formation in presence of nonlocality for (aj)j=1,2: $\beta_{6}=-0.1$ ps6mm-1,
$\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1 and (bj)j=1,2:
$\beta_{6}=-0.5$ ps6mm-1, $\beta_{4}=-0.8$ ps4mm-1, $\beta_{2}=-1.5$ ps2mm-1,
with $\gamma=0.01$, $\Omega=0.75$ THz, $P_{0}=1$W, and $\alpha=0.5$ W-1.mm-1.
The upper line shows the evolution of the perturbed plane wave, while the
lower line displays the spectral amplitude versus the propagation distance $z$
and the frequency shift.

More recently, experimental and theoretical investigations of localized waves
on plane-wave background have demonstrated that their dynamics is closely
related to RWs, mainly including the AB AkhmedievTMP1986 , Peregrine rogue
wave PeregrineJAMS1983 , and Kuznetsov-Ma breather Kuznetsov1977 . RWs are
relatively large and spontaneous waves whose appearance may result in
catastrophic damage DraperMObs1965 ; SolliNat2007 ; KharifBook2009 . It is
known that the generation of RWs is closely related to modulation instability
(MI), which is a complex nonlinear process exhibiting emergent behavior and
strong sensitivity to initial conditions OnoratoPR2013 ; DudleyNatPhot2014 ;
DudleyOptExp2009 ; AkhmedievJOpt2016 . The MI is frequently studied within the
NLS equation, which models the evolution of weakly nonlinear dispersive wave
packets. One can therefore study MI’s initial (i.e., linear) stage by
linearizing the NLS equation around the CW background. It is a well-known fact
that the linear stage of the MI is characterized by the exponential growth of
all the perturbations falling in the region of the Fourier spectrum below a
certain threshold through the amplification of sideband frequencies and is
closely associated with the concept of self-localized waves, or solitons. The
linearization ceases to be valid when the amplitude of the growing
perturbation becomes comparable to the CW background, that is, at the
nonlinear stage of MI, corresponding to the long-term spatiotemporal dynamics.
In this respect, a particular scenario of the MI development strongly depends
on the type of initial conditions considered, namely noisy perturbations of
the plane wave, localized perturbations of the plane wave, and harmonic
perturbations of the plane wave, respectively. For an initial condition
consisting of a CW field perturbed by a sine-modulation at the frequency of
maximum MI gain, the dynamics exhibits a recurrent behavior where AB-like
structures are formed periodically. In this context, the nonlinear stage of
the MI is described by an exact solution of the NLS, known as AB
AkhmedievJETP1985 ; AkhmedievMP1986 ; AkhmedievTMP1987 ; AkhmedievJETP1988 ;
AkhmedievPLA2009 ; EstellePRE2015 ; EstellePRE2016 ; EstellePRE2018 . For
noisy initial conditions, structures close to the AB, Kuznetsov-Ma solitons,
and Peregrine solitons, which are breather solutions of the 1D NLS equation,
emerge locally from the propagation of a perturbed modulational unstable CW
background OnoratoPR2013 ; DudleyNatPhot2014 ; DudleyOptExp2009 ;
AkhmedievJOpt2016 . In fact, the simplest first-order solutions that are
either periodic in space and localized in time or periodic in time and
localized in space are referred to as Kuznetsov-Ma breathers and AB,
respectively. Taking both solutions’ periods towards infinity makes it
possible to approach the limit of the doubly localized Peregrine breather.

The Darboux transformation method has explored interaction properties between
rogue waves and breathers LiCSF2022 . The correspondence relation between MI
and types of fundamental nonlinear excitation in optical fibers with both TOD
and FOD terms have been reported, including mainly the AB, RW, Kuznetsov-Ma
breather, periodic wave, W-shaped soliton train, rational W-shaped soliton,
anti-dark soliton, and nonrational W-shaped soliton DuanPRE2017 . It has been
shown that the dynamical process of Kuznetsov-Ma breather involves at least
two distinctive mechanisms, where the MI plays dominant roles in the mechanism
of Kuznetsov-Ma breather admitting weak perturbations, while the interference
effect plays a dominant role for the Kuznetsov-Ma breather admitting strong
perturbations ZhaoPRE2018 . Predictable rogue wave solutions of the (2
+1)-dimensional B-type Kadomtsev-Petviashvili equation by using symbolic
calculation methods PengPLA2018 . It has been reported that different MI gain
distribution can induce different rogue wave excitation patterns in a two-mode
nonlinear fiber whose dynamics is described by two-component coupled NLS
equations ZhaoPRE2014 .

##  V Concluding remarks

Based on recent experimental findings, the importance of wave propagation in
Kerr media in the presence of quadratic, quartic, and sextic dispersions has
gained significant interest. While some orientations pay attention to finding
exact soliton solutions to such models, we have devoted the present work to
the study of MI in presence of weakly nonlocal nonlinearity. In this context,
considering the nonlocal response as a perturbation to the Kerr nonlinearity,
we have evaluated the development of MI under different contexts of
dispersions both analytically and numerically. In order to find an expression
for the MI gain, the linear stability analysis of a CW has been used, from
which we have further derived the optimum modulation frequency. Using the
latter, the salient effect of the quartic and the sextic dispersions, and
their competition against the nonlocal nonlinearity, on the MI spectra have
been comprehensively addressed. In many cases, the dynamics of systems
exhibiting MI is governed to leading order by the one-dimensional cubic NLS
equation. Indeed, as noticed in this work, the NLS equation admits a plane
wave solution whose condition of instability has been widely discussed in the
literature. When the plane wave is unstable, its disintegration into solitonic
structures takes place. Fortunately, the condition for instability, also known
as the Benjamin-Feir criterion, gives rise to regions of parameters where
bright-type soliton (localized envelope) exact solution exists with a
vanishing amplitude at $|x|\rightarrow\infty$. With rare exceptions
AkhmedievSpringe2005 , dissipative systems are non-integrable, with a good
example being the cubic complex Ginzburg-Landau (CGL) equation, which also
admits a plane wave solution. The latter becomes unstable under the Lange-
Newell criterion, a generalized version of the Benjamin-Feir condition, due to
the competing effects of gains and losses PelapJPSJ2001 ; KenfackJPSJ2003 .
Under such a condition, the plane wave also breaks up into solitonic
structures, albeit the CGL equation is non-integrable. This also explains why,
although nonintegrable, the pure quartic NLS equation has been found to
support areas of instability under the strong action of nonlocality, both for
negative and positive values of $|\beta_{4}|$.

For OMF involving higher-order polynomials, the spectrum of MI has been
analyzed numerically, showing the strong input of the nonlocality in the
emergence of MI sidebands when the sextic dispersion is involved, under the
strong effect of the quartic and quadratic dispersions. Using such
predictions, the MI gain has been comprehensively exploited in finding regions
of parameters where MI could take place. The calculations revealed the
existence of extended tails of the MI gain due to the nonlocality and well-
balanced higher-order even dispersions. To confirm such analytical
predictions, direct numerical simulations have been carried out on the generic
model, where investigations have been based on various combinations of the
dispersions terms, the nonlinearity, and the nonlocality. Under pure quartic
conditions ($\beta_{2}=0$ and $\beta_{6}=0$) the CW was stable in the absence
of nonlocality, while the latter ensured the appearance of nonlinear
structures with solitonic form, showing locally and periodically AB-like
evolutions. The joint presence of the quadratic and quartic terms was found to
support CW disintegration, but including the nonlocal terms brought about
extended sneaky breathers that were spatially more extended with a broader
frequency shift. However, strong quartic dispersion brought about early MI
development but the frequency shift was too extended as a manifestation of
erratic patterns, under strong output signal intensity. With the right choice
of $\beta_{2}$ and $\beta_{4}$, it was also possible to generate extended
frequency shift modulation mediated by single- and double-humped breathers,
with strong input from nonlocal nonlinearity. More interestingly, the presence
of the sextic dispersion in the previous context brought about other features
of instability, among which the possibility of AB generated via the activation
of MI, with exotic behaviors highly dependent on the nonlocality. For example,
one could generate highly localized repetitive impulses by well-balancing the
sextic dispersion and the nonlocality term.

Solitons or other nonlinear objects are of relevant importance in physics in
general since, as mathematical models, they provide a better understanding of
physical models, and they may lead to physical applications. So, finding exact
solutions of the sixth-order dispersive NLS equation with a weak nonlocal
nonlinearity is a task of relative importance. For example, exact solutions
may (i) help to choose appropriate experimental parameters, (ii) provide a way
for probing the validity of the nonlinear evolution equations, (iii) help to
analyze the stability of solutions, (iv) check the numerical analysis of the
nonlinear evolution equations, and (v) help to explain the formation and the
propagation of different kinds of patterns in the system under consideration
as well as their long-time evolution. However, beyond the limitations inherent
to exact methods for the derivation of exact soliton solutions, MI can produce
a soliton train formation with 26 solitons formed in an elongated attractive
Bose-Einstein condensate in an axisymmetric harmonic trap CarrPRL2004 , which
is a groundbreaking result. On the contrary, using the inverse scattering
methods or the Hirota bilinear method to generate 26 solitons is quite
cumbersome, if not possible. Additionally, the spontaneous development of MI
seeded from noise has also been shown to underlie the initial stages of fiber
supercontinuum generation seeded by continuous wave (CW) radiation or
picosecond or nanosecond pulses, which remains a challenging ambition of exact
solution seekers.

## Acknowledgements

CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of
California Santa Barbara (USA), where this work was supported in part by the
National Science Foundation Grant no.NSF PHY-1748958, NIH Grant
no.R25GM067110, and the Gordon and Betty Moore Foundation Grant no.2919.01.

## References

  * (1) G. L. Lamb Jr, Elements of Soliton Theory (Wiley, New York, 1980).
  * (2) M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981).
  * (3) V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method (Consultants Bureau, New York, 1984 ).
  * (4) R. K. Dodd, J. C. Eilbeck, J. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, New York, 1982 ).
  * (5) Y. S. Kivshar and G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Academic, New York, 2003).
  * (6) Y. Silberberg, Opt. Lett. 15, 1282 (1990).
  * (7) C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean ( Springer, New York, 2009).
  * (8) D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
  * (9) D. Peregrine, J. Austr. Math. Soc. Series 25, 16 (1983).
  * (10) E. A. Kuznetsov, Sov. Phys.-Doklady 22, 575 (1977) .
  * (11) Y. C. Ma, Stud. Appl. Math. 60, 43 (1979).
  * (12) N. Akhmediev, V. Eleonskii, and N. Kulagin, Theor. Math. Phys. 72, 809 (1987).
  * (13) G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989).
  * (14) P. K. A. Wai, H. H. Chen, and Y. C. Lee, Phys. Rev. A 41, 426 (1990).
  * (15) J. N. Elgin, Phys. Rev. A 47, 4331 (1993).
  * (16) V. I. Karpman, Phys. Rev. E 47, 2073 (1993).
  * (17) Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, Opt. Lett. 19, 165 (1994).
  * (18) N. Akhmediev and M. Karlsson, Phys. Rev. A 51, 2602 (1995).
  * (19) J. N. Elgin, T. Brabec, and S. M. J. Kelly, Opt. Commun. 114, 321 (1995).
  * (20) S. Roy, S. K. Bhadra, and G. V. Agrawal, Phys. Rev. A 79, 023824 (2009).
  * (21) A. Hook and M. Karlsson, Opt. Lett. 18, 1388 (1993).
  * (22) N. N. Akhmediev, A. V. Buryak, and M. Karlsson, Opt. Comm. 110, 540 (1994).
  * (23) I. P. Christov, Margaret M. Murnane, Henry C. Kapteyn, Jianping Zhou, and C.-P. Huang, Opt. Lett. 19, 1465 (1994).
  * (24) M. Karlsson and A. Höök, Opt. Commun. 104, 303 (1994).
  * (25) F. Kh.Abdullaev, S. A. Darmanyan, S. Bischoff, P. L. Christiansen, and M. P. Sørensen, Opt. Commun. 108, 60 (1994).
  * (26) N. N. Akhmediev, and A. V. Buryak, Opt. Comm. 121, 109 (1995).
  * (27) M. Piche, J.-F. Cormier, and X. Zhu, Opt. Lett. 21, 845 (1996).
  * (28) V. E. Zakharov and E. A. Kuznetsov, J. Exp. Theor. Phys. 86, 1035 (1998).
  * (29) B. Kalithasan, K. Porsezian, and P. Tchofo Dinda, J. Opt. 12, 035210 (2010).
  * (30) A. Armaroli and F. Biancalana, Opt. Lett. 39, 4804 (2014).
  * (31) A. Blanco-Redondo, C. Martijn de Sterke, J. E. Sipe, T. F. Krauss, B. J. Eggleton, and C. Husko, Nature Commun. 7, 10427 (2016).
  * (32) C. Martijn de Sterke, A. F. J. Runge, D. D. Hudson, and A. Blanco-Redondo, APL Photon. 6, 091101 (2021).
  * (33) C.-W. Lo, A Stefani, C Martijn de Sterke, and A. Blanco-Redondo, Opt. Express 26, 7786 (2018).
  * (34) H. Taheri, and A. B. Matsko, Opt. Lett. 44, 3086 (2019).
  * (35) V. I. Kruglov and J. D. Harvey, Phys. Rev. A 98, 063811 (2018).
  * (36) K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, and C. Martijn de Sterke, Opt. Lett. 44, 3306 (2019).
  * (37) P. Gao, C. Liu, Li-C. Zhao, Z.-Y. Yang, and W.-Li Yang, Phys. Rev. E 102, 022207 (2020).
  * (38) K K. K. Tam and T. J. Alexander, Phys. Rev. A 101, 043822 (2020).
  * (39) R. Parker and A. Aceves, Physica D 422, 132890 (2021).
  * (40) R. I. Bandara, A. Giraldo, N. G. R. Broderick, and B. Krauskopf, Phys. Rev. A 103, 063514 (2021).
  * (41) K. Zhao, C. Gao, X. Xiao, and C. Yang, Opt. Lett. 46, 761 (2021).
  * (42) Z.-X. Zhang, M. Luo, J.-X. Chen, L.-H. Chen, M. Liu, A.-P. Luo, W.-C. Xu, and Z.-C. Luo, Opt. Lett. 47, 1750 (2022).
  * (43) Z.-C. Qian, M. Liu, A.-P. Luo, Z.-C. Luo, and W.-C. Xu, Opt. Express 30, 22066 (2022).
  * (44) X. Yao, C. Liu, Z.-Y. Yang, and W.-L. Yang, Phys. Rev. Res. 4, 013246 (2022).
  * (45) T. J. Alexander, G. A. Tsolias, A. Demirkaya, R. J. Decker, C. Martijn de Stecrke, and P. G. Kevrekidis, Opt. Lett. 47, 1174 (2022).
  * (46) Y. Li, J. Dai, J. Liu, Z. Weng, W. Hu, D. Lu, Opt. Commun. 128790 (2022). DOI: 10.1016/j.optcom.2022.128790.
  * (47) Y. L. Qiang, T. J. Alexander, and C. M. de Sterke, Phys. Rev. A 105, 023501 (2022).
  * (48) M. Salerno and B. Baizakov, Phys. Rev. E 98, 062220 (2018).
  * (49) D. Zanga, S. I. Fewo, C. B. Tabi, and T. C. Kofané, Commun. Nonl. Sci. Numer. Simul. 80, 104993 (2020).
  * (50) P. G. Kevrekidis, R. Carretero-Gonzalez, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed, Phys. Rev. A 68, 035602 (2003).
  * (51) W. X. Zhang, D. L. Zhou, M.- S. Chang, M. S. Chapman, and L. You, Phys. Rev. Lett. 95, 180403 (2005).
  * (52) A. Smerzi, A. Trombettoni, P. G. Kevrekidis, D. J. Frantzeskakis, and A. R. Bishop, Phys. Rev. Lett. 89, 170402 (2002).
  * (53) A. F. J. Runge, Y. L. Qiang, T. J. Alexander, M. Z. Rafat, D. D. Hudson, A. Blanco-Redondo, and C. Martijn de Sterke, Phys. Rev. Res. 3, 013166 (2021).
  * (54) W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, J. Opt. B 6, S288 (2004).
  * (55) H. Tagwo, A. Mohamadou, C. G. Latchio Tiofack, and T. C. Kofane, Optik 144, 246 (2017).
  * (56) C. B. Tabi, H. Tagwo, C. G. L. Tiofack, and T. C. Kofané, Opt. Lett. 47, 5557 (2022).
  * (57) C. B. Tabi, H. Tagwo, and T. C. Kofane, Phys. Rev. E 106, 054201 (2022).
  * (58) O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, Phys. Rev. E 66, 046619 (2002).
  * (59) M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, Opt. Lett. 27, 1460 (2002).
  * (60) N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, Opt. Lett. 29, 286 (2004).
  * (61) Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Lett. 30, 3171 (2005).
  * (62) A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
  * (63) A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, Phys. Rev. E 71, 065603 (2005).
  * (64) D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Opt. Express 13, 435 (2005).
  * (65) A. S. Desyatnikov and Yu. S. Kivshar, Phys. Rev. Lett. 88, 053901 (2002).
  * (66) A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005).
  * (67) K. S. Bezuhanov, A. A. Dreischuh, and W. Krolikowski, Phys. Rev. A 77, 033825 (2008).
  * (68) S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, Phys. Rev. E 73, 066603 (2006).
  * (69) W. Chen, Q. Wang, J. Shi, Q. Kong, Y. Chen, and M. Shen, J. Phys. B: At. Mol. Opt. Phys. 50, 135401 (2017).
  * (70) X. X. Li, R. J. Cheng, A. X. Zhang, and J. K. Xue, Phys. Rev. E 100, 032220 (2019).
  * (71) C. B. Tabi, P. Otlaadisa, and T. C. Kofané, Phys. Lett. A 449, 128334 (2022).
  * (72) C. B. Tabi, S. Veni, and T. C. Kofané, Phys. Rev. A 104, 033325 (2021).
  * (73) C. B. Tabi, S. Veni, and T. C. Kofané, Phys. Lett. A 442, 128192 (2022).
  * (74) P. Tchofo Dinda, C.M. Ngabireng, K. Porsezian, and B. Kalithasan, Optics Commun. 266, 142 (2006) 142.
  * (75) H. Triki, A. Pan, and Q. Zhou, Phys. Lett. A 29, 128608 (2023).
  * (76) N. Akhmediev and V. I. Korneev, Theor. Math. Phys. 69, 1089 (1986).
  * (77) ] D. H. Peregrine, J. Aust. Math. Soc. Ser. B 25, 16 (1983).
  * (78) E. Kuznetsov, Sov. Phys. Dokl. 22, 507 (1977); Y. C. Ma, Stud. Appl. Math. 60, 43 (1979).
  * (79) L. Draper, Mar. Obs. 35, 193 (1965).
  * (80) D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature 450, 1054 (2007).
  * (81) C. Kharif, E. Pelinovsky, and A. Slyunyaev, Quasi-linear wave focusing. In: Rogue waves in the ocean. Berlin: Springer; 2009. p. 63-89.
  * (82) M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, F. T. Arecchi, Phys. Rep. 528, 47 (2013).
  * (83) J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Nature Photon. 8, 756 (2014).
  * (84) J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Opt. Express 17, 21497 (2009).
  * (85) N. Akhmediev, B. Kibler, F. Baronio, M Belić, W.-P. Zhong, and Y. Zhang et al., J. Opt. 18, 063001 (2016).
  * (86) N. Akhmediev, V. Eleonskii, and N. Kulagin, Sov. Phys. JETP 62, 894 (1985).
  * (87) N. Akhmediev and V. Korneev, Theor. Math. Phys. 69, 1089 (1986).
  * (88) N. N. Akhmediev, V. I. Korneev, and N. V. Mitskevich, J. Experiment. Theor. Phys. 67, 159 (1988).
  * (89) N. Akhmediev, A. Ankiewicz, and M. Taki, Phys. Lett. A 373, 675 (2009).
  * (90) D. D. Estelle Temgoua and T. C. Kofané, Phys. Rev. E 91, 063201 (2015).
  * (91) D. D. Estelle Temgoua and T. C. Kofané, Phys. Rev. E 93, 062223 (2016).
  * (92) D. D. Estelle Temgoua, M. B. Tchoula Tchokonte, and T. C. Kofané, Phys. Rev. E 97, 042205 (2018).
  * (93) B.-Q. Li and Y.-L. Ma, Chaos Solit. Fract. 156, 111832 (2022).
  * (94) L. Duan, L.-C. Zhao, W.-H. Xu, C. Liu, Z.-Y. Yang, and W.-L. Yang, Phys. Rev. E 95, 042212 (2017).
  * (95) L.-C. Zhao, L. Ling, and Z.-Y. Yang, Phys. Rev. E 97, 022218 (2018).
  * (96) W.-Q. Peng, S.-F. Tian, and T.-T. Zhang, Phys. Lett. A 382, 2701 (2018).
  * (97) L.-C. Zhao, G.-G. Xin, and Z.-Y. Yang, Phys. Rev. E 90, 022918 (2014).
  * (98) Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz (Springer, Heidelberg, 2005).
  * (99) F. B. Pelap, T. C. Kofané, N. Flytzanis, and M. Remoissenet, J. Phys. Soc. Jpn. 70, 2568 (2001).
  * (100) A. Kenfack Jiotsa and T. C. Kofané, J. Phys. Soc. Jpn. 72, 1800 (2003).
  * (101) L. D. Carr and J. Brand, Phys. Rev. Lett. 92, 040401 (2004).