# Dipole and quadrupole nonparaxial solitary waves

Naresh Saha<EMAIL_ADDRESS>Barnana Roy<EMAIL_ADDRESS>Physics
and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108,
India.  Avinash Khare<EMAIL_ADDRESS>Department of Physics,
Savitribai Phule Pune University, Pune - 411007, India.

###### Abstract

The cubic nonlinear Helmholtz equation with third and fourth order dispersion
and non-Kerr nonlinearity like the self steepening and the self frequency
shift is considered. This model describes nonparaxial ultrashort pulse
propagation in an optical medium in the presence of spatial dispersion
originating from the failure of slowly varying envelope approximation. We show
that this system admits periodic (elliptic) solitary waves with dipole
structure within a period and also transition from dipole to quadrupole
structure within a period depending on the value of the modulus parameter of
Jacobi elliptic function. The parametric conditions to be satisfied for the
existence of these solutions are given. The effect of the nonparaxial
parameter on physical quantities like amplitude, pulse-width and speed of the
solitary waves are examined. It is found that by adjusting the nonparaxial
parameter, the speed of solitary waves can be decelerated. The stability and
robustness of the solitary waves are discussed numerically.

The propagation of broad self-trapped beams (pulses) in Kerr-type nonlinear
media in the presence of spatial dispersion arising from the nonparaxial
effect, is described by the nonlinear Helmholtz equation. Such an effect is
important in the case when the wavelength is comparable to the beam (pulse)
width, e.g. in the progressive miniaturization of the optical devices. For
studying the propagation of ultrashort nonparaxial optical beams (pulses), the
higher order dispersion/diffraction terms as well as non-Kerr nonlinear terms
like the self steepening and the self frequency shift should be included in
the nonlinear Helmholtz equation. We present analytical periodic (elliptic)
solitary waves which exhibit the dipole and the quadrupole structure within a
period. Importantly, transition from the dipole to the quadrupole structure is
possible depending on the value of the modulus parameter of Jacobi elliptic
function. Numerical simulations validate the stability and robustness of the
solitary wave solutions.

##  1 Introduction:

In nonlinear optics, the study of solitons has been a subject of intense
investigations since their first theoretical prediction [1]. It is widely
accepted that the cubic nonlinear Schrodinger equation (NLSE) is the generic
model for the description of dynamics of optical solitons in Kerr type
nonlinear media [2]. Hasegawa and Tappert derived the NLSE by employing
paraxial approximation or slowly varying envelope approximations (SVEA) [3],
in which the width of a sufficiently low intensity pulse (beam) is larger than
the carrier wavelength and it propagates along (or at negligible angles) a
reference axis. If at least one of the conditions is not met, the pulse (beam)
is referred to as nonparaxial [4]. Interest on nonparaxial beams has started
attracting interest after the pioneering work [5] in which the propagation of
ultra narrow optical beams is considered. On the other hand, Helmholtz
nonparaxiality occurs when the broad pulse (beam) propagates at arbitrary
angles to the reference direction [6, 4, 7]. The governing equation is then
known as the Helmholtz-Manakov (HM) equation [8]. In both of the above
scenarios, the equivalence of the transverse and longitudinal dimensions in
uniform media is achieved by ignoring SVEA.
The use of ultra narrow soliton beams in miniaturized optical devices is
proposed in [6]. The analytical soliton solutions to the scalar NLH equation
are derived in [4, 12] for focusing and defocusing Kerr nonlinearity, while
those for the power law, polynomial, and the saturable nonlinearity are
obtained in [11, 13, 14], those in the anomalous and normal group velocity
dispersion regimes are obtained in [15]. Further, exact soliton solutions are
studied in the relativistic and pseudo relativistic formulation of the scalar
NLH equaion with cubic, quintic and saturable nonlinearity [17, 16]. Quasi
soliton behavior of the exact as well as approximate solutions to the scalar
and vector nonparaxial evolution equations developed for describing
propagation in two dimensions, are shown in [18]. Vortex solitons in two
dimensions [19], rogue waves [21], and numerical solutions to the nonparaxial
nonlinear Schödinger equation are obtained in [20]. The study of
nonparaxiality in $\cal{PT}$-symmetric optical context is found in [22].
Solitary wave solutions to the coupled NLH equation in $(1+1)$ dimensions and
to the coupled nonparaxial NLSE in $(2+1)$ dimensions are obtained in [23] and
[24] respectively. The effect of nonparaxiality on the amplitude, pulse-width
and the speed of the solitary waves are studied too [23]. Modulational
instability of the cubic quintic NLH system is studied in [25] and the chirped
elliptic and hyperbolic solitary wave solutions are presented while those for
the cubic coupled NLH equation in the presence of the self steepening (SS) and
the self frequency (SFS) shift are studied in [26].

However, for modeling the propagation of sub-picosecond (femtosecond) optical
pulses (beams) in non-paraxial regime, it is required to incorporate higher
order effects, such as the third order dispersion (TOD), the fourth order
dispersion (FOD) and non-Kerr nonlinear terms like the SS and the SFS into the
scalar NLH equation. It is relevant to mention that different higher order
effects appear in optical materials for various applications such as, in
ultrahigh-bit-rate optical communication systems, ultrafast physical processes
etc.[28]. TOD plays an important role in the propagation of short pulses whose
widths are nearly $50$ fs [29, 30]. When pulse widths are shorter than $10$
fs, FOD becomes essential [29, 30]. The wave dynamics in such a situation in
paraxial regime, is governed by higher order NLSE (HNLSE) [31]. Recent
researches have focused on seeking more complex, such as, dipole, multipole
structures [32] than nodeless amplitude distributions featured by fundamental
solitons for HNLSE. It is relevant to mention that a bright solitary wave
(having a $sech$-profile) has a localized intensity peak over a continuous
wave background and the intensity profile vanishes at $\pm\infty$ whereas a
dark solitary wave (having $\tanh$-profile) exhibits a dip in a continuous
wave background and the intensity profile increases monotonically from the dip
at the center and approaches a constant value at $\pm\infty$. When the
intensity profile of a solitary wave has two peaks and one dip at the centre
(where the light intensity is zero) in a continuous wave background, it is
referred to as dipole solitary wave. It can be viewed as an inter-modulation
of bright and dark components with the same pulse width and velocity [37].
When the intensity profile of a solitary wave has four peaks and three dips in
a continuous wave background, it is referred to as quadrupole solitary wave.
As mentioned in [38, 39, 46] multipole solitary waves can be viewed as
composed of several fundamental solitary waves with alternating phases. This
opens up the possibility of encoding complex solitary images and letters by
packing an arbitrary number of fundamental solitary waves into a stable matrix
[38]. Dipole solitons occur in many optical media, like, nonlocal nonlinear
media [33, 34, 35, 36], thermal nonlinear media [39] and photonic lattices
[40, 41]. Stationary two dimensional multipole mode solitons have been
observed experimentally in a medium with thermal nonlinearity [42]. Efficient
ansatz approaches for example, sum and product of bright and dark solitons
have been used to derive the so-called dark-in-the -bright, dipole, tripole,
and the multipole solutions [43, 59, 44, 45, 46, 47, 48, 49]. Absence of such
type of study in the nonparxial regime, so far as we are aware of, has
motivated us to look for dipole and multipole solitary waves in NLH system
with higher order dispersion and non-Kerr nonlinearity. As it extends the
known classification of solitary wave solution namely dipole and multipole
solitary wave to the nonparaxial regime, this is a problem well worth
investigating.
The objective of the present study is to investigate the solitary wave
propagation for higher order NLH systems with cubic Kerr nonlinearity and
spatial dispersion which arises when SVEA fails. By considering the effect of
TOD, FOD, SS and SFS, we show that the considered system admits dipole soliton
and periodic solitary waves with the dipole and quadrupole structure within a
period under certain parametric conditions. The novelty of the present study
lies not only in the existence of dipole structure within a period of periodic
(elliptic) solitary wave but also in the transition from dipole to quadrupole
structure within a period, depending on the value of the modulus parameter $m$
of the Jacobi elliptic function [61]. So far as we are aware of, these
interesting propagation characteristics of the nonparaxial periodic waves is
uncovered for the first time in this article. The evolution and stability of
these waves are analyzed by direct numerical simulations.
The organization of the present article is as follows. In Section II we
present the theoretical model of the cubic nonlinear Helmholtz equation in the
presence of TOD, FOD, SS and SFS. This model describes propagation of
ultrashort nonparaxial solitary waves in optical waveguide/fibre with the
spatial dispersion arising from the failure of SVEA. In Section III, we obtain
the dipole and quadrupole solitary waves. The characteristic features of these
solitary waves and their physical significance are studied too. In Section IV
we discuss the stability and robustness of the four solutions obtained here.
Finally, in Section V, we summarize our results and indicate their relevance.

##  2 Theoretical model

We start with the following higher order dimensionless NLH equation describing
pulse propagation in the nonparaxial regime [16]

$\begin{array}[]{lcl}i(\frac{\partial\psi}{\partial\zeta}+\alpha\frac{\partial\psi}{\partial\tau})+\frac{a_{1}}{2}\frac{\partial^{2}\psi}{\partial\tau^{2}}+a_{2}|\psi|^{2}\psi+i[a_{3}\frac{\partial^{3}\psi}{\partial\tau^{3}}+a_{4}\frac{\partial(|\psi|^{2}\psi)}{\partial\tau}+a_{5}\psi\frac{\partial(|\psi|^{2})}{\partial\tau}]+a_{6}\frac{\partial^{2}\psi}{\partial\zeta^{2}}+a_{7}\frac{\partial^{4}\psi}{\partial\tau^{4}}=0\end{array}$
(1)

where $\psi(\zeta,\tau)$ is the complex pulse envelope, $\zeta$ and $\tau$ are
dimensionless space and time variables respectively. The coefficients
$a_{i},i=1,2,3,7$ are respectively the group velocity dispersion (GVD) , the
self phase modulation (SPM), the TOD, and the FOD. $a_{2}=1$, $a_{1}=\pm 1$
indicates anomalous and normal GVD respectively. The term proportional to
$a_{4}$ corresponds to SS [50]. This causes a temporal pulse distortion, which
leads to the formation of an optical shock at the trailing edge of the pulse
[51] unless balanced by the dispersion. The term proportional to $a_{5}$
accounts for the delayed Raman response [52] which is responsible for the
pulse undergoing a frequency shift known as SFS [53]. The term proportional to
$a_{6}(>0)$ (nonparaxial parameter) refers to the spatial dispersion arising
from the nonparaxial effect.
The term $i\alpha\frac{\partial\psi}{\partial\tau}$ in (1) can be transformed
away by the Galilean boost

$t=\tau-\alpha\zeta,~{}~{}~{}~{}x=\zeta$ (2)

i.e. the solutions to (1) describe waves in a frame moving with speed
$\frac{1}{\alpha}$ along the $\zeta$ axis. Thus Eq. (1) is transformed into

$\begin{split}i\frac{\partial\psi}{\partial
x}+\frac{1}{2}(a_{1}+2a_{6}\alpha^{2})\frac{\partial^{2}\psi}{\partial
t^{2}}-2\alpha a_{6}\frac{\partial^{2}\psi}{\partial x\partial
t}+a_{2}|\psi|^{2}\psi+i[a_{3}\frac{\partial^{3}\psi}{\partial
t^{3}}+a_{4}\frac{\partial(|\psi|^{2}\psi)}{\partial
t}+a_{5}\psi\frac{\partial(|\psi|^{2})}{\partial t}]\\\
+a_{6}\frac{\partial^{2}\psi}{\partial
x^{2}}+a_{7}\frac{\partial^{4}\psi}{\partial t^{4}}=0\end{split}$ (3)

Now we restrict to solutions for which $\alpha a_{6}<<O(1)$ and
$a_{6}\alpha^{2}<<O(1)$. Also the terms $\frac{\partial^{2}\psi}{\partial
t^{2}}$ and $\frac{\partial^{2}\psi}{\partial x\partial t}$ are assumed to be
both $O(1)$. These solutions will satisfy the equation

$\begin{array}[]{lcl}i\frac{\partial\psi}{\partial
x}+\frac{a_{1}}{2}\frac{\partial^{2}\psi}{\partial
t^{2}}+a_{2}|\psi|^{2}\psi+i[a_{3}\frac{\partial^{3}\psi}{\partial
t^{3}}+a_{4}\frac{\partial(|\psi|^{2}\psi)}{\partial
t}+a_{5}\psi\frac{\partial(|\psi|^{2})}{\partial
t}]+a_{6}\frac{\partial^{2}\psi}{\partial
x^{2}}+a_{7}\frac{\partial^{4}\psi}{\partial t^{4}}=0\end{array}$ (4)

In what follows we shall take $a_{1},a_{2}=1$. It is important to note that
the term $a_{6}\frac{\partial^{2}\psi}{\partial x^{2}}$ in (4) arising from
the absence of SVEA, leads to a dispersion relation that supports both forward
and backward propagating waves. This is in sharp contrast to NLS [9] and
Manakov [10] equations where the backward propagating waves are not allowed
[8].

To obtain the dipole solitary waves we use the following traveling wave ansatz

$\psi=u(\xi)e^{i\phi(x,t)},~{}~{}\phi(x,t)=kx-\omega
t+\phi_{0},~{}~{}\xi=\beta(t-cx),$ (5)

where $u$ is the amplitude and $\phi$ is the phase of the solitary wave. These
are real functions of $\xi$. The parameter $c,k,\phi_{0}$ are the inverse
velocity, wave number and real constant respectively. On substituting this
ansatz in Eq. (4) and separating the real and imaginary parts we obtain

$\begin{array}[]{lcl}a_{7}\beta^{4}u^{\prime\prime\prime\prime}+(\frac{1}{2}a_{1}+a_{6}c^{2}+3a_{3}\omega-6a_{7}\omega^{2})\beta^{2}u^{\prime\prime}+(a_{2}+\omega
a_{4})u^{3}-(k+k^{2}a_{6}+\frac{1}{2}\omega^{2}a_{1}+\omega^{3}a_{3}-\omega^{4}a_{7})u=0\,,\end{array}$
(6)

and further

$\begin{array}[]{lcl}(a_{3}-4\omega
a_{7})\beta^{3}u^{\prime\prime\prime}+(3a_{4}+2a_{5})\beta
u^{2}u^{\prime}-(c+2kca_{6}+\omega
a_{1}+3\omega^{2}a_{3}-4\omega^{3}a_{7})\beta u^{\prime}=0\end{array}$ (7)

##  3 Dipole/quadrupole solitary waves

Solution I. Dipole Soliton. It is easily checked by direct substitution that
Eqs. (6) and (7) admit the following dipole solitary wave solution

$u(\xi)=Asech(\xi)\tanh(\xi)\,,$ (8)

provided

$\displaystyle 2a_{5}+3a_{4}$ $\displaystyle=0$ (9a) $\displaystyle
a_{3}-4\omega a_{7}$ $\displaystyle=0$ (9b) $\displaystyle(1+2ka_{6})c$
$\displaystyle=-\omega(a_{1}+8\omega^{2}a_{7})$ (9c) $\displaystyle\
(a_{2}+\omega a_{4})A^{2}$ $\displaystyle=120a_{7}\beta^{4}$ (9d)

and further

$\begin{array}[]{lcl}\frac{1}{2}a_{1}+c^{2}a_{6}=-2(3\omega^{2}-5\beta^{2})a_{7}\\\
k+k^{2}a_{6}+\frac{1}{2}\omega^{2}a_{1}=(11\beta^{4}-3\omega^{4})a_{7}\end{array}$
(10)

It is relevant to mention that similar constraint relation between the SS and
the SFS (Eqn.9(a)) has also been obtained in [43, 54, 46, 47, 55] for
obtaining different solitary wave solutions of nonlinear Schödinger equation
containing higher order dispersion and non-Kerr nonlinearities and also for
obtaining chirped solitary wave solutions of coupled cubic NLH equation in the
presence of SS and SFS [26].
We now show that the dipole solitary wave (8) is indeed a dipole soliton by
considering the collision dynamics as depicted in Fig. 1. For this the initial
condition chosen is

$\begin{array}[]{rcr}\Psi(0,t)=\psi(0,t)-1.5~{}sech(0.2t+5)~{}\tanh(0.2t+5)\displaystyle
e^{i(-\omega t+\phi_{0})}\end{array}$ (11)

The two dipole solitary waves converge during propagation and at a distance of
about $x=10~{}\rm units$ they collide as shown in Fig.1. It is quite evident
that the two dipole solitary waves collide elastically and continue their
propagation after the collision. The intensity profile of the dipole soliton
(8) with respect to $t$ is shown in Fig.2(a) while the numerical simulation of
the intensity profile is shown in Fig.2(c) demonstrating its stable evolution
for the chosen values of the parameters. From Fig.2(b) we can see that as the
nonparaxial parameter $a_{6}$ increases, the amplitude $A$ increases but the
speed $|c|$ and the pulse width $\frac{1}{\beta}$ both decrease.

Figure 1: Evolution of the intensity profile and collision dynamics of (8) as
a function of propagation distance $x$ for
$a_{3}=\frac{1}{60},~{}a_{4}=0.126,~{}a_{5}=-0.189,~{}a_{6}=0.5,~{}a_{7}=0.15,\beta=0.577914,~{}\omega=0.027778,~{}\phi_{0}=0.5,~{}c=0.023777,~{}k=-2.169328,~{}A=1.4195$.

(a)  (b)  (c)

(d)  (e)  (f)

Figure 2: (a) plot of $|\psi|^{2}$ versus $t$ at $x=0$ (c), (d) numerical
simulations of the intensity profile without and with $10\%$ white noise
respectively of the solution (8) for
$a_{3}=\frac{1}{60},~{}a_{4}=0.126,~{}a_{5}=-0.189,~{}a_{6}=0.5,~{}a_{7}=0.15,\beta=0.577914,~{}\omega=0.027778,~{}\phi_{0}=0.5,~{}c=0.023777,~{}k=-2.169328,~{}A=1.4195$
respectively, (b) plots of $|c|$, $\frac{1}{\beta}$, $A$ versus $a_{6}$ for
the solution (8) for $a_{3}=-0.18,~{}a_{4}=0.126,~{}a_{7}=0.15$, (e) plot of
the numerical simulation of the intensity profile by adding $0.003$ to the
parameters $a_{i},i=1,2,\cdots 7$ and (f) plot of the numerical simulation of
the intensity profile by adding the overall factor $0.001$ to the parametric
conditions (9) & (10) respectively.

Solution II. Periodic wave exhibiting dipole/quadrupole structure within a
period

Eqns.(6) and (7) also admit the following elliptic solitary wave solution

$u(\xi)=A\sqrt{m}sn(\xi,m)dn(\xi,m)\,,$ (12)

($m$ being the modulus of the Jacobi elliptic functions $sn,dn$, $0<m<1$ [61])
provided Eqs.(6a) - (6d) are satisfied and in addition, the following two
relations are also satisfied

$\begin{array}[]{lcl}\frac{1}{2}a_{1}+c^{2}a_{6}=-2[3\omega^{2}-5(2m-1)\beta^{2}]a_{7}\\\
k+k^{2}a_{6}+\frac{1}{2}\omega^{2}a_{1}=[(11+64m-64m^{2})\beta^{4}-3\omega^{4}]a_{7}\end{array}$
(13)

It is easily checked that this solution exhibits dipole structure within a
period so long $m\leq 0.5$. On the other hand, for values of $m>0.5$, it
exhibits quadrupole structure within a period. Fig.3(a) and 4(a) respectively
depict the dipole and the quadrupole intensity profiles of (12) respectively
while Figs. 3(c) and 4(c) depict the numerical simulation of the intensity
profiles of the dipole and quadrupole structure respectively. As is evident
from the figures, both the dipole and the quadrupole structures exhibit stable
evolutions for the chosen parameters. From Figs.3(b) and 4(b), we can see that
the speed $|c|$ of both the elliptic waves having dipole and quadrupole
structure within a period decreases as $a_{6}$ increases. Fig.3(b) shows that
both the amplitude and the pulse width of the elliptic wave having dipole
structure within a period increase for increasing $a_{6}$ but for elliptic
wave having quadrupole structure within a period, while the amplitude
increases, the pulse width decreases as $a_{6}$ increases as shown in Fig.
4(b) . It is worth noting that the solution (8) is the $m=1$ limit of the
elliptic wave solution (12).

(a)  (b)

(c)  (d)

(e)  (f)

Figure 3: (a) plot of $|\psi|^{2}$ versus $t$ at $x=0$ (c), (d) numerical
simulation of the intensity profile without and with $10\%$ white noise
respectively of the solution (12) (dipole) for
$a_{3}=-0.165022,~{}a_{4}=0.1671553,~{}a_{5}=0.250733,~{}a_{6}=4.31016,~{}a_{7}=-0.006895,~{}\beta=0.0103625,~{}\omega=5.982461,~{}\phi_{0}=0.5,~{}c=0.476998,~{}k=1.301437,~{}A=0.2141115,~{}m=0.5$
respectively, (b) plots of $|c|$ , $\frac{1}{\beta}$, $A$ versus $a_{6}$ for
the solution (12) for $a_{3}=0.4,~{}a_{4}=0.5,~{}a_{7}=-0.04,~{}m=0.4$, (e)
plot of numerical simulation of the intensity profile by adding $0.003$ to the
parameters $a_{i},i=1,2,\cdots 7$ and (f) plot of numerical simulation of the
intensity profile by adding the overall factor $0.001$ to the parametric
conditions (9) & (13) respectively.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 4: (a) plot of $|\psi|^{2}$ versus $t$ at $x=0$ (c), (d) numerical
simulation of the intensity profile without and with $10\%$ white noise
respectively of the solution (12) (quadrupole) for
$a_{3}=\frac{1}{60},~{}a_{4}=0.126,~{}a_{5}=-0.189,~{}a_{6}=0.95,~{}a_{7}=0.15,~{}\beta=1.291913,~{}\omega=0.027778,~{}\phi_{0}=0.5,~{}c=-0.004247,~{}k=2.919107,~{}A=7.0688,~{}m=0.6$
respectively, (b) plots of $|c|$, $\frac{1}{\beta}$ , $A$ versus $a_{6}$ of
the solution (12) for $a_{3}=0.4,~{}a_{4}=0.126,~{}a_{7}=0.4,~{}m=0.75$, (e)
plot of numerical simulation of the intensity profile by adding $0.003$ to the
parameters $a_{i},i=1,2,\cdots 7$ and (f) plot of numerical simulation of the
intensity profiles by adding the overall factor $0.001$ to the parametric
conditions (9) & (13) respectively.

Solution III. Periodic wave exhibiting dipole structure within a period

Eqns.(6) and (7) admits yet another elliptic solitary wave

$u(\xi)=Amsn(\xi,m)cn(\xi,m)\,,$ (14)

($m,0<m<1$ being the modulus of the Jacobi elliptic functions $sn,cn$, [61])
provided Eqs.(6a) - (6d) are satisfied and further the following relations are
also satisfied

$\begin{array}[]{lcl}\frac{1}{2}a_{1}+c^{2}a_{6}=-2[3\omega^{2}-5(2-m)\beta^{2}]a_{7}\\\
k+k^{2}a_{6}+\frac{1}{2}\omega^{2}a_{1}=[(11m^{2}+64m-64)\beta^{4}-3\omega^{4}]a_{7}\end{array}$
(15)

It is worth noting that when m=1, this periodic solution becomes the solution
(8). The intensity profile and numerical simulation of the intensity profile
of the elliptic wave (14) are shown in Figs.5(a) and (c) respectively. Stable
evolution for the chosen values of the parameters are evident from the Fig.
5(c). From Figs.5(b), it is clear that the speed $|c|$ and the amplitude $A$
of this elliptic wave decrease as $a_{6}$ increases while the pulse-width
$\frac{1}{\beta}$ increases as $a_{6}$ increases.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 5: (a) plot of $|\psi|^{2}$ versus $t$ at $x=0$ (c), (d) numerical
simulation of the intensity profile without and with $10\%$ white noise
respectively of the solution (14) for
$a_{3}=\frac{1}{60},~{}a_{4}=0.126,~{}a_{5}=-0.189,~{}a_{6}=0.1,~{}a_{7}=0.15,~{}\beta=0.506763,~{}\omega=0.027778,~{}\phi_{0}=0.5,~{}c=0.0285981,~{}k=-9.861067,~{}A=1.0876,~{}m=0.7$
respectively, (b) plots of $|c|$, $\frac{1}{\beta}$, $A$ versus $a_{6}$ of the
solution (14) for $a_{3}=0.8,~{}a_{4}=-0.5,~{}a_{7}=0.4,~{}m=0.7$, (e) plot of
numerical simulation of the intensity profile by adding $0.003$ to the
parameters $a_{i},i=1,2,\cdots 7$ and (f) plot of numerical simulation of the
intensity profile by adding the overall factor $0.001$ to the parametric
conditions (9) & (15) respectively.

Solution IV. Periodic wave with arbitrary amplitude having dipole structure
within a period
Remarkably, Eqns.(6) and (7) also admit the following trigonometric solitary
wave solution with arbitrary amplitude $A$

$u(\xi)=A{\rm sin}(\xi)\,,$ (16)

provided following relations are also satisfied

$\begin{array}[]{lcl}3a_{4}+2a_{5}=0\\\ 4(a_{3}-4\omega
a_{7})\beta^{2}+(1+2ka_{6})c+(\omega
a_{1}+3\omega^{2}a_{3}-4\omega^{3}a_{7})=0\\\ \omega a_{4}+a_{2}=0\\\
a_{7}(16\beta^{4}+\omega^{4}+24\beta^{2}\omega^{2})-a_{3}\omega(12\beta^{2}+\omega^{2})-2\beta^{2}(a_{1}+2a_{6}c^{2})-(k+k^{2}a_{6}+\frac{1}{2}\omega^{2}a_{1})=0\\\
\end{array}$ (17)

The intensity profile and numerical simulation of the intensity profile of the
solitary wave (16) exhibiting stable evolution for the chosen parameters are
shown in Figs.6(a) and (c) respectively. Fig. 6(b) shows that the pulse-width
$\frac{1}{\beta}$ increases as $a_{6}$ increases but $|c|$ decreases with
increasing $a_{6}$.

(a)  (b)

(c)  (d)

(e)  (f)

Figure 6: (a) plot of $|\psi|^{2}$ versus $t$ at $x=0$ (c), (d) numerical
simulation of the intensity profile without and with $10\%$ white noise
respectively of the solution (16) for
$a_{3}=0.1,~{}a_{4}=-0.4,~{}a_{5}=0.6,~{}a_{6}=0.5,~{}a_{7}=0.04,~{}\beta=1.25169,~{}\omega=2.5,~{}\phi_{0}=0.5,~{}c=0.005,~{}k=0.0125112,~{}A=0.1$
respectively, (b) plots of $|c|$, $\frac{1}{\beta}$ versus $a_{6}$
respectively of the solution (16) for
$a_{3}=0.1,~{}a_{4}=0.5,~{}a_{6}=0.04,~{}a_{7}=0.8,~{}k=0.1$, (e) plot of
numerical simulation of the intensity profile by adding $0.003$ to the
parameters $a_{i},i=1,2,\cdots 7$ and (f) plot of numerical simulation of the
intensity profile by adding the overall factor $0.001$ to the parametric
conditions (9) & (17) respectively.

##  4 Stability analysis

The physical importance of the solitary wave solutions rests on their
stability. In this regard, it is worth noting that Figures 2(c), 3(c), 4(c),
5(c), and 6(c) all refer to an ideal setting, which is not always the case in
real-world situations. There are a variety of factors that might induce
instability in propagating structures [27]. As a result, it becomes vital to
investigate the stability of solutions in the presence of external noise or
perturbations. For this, $10\%$ white noise perturbation is added to the
initial input, as $\psi(x=0,t)=u(\beta t)e^{i\phi(x=0,t)}(1+\epsilon)$, where
$u(\beta t)$ are exact solutions which are given by (8), (12), (14), (16) and
the white noise $\epsilon$ is realized by utilizing a random matrix given by
[57],

$\epsilon={\rm 10\%noise}=0.1\sqrt{2}(rand(numel(t),1)-0.5)(1+i)$ (18)

In MATLAB program, $numel(t)$ represents the number $N$ of discretized values
of $t$ (refer Appendix) and $rand(numel(t),1)$ returns a $N\times 1$ matrix of
uniformly distributed random numbers on the open interval $(0,1)$. The term
$(1+i)$ in (18) signifies that $\psi(x=0,t)$ is perturbed by complex broadband
perturbation. The results of numerical simulation of the intensity profiles of
the dipole soliton (8), elliptic solitary wave (12) having the dipole
structure within a period for $m=0.5$ and having the quadrupole structure
within a period for $m=0.6$ respectively, elliptic solitary wave (14) having
the dipole structure within a period and trigonometric solitary wave (16)
exhibiting the dipole structure within a period, by adding $10\%$ white noise
are shown in Figs.2(d), 3(d), 4(d), 5(d), 6(d) respectively. The results
reveal that the elliptic solitary wave (12) having the dipole structure within
a period for $m=0.5$ of the Jacobi elliptic function, has very weak
instability in its propagation under $10\%$ ((Fig.3(d)) white noise but the
quadrupole structure within a period for $m=0.6$ (Fig.4(d)) shows stable
evolution for the values of the parameters considered herein. But the other
dipole solitary waves (14) and (16) propagate appreciably in a stable manner
for the chosen parameters under finite initial white noise perturbations. It
is also important to test the robustness of the solutions by violation of the
parametric conditions. This is especially essential for experimental
implementations where the values of the parameters are always approximate and
may also fluctuate [47]. Here, the violations of the parametric conditions are
done in two ways: (i) by adding $0.003$ to all the parameters
$a_{i},i=1,2\cdots 7$ (ii) by adding an overall factor $0.001$ to the right
side of the parametric conditions 9(a) - (d), (10), (13), (15), (17). The
resulting numerical simulations are shown in Fig.2(e)-(f), Fig.3(e)-(f) $\&$
Fig.4(e)-(f), Fig.5(e)-(f), Figs.6(e)-(f) for the solitary waves (8), (12) for
$m=0.5~{}\&~{}m=0.6$, (14), (16) respectively. It is evident from the above
mentioned figures that the solitary waves are quite robust in both the cases
of violations of the parametric conditions. It is to be noted that the
nonlinear modes are stable under $10\%$ white noise perturbations and also
under violations of the parametric conditions for both positive (Figs. 3, 4, 6
) and negative (Figs. 1, 2, 5) $k$ values (similar to [23]) when the other
parameters are kept fixed at the chosen values. The positive and negative $k$
values signify that the direction of the wave vector $\vec{k}$ (in one
dimension it is a scalar) for forward propagating wave is opposite to that of
the backward propagating wave.

##  5 Conclusion

In this article, we have studied the cubic NLH system in the presence of TOD,
FOD and also non-Kerr nonlinearity like the SS and the SFS. This system models
ultrashort nonparaxial pulse propagation with Kerr and non-Kerr
nonlinearities, higher order dispersions, and spatial dispersion which becomes
important when SVEA fails. We have demonstrated that this system features
dipole soliton and also elliptic waves having the dipole/quadrupole structure
within a period. The novelty of the propagation dynamics arising from
incorporating the higher order dispersion and non-Kerr nonlinearity lies in
the appearance of dipole structure within a period of these elliptic waves.
Most interestingly, for one of the elliptic waves, there is a transition from
the dipole to the quadrupole structure when the value of the modulus parameter
$m$ of the Jacobi elliptic function $m>0.5$. Also an arbitrary amplitude
sinusoidal periodic wave is obtained which exhibits dipole structure within a
period. So far as we are aware of, this interesting characteristic feature of
the propagation dynamics of nonparaxial periodic waves has not been unveiled
till now. These solitary wave solutions exist when the model parameters
satisfy certain parametric conditions. These conditions exhibit a subtle
balance between dispersion, Kerr, and non-Kerr nonlinearities, which is
essential to control the dynamics of these solitary waves. To determine the
physical significance of the obtained solutions, the effect of the nonparaxial
parameter on the amplitude, pulse-width and speed of solitary waves are
investigated. It has been observed that the speed of the solitary waves can be
reduced by tuning the nonparaxial parameter. To test the stability of the
solitary waves, numerical simulations have been performed by adding $10\%$
white noise to the initial stationary mode. It has been found that the
elliptic wave having the dipole structure (depending on the value of the
modulus parameter of Jacobi elliptic function) within a period has very weak
instability in its propagation pattern under these noises for the chosen
values of the parameters while the corresponding quadrupole structure
propagate stably. All the other dipole solitary waves has stable propagation
under these white noise perturbations. Furthermore, robustness of the
solutions are examined by violation of the parametric conditions in two
different ways namely, by adding $.003$ to all the parameters
$a_{i},i=1,2\cdots 7$ and also by adding an overall factor $.001$ to the
parametric conditions. The solitary waves are found to be quite robust. The
applications of dipole and quadrupole solitary wave solutions presented here
may be found in nonparaxial optical contexts involving sub picosecond or
femtosecond pulses, such as in ultrahigh-bit-rate optical communication
systems [28]. The results presented in this article constitute a step forward
to the exploration of dipole, multipole, kink, anti-kink etc. solitary waves
for inhomogeneous higher order NLH system as is done in [59] for HNLSE . Also,
it would be interesting to construct these types of solitary wave soluions in
the presence of quintic, septic Kerr and non-Kerr nonlinearity as obtained in
[58] and [48, 47] respectively for HNLSE . This is important in order to keep
up with current developments in high-repetition-rate (beyond ultrashort)
fiber-based pulse sources [60]. Some of these issues are currently being
investigated, and we hope to report these in the near future.
Data Availability
Data sharing is not applicable to this article as no new data were created or
analyzed in this study.
Conflict of Interest
The authors have no conflicts to disclose.

Appendix :
For checking the dynamical evolution of the derived solutions, the finite
difference method [62] is used with $\delta t=0.1$, $\delta x=10^{-4}$. and
the boundary condition is given by:

$\psi(x,t)=0,~{}\text{at the boundary.}$

A one-dimensional uniform spatial grid is taken consisting of $N$ points
labeled by $t_{j}=-L+(j-1)\delta t$, $j=1,2,..N$, where $\delta
t=\frac{2L}{N-1}$ is the lattice spacing and $L$ is the half-width. The left
and right boundary points are located at $j=1$ and $j=N$ respectively. So, the
boundary conditions at the edges of the spatial grid are

$\psi(x,\pm L)=0.$

The derivatives are discretized as [62]

$\displaystyle\psi_{x}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i+1,j}-\psi_{i-1,j}}{2\delta
x},$
$\displaystyle\psi_{xx}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i+1,j}-2\psi_{i,j}+\psi_{i-1,j}}{\delta
x^{2}},$
$\displaystyle\psi_{t}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i,j+1}-\psi_{i,j-1}}{2\delta
t},$
$\displaystyle\psi_{tt}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i,j+1}-2\psi_{i,j}+\psi_{i,j-1}}{\delta
t^{2}},$
$\displaystyle\psi_{ttt}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i,j+2}-2\psi_{i,j+1}+2\psi_{i,j-1}-\psi_{i,j-2}}{2\delta
t^{3}},$
$\displaystyle\psi_{tttt}(x,t)|_{(x_{i},t_{j})}=\frac{\psi_{i,j+2}-4\psi_{i,j+1}+6\psi_{i,j}-4\psi_{i,j-1}+\psi_{i,j-2}}{\delta
t^{4}},$

where $\psi(x_{i},t_{j})=\psi_{i,j}.$

## References

  * [1] R.Y. Chiao, E. Garmire and C.H. Townes, Phys.Rev.Lett. 13, 479 (1964)
  * [2] A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Oxford University Press, New York, 1995
  * [3] A. Hasegawa and F. Tappert Appl.Phys.Lett. 23, 142 (1973); Appl.Phys.Lett. 23, 171 (1973)
  * [4] P. Chamorro-Posada, G.S. McDonald and G.H.C. New, J.Opt.Soc.Am.B 19, 1216 (2002)
  * [5] M. Lax, W.H. Louisell and W.B. McKnight Phys.Rev.A. 11, 1365 (1975)
  * [6] P. Chamarro-Posada, G.S. McDonald, G.H.C. New, J.Mod.Opt. 45, 1111 (1998)
  * [7] P. Chamorro-Posada and G.S. McDonald Phys.Rev.E. 74, 036609 (2006)
  * [8] J.M. Christian, G.S. McDonald,P. Chamorro-Posada Phys.Rev.E 74, 066612 (2006)
  * [9] V.E.Zakharov and A.B.Shabat, Sov. Phys. JETP 34, 62 (1972); ibid 37, 823 (1973)
  * [10] S.V.Manakov, Sov. Phys. JETP 38, 248 (1974)
  * [11] J.M. Christian, G.S. McDonald, R.J. Potton, P. Chamorro-Posada, Phy.Rev.A. 76, 033834 (2007); J.M. Christian, G.S. McDonald, P. Chamorro-Posada, Phys.Rev.A. 81, 053831 (2010); J.M. Christian, E.A. McCoy, G.S. McDonald, P. Chamorro-Posada, J.Atom.Mol.Opt.Phys. 2012, 137967 (2012)
  * [12] P. Chamorro-Posada and G.S. McDonald, Opt.Lett. 28, 825 (2003)
  * [13] J.M. Christian, G.S. Mcdonald, P. Chamorro-Posada, J.Phys.A: Math.Theor. 40, 1545 (2007)
  * [14] J.M. Christian, G.S. Mcdonald, P. Chamorro-Posada, Jour.Opt.Soc.Am.B 26, 2323 (2009)
  * [15] J.M. Christian, G.S. McDonald, T.F. Hodgkinson and P. Chamorro-Posada, Phys.Rev.A 86, 023838 (2012); J.M. Christian, G.S. McDonald, T.F. Hodgkinson and P. Chamorro-Posada, Phys.Rev.A 86, 023839 (2012)
  * [16] J.M. Christian, G.S. McDonald, T.F. Hocgkinson, and P. Chamorro-Posada, Phys.Rev.Lett 108, 034101 (2012)
  * [17] J.M. Christian, G.S. McDonald and A. Kotsampaseris, Phys.Rev.A. 98, 053842 (2018); J.M. Christian, G.S. McDonald, M.J. Lundie and A. Kotsampaseris, Phys.Rev.A 98, 053843 (2018)
  * [18] S. Blair Chaos 10, 570 (2000)
  * [19] H. Wang, W. She Opt.Exp. 14, 1590 (2006)
  * [20] K. Malakuti, E. Parilov, Appl.Num.Math. 61, 891 (2011)
  * [21] D.D. Estelle Temgoua and T.C. Kofane, Phys.Rev.E. 91, 063201 (2015)
  * [22] C. Huang, F. Ye, Y.V. Kartashov, B.A. Malomed Opt.Lett. 39, 5443 (2014)
  * [23] K.Tamilselvan, T. Kanna, A. Khare, Commun.Nonl.Sci.Num.Sim. 39, 134 (2016)
  * [24] M. Kumar, K. Nityanandan, H. Triki, K. Porsezian, Optik 182, 1120 (2019)
  * [25] K. Tamilselvan, T. Kanna, A. Govindrajan, Chaos 29, 063121 (2019)
  * [26] N. Saha, B. Roy, A. Khare, Chaos 31, 113104 (2021); Europhys. Letts. DOI: 10.1209/0295-5075/ac58ba
  * [27] H.Triki, E.Porsezian, K.Senthilnathan, K.Nityanandan, Phys.Rev.E 100, 042208 (2019)
  * [28] H. Triki and V. Kruglov, Phys.Rev.E 101, 042220 (2020)
  * [29] S.L. Palacios and J.M. Fernandez-Diaz, Opt.Commun. 178, 457 (2000)
  * [30] S.L. Palacios, J.Opt.A:Pure Appl.Opt. 5, 180 (2003)
  * [31] Y. Kodama, J.Stat.Phys. 39, 597 (1985); Y.Kodama and A.Hasegawa, IEEE J.Quantum. Electron. 23, 510 (1987)
  * [32] Y.J. He and H.Z. Wang, Opt.Exp. 14, 9833 (2006) and references cited therein
  * [33] L. Ge et al, Opt.Coommun. 284, 2351 (2011)
  * [34] W. Hong, Y. Jung, Phys.Lett.A 377, 1701 (2013)
  * [35] C. Huang, Opt.Commun. 284, 5786 (2011)
  * [36] M. Shen et al, Opt.Commun. 282, 4805 (2011)
  * [37] X. Min et al, J.Mod.Opt. 63, S44 (2016)
  * [38] Y.V.Kartashov, A.A.Egorov, L.Torner, D.N.Christodoulides, Opt.Lett. 29, 1918 (2004)
  * [39] L. Dong and F. Ye, Phys.Rev.A 81, 013815 (2010)
  * [40] H. Susanto et al, Physica D 237, 3123 (2008)
  * [41] Z. Xu. Phys.Rev.A 80, 053827 (2009)
  * [42] C. Rotschild et al, Opt Letts. 31, 3312 (2009)
  * [43] Z.H. Li, L. Li, H.P. Tian, G.S. Zhou Phys.Rev.Lett. 84, 4096 (2000)
  * [44] W.P. Hong, Opt.Commun. 194, 217 (2001)
  * [45] F. Azzouzi, H. Triki, K. Mezghiche, A.E. Akrmi, Chaos Solitons Fractals 39, 1304 (2009)
  * [46] A. Choudhuri, K. Porsezian, Opt.Commun. 285, 364 (2012)
  * [47] H. Triki, F. Azzouzi, P. Grelu, Opt.Commun. 309, 71 (2013)
  * [48] F. Azzouzi, H. Triki, P. Grelu, Appl.Math.Modell. 39, 1300 (2015)
  * [49] J.P. Tian, H.P. Tian, Z.H. Li Phys.Scr. 7, 325 (2003)
  * [50] D. Anderson and M. Lisak, Opt.Lett. 7, 394 (1982)
  * [51] J.R. de Oliveira and M.A. de Moura, Phys.Rev.E 57, 4751 (1998)
  * [52] Y.S. Kivshar and G.P. Agrawal, Optical Solitons: From fibres to photonic crystals, (Academic Press, San Diego, USA, 2003)
  * [53] F.M. Mitschke and L.F. Mollenauer, Opt.Lett. 11, 659 (1986); J.P.Gordon, Opt.Lett. 11, 662 (1986)
  * [54] W. Liu, B. Tian, H. Zhang, L. Li, Y. Xue, Phys. Rev. E 77, 066605 (2008)
  * [55] Y. Jiang, B. Tian, M. Li, Pan Wang, Nonlinear Dyn., 74, 1053 (2013)
  * [56] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover Publications, INC., New York 1970
  * [57] Y. Chen and Z. Yan Phys.Rev.E 95, 012205 (2017)
  * [58] A.K. Sarma, Commun.Nonl.Sci.Num.Simul. 14, 3215 (2009)
  * [59] R.C. Yang, L. Li, R.Y. Hao, Z.H. Li, G.S. Zhou, Phys.Rev.E. 71, 036616 (2005)
  * [60] S. Hadrich et al, Opt.Exp. 18, 0242 (2010); Y. Song et al, Opt.Exp, 19, 14518 (2011)
  * [61] M.Abramowitz and I.A.Stegun, Handbook of Mathematical Functions, Dover Publications, INC, New York 1970
  * [62] L.N.Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Cornell University, Ithaca, N.Y. 1994