# Management of solitons in medium with competing cubic and quadratic
nonlinearities

F. Kh. Abdullaev1,2 J. S. Yuldashev1,2 M. Ögren3,4 1 Physical-Technical
Institute, Uzbek Academy of Sciences, 100084 Tashkent, Uzbekistan  2
Theoretical Physics Department, National University of Uzbekistan, 100174
Tashkent, Uzbekistan  3School of Science and Technology, Örebro University,
70182 Örebro, Sweden  4Hellenic Mediterranean University, P.O. Box 1939,
GR-71004, Heraklion, Greece

###### Abstract

Management of solitons in media with competing quadratic and cubic
nonlinearities is investigated. Two schemes, using rapid modulations of a
mismatch parameter, and of the Kerr nonlinearity parameter are studied. For
both cases, the averaged in time wave equations are derived. In the case of
mismatch management, the region of the parameters where stabilization is
possible is found. In the case of Kerr nonlinearity management, it is shown
that the effective $\chi^{(2)}$ nonlinearity depends on the intensity
imbalance between fundamental (FH) and second (SH) harmonics. Predictions
obtained from the averaged equations are confirmed by numerical simulations of
the full PDE’s.

###### keywords:

Solitons, mismatch management, Kerr nonlinearity management, competing
nonlinearities.

††journal: Optik

##  1 Introduction

Solitons in media with quadratic ($\chi^{(2)}$) and cubic ($\chi^{(3)}$)
nonlinearaties have been studied intensively experimentally as well as
theoretically. Special interest is attracted to optical dispersive media with
both $\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities present, i.e., to the case
of so-called competing nonlinearities. Such systems also appears from the
mean-field description of matter waves in atomic-molecular condensates
(AMBEC). In the latter case the role of cubic nonlinearities is played by the
two-body interactions for atomic and molecular fields, respectively. The role
of the quadratic nonlinearities is played by conversion of pairs of atoms into
molecules and vice versa. The analysis of the existence of optical solitons
and their stability in such systems has been performed in [1], and for the
case of matter waves in [2]. In particular it have been shown in [3] that
stable bright solitons can exists in a media with competing nonlinearities,
and that the strength of the $\chi^{(2)}$ non-linearity plays an important
role for the instability of the solitons.

Hence, it is of importance to develope methods for stabilization and control
of solitons in these systems. The method of dynamical stabilization of
solitons by using rapid variations in time of parameters in the media have
been suggested in [4]. Reviews of these results are contained in the books [5,
6], and a more recent review is [7].

One of the possible methods of dynamical stabilization can be mismatch
management. Mismatch management in a pure $\chi^{(2)}$ system has been
considered in [8, 9] where regions of stability and instability has been
analyzed for the slowly varying and resonant cases mainly, and in [10, 11] for
quasi-phase-matching (QPM) regime. Also the influence of the management of the
cubic- and quadratic-nonlinearities on soliton stability and continuous wave
(CW) dynamics have been studied in [9, 12], where the resonant responses of
solitons and CW have been analyzed.

Here we will study the possible roles of the rapidly varying periodic mismatch
and the Kerr non-linearity for the dynamical stabilization and the control of
optical and matter-wave solitons in media with competing cubic- and quadratic-
nonlinearities.

The structure of the article is the following:

In section 2 we describe the model used for the propagation of the waves in
media with competing cubic and quadratic non-linearities. The system obtained
by averaging over rapid modulations of the mismatch parameter, and a condition
for dynamical stabilization of solitons, are obtained in section 3. The
averaged equations and solitons for a system with a periodically modulated
Kerr non-linearity parameter, and the corresponding full numerical
simulations, are considered in section 4. Finally, we present the conclusions
of the investigations in section 5.

##  2 The model

The system, describing the propagation of the fundamental- (FH) and second-
(SH) harmonics in a quadratic nonlinear media with a cubic nonlinearity have
in standard optics dimensionless variables the form [3]

$\displaystyle i\frac{\partial u}{\partial z}+r\frac{\partial^{2}u}{\partial
x^{2}}-u+u^{*}w+\chi\left(\frac{1}{2\sigma}\lvert u\rvert^{2}+\rho\lvert
w\rvert^{2}\right)u=0,$ $\displaystyle i\sigma\frac{\partial w}{\partial
z}+s\frac{\partial^{2}w}{\partial
x^{2}}-qw+\frac{1}{2}u^{2}+\chi\left(2\sigma\lvert w\rvert^{2}+\rho\lvert
u\rvert^{2}\right)w=0.$ (1)

Here $u,w$ are the fields of the FH and the SH respectively. The parameter
$\chi\sim\chi^{(3)}/\chi^{(2)^{2}}$ characterize a balance of contributions
that are due to the $\chi^{(2)}$ and $\chi^{(3)}$ nonlinearities. We consider
spatial bright solitons ($r=s=+1$) and select the same parameter values
($\rho=\sigma=2$) as in [3]. Below we consider two cases separately, a
longitudinally modulated linear mismatch parameter, $q(z)$, and a modulated
cubic nonlinearity, $\chi(z)$. We decompose the modulation of the mismatch
parameter $q(z)$ (cubic nonlinearity $\chi(z)$), into a mean-value part
$q_{0}$ ($\chi_{0}$), and a fastly varying part $q_{1}(z)$ ($\chi_{1}(z)$)
with a large amplitude, as follows:

$q(z)=q_{0}+q_{1}(z)=q_{0}+q_{1}\cos(\omega
z),\,\,\,\Big{(}\chi(z)=\chi_{0}+\chi_{1}(z)=\chi_{0}+\chi_{1}\cos(\omega
z)\Big{)}.$ (2)

##  3 Longitudinally modulated mismatch parameter

First we obtain the averaged equations in the case of $q\rightarrow q(z)$ and
$\chi=\chi_{0}$. We consider the two cases of a small parameter mismatch and a
large parameter mismatch. In the former case (small $q_{0}$) we use the
following transformation for the second-harmonic field for deriving so-called
averaged equations

$w=ve^{-i\frac{\varGamma(z)}{\sigma}},$ (3)

where $\varGamma_{z}(z)=q_{1}(z)$, i.e., from Eq. (2) we have
$\varGamma(z)=\frac{q_{1}}{\omega}\sin(\omega z)$. This transformation permits
us to eliminate the strong rapid varying terms [4]. By using mismatch
parameters in the form of (2) and substituting the new second-harmonic field
$v$ into Eqs. (2) we get similar equations to Eqs. (2), but with the quadratic
nonlinear terms with exponential factors

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u+u^{*}ve^{-\frac{i}{\sigma}\varGamma(z)}+\chi_{0}\left(\frac{1}{4}\lvert
u\rvert^{2}+2\lvert v\rvert^{2}\right)u$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial
x^{2}}-\frac{q_{0}}{2}v+\frac{1}{4}u^{2}e^{\frac{i}{\sigma}\varGamma(z)}+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v$ $\displaystyle=$ $\displaystyle 0.$
(4)

We then expand the exponential factors in Eqs. (3) into Fourier series and
restrict the expansion coefficients to the average values (zero-order term) in
obtaining an averaged equation, i.e.

$e^{\pm
i\frac{q_{1}}{\sigma\omega}\sin(z^{\prime})}=\sum_{n=-\infty}^{\infty}c_{n}^{\pm}e^{inz^{\prime}}=\sum_{n=-\infty}^{\infty}J_{n}\left(\pm\frac{q_{1}}{\sigma\omega}\right)e^{inz^{\prime}},$
(5)

where $z^{\prime}=\omega z$. The averaged equations can now be written in the
final form:

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u+u^{*}vJ_{0}\left(\frac{q_{1}}{\sigma\omega}\right)+\chi_{0}\left(\frac{1}{4}\lvert
u\rvert^{2}+2\lvert v\rvert^{2}\right)u$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial
x^{2}}-\frac{q_{0}}{2}v+\frac{1}{4}u^{2}J_{0}\left(\frac{q_{1}}{\sigma\omega}\right)+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v$ $\displaystyle=$ $\displaystyle 0,$
(6)

where $J_{0}(\cdot)$ is the zero-order Bessel function.

In the latter case (large $q_{0}$) we use instead a similar transformation,
including the large mismatch parameter in the exponential, as was done for the
quasi-phase-matching (QPM) scheme in [13]

$w=ve^{-i\frac{\varGamma(z)}{\sigma}-i\frac{q_{0}}{\sigma}z}.$ (7)

By using the transformation (7), we get the following equations

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u+u^{*}ve^{-\frac{i}{\sigma}\varGamma(z)}e^{-i\frac{q_{0}}{\sigma}z}+\chi_{0}\left(\frac{1}{4}\lvert
u\rvert^{2}+2\lvert v\rvert^{2}\right)u$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial
x^{2}}+\frac{1}{4}u^{2}e^{\frac{i}{\sigma}\varGamma(z)}e^{i\frac{q_{0}}{\sigma}z}+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v$ $\displaystyle=$ $\displaystyle 0.$
(8)

We now do the same mathematical procedure as in deriving Eq. (3), but we also
take into account first order terms ($c_{1}$ and $c_{-1}$) in the Fourier
expansion of Eq. (5), that is

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u+u^{*}v\left[J_{0}\left(\frac{q_{1}}{\sigma\omega}\right)-J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{i\omega
z}+J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{-i\omega
z}\right]e^{-i\frac{q_{0}}{\sigma}z}$
$\displaystyle+\chi_{0}\left(\frac{1}{4}\lvert u\rvert^{2}+2\lvert
v\rvert^{2}\right)u=0,$ $\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial
x^{2}}+\frac{1}{4}u^{2}\left[J_{0}\left(\frac{q_{1}}{\sigma\omega}\right)+J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{i\omega
z}-J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{-i\omega
z}\right]e^{i\frac{q_{0}}{\sigma}z}$ $\displaystyle+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v=0,$ (9)

where $J_{1}(\cdot)=-J_{-1}(\cdot)$ is the first-order Bessel function. To
obtain the averaged equation we omit the fast oscillating terms (with large
exponentials) in Eq. (3)

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u-u^{*}vJ_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{-i(\frac{q_{0}}{\sigma}-\omega)z}+\chi_{0}\left(\frac{1}{4}\lvert
u\rvert^{2}+2\lvert v\rvert^{2}\right)u$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial
x^{2}}-\frac{1}{4}u^{2}J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)e^{i(\frac{q_{0}}{\sigma}-\omega)z}+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v$ $\displaystyle=$ $\displaystyle 0.$
(10)

By applying the following transformation for the second-harmonic:
$v\rightarrow\bar{v}e^{i\delta qz}$, we can again write averaged equations
similar to Eq. (2), but with re-normalized terms

$\displaystyle i\frac{\partial u}{\partial z}+\frac{\partial^{2}u}{\partial
x^{2}}-u-u^{*}vJ_{1}\left(\frac{q_{1}}{\sigma\omega}\right)+\chi_{0}\left(\frac{1}{4}\lvert
u\rvert^{2}+2\lvert v\rvert^{2}\right)u$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle i\frac{\partial v}{\partial
z}+\frac{1}{2}\frac{\partial^{2}v}{\partial x^{2}}-\delta
qv-\frac{1}{4}u^{2}J_{1}\left(\frac{q_{1}}{\sigma\omega}\right)+\chi_{0}\left(2\lvert
v\rvert^{2}+\lvert u\rvert^{2}\right)v$ $\displaystyle=$ $\displaystyle 0,$
(11)

where $\delta q=q_{0}/\sigma-\omega$, and we have dropped the bar-sign for
convenience.

Stability of soliton solutions of Eqs. (2) in the case of small mismatch
parameters ($q_{0}=2$ and $q_{1}=0$) was investigated numerically in [3], and
the three most physically important soliton families of low energy were
presented. Here the energy is defined by [3]

$P=|\chi_{0}|\int\left(|u|^{2}+4|w|^{2}\right)dx.$ (12)

According to that investigation, solitons with energy higher than $8\sigma$
($W$ - type solitons) are unstable for any value of the parameter $\chi_{0}$,
while solitons $v(x)=\sqrt{q_{0}/(\sigma\chi_{0})}\mbox{sech}(\sqrt{q_{0}}x)$
with energy $4\sqrt{2}q_{0}/\sigma$ ($V$ \- type solitons) are stable for
$\chi_{0}\geq 8.76$, but unstable for $0<\chi_{0}\lessapprox 8.76$. It is
apparent from the averaged Eqs. (3) and (3) that the contribution of the
quadratic nonlinearities can be controlled by changing the parameters $q_{1}$
or $\omega$. Therefore, stabilization of $W$ and $V$ type solitons may be
possible as we can reduce the effect of the quadratic nonlinearity by changing
modulation parameters in the unstable range of $\chi_{0}$.

Figure 1: (Color online) $W$ \- type solitons, stationary solutions of Eqs.
(2). a) Soliton profile with total energy $P=16.8182$ and Kerr parameter
$\chi_{0}=8$; b) Soliton profile with total energy $P=17.0947$ and Kerr
parameter $\chi_{0}=5$. Solid (blue) curves corresponds to FH and dashed (red)
curves corresponds to SH. The other parameters, $q_{0}=2$ and $q_{1}=0$, are
the same for both cases.

###  3.1 Numerical analysis

We have carried out numerical calculations to check the stability of $W$ and
$V$ type solitons for different parameter values of $\chi_{0}$ and the
energies, for which these solitons are unstable. It has been noted that $W$ \-
type solitons are supported mainly by the cubic nonlinearity, but the second-
harmonic, which has two bumps in the profile as shown in Fig. 1, is also
generated as the result of parametric coupling. The numerical analysis of the
original equations (2), shows that modulation of the mismatch parameter along
the direction of propagation leads to stabilization of the unstable $W$ \-
type (Figs. 2 and 3), and of the $V$\- type solitons (Fig. 4). In Fig. 2 and
Fig. 3, the dynamics of the intensity profiles ($|u(x,z)|^{2}$ and
$|w(x,z)|^{2}$) and the squared central amplitudes ($A_{u}^{2}=|u(0,z)|^{2}$
and $A_{w}^{2}=|w(0,z)|^{2}$) of the stationary states [Fig. 1, a) and b)] of
Eqs. (2) in the case of $\chi_{0}=8$ and $\chi_{0}=5$, with total energy
$P=16.8182$ and $17.0947$, respectively, are shown.

For the $V$ \- type solitons with small mismatch parameter, numerical
simulations of the full equations (2), shows that stabilization of stationary
states in the unstable region, $0<\chi_{0}\lessapprox 8.76$ and $P=4\sqrt{2}$,
can be achieved by modulating the mismatch parameter with $q_{1}=32$, and
$q_{1}=25$, in the case of $\chi_{0}=2$, and $\chi_{0}=5$, respectively, see
Fig. 4.

Figure 2: (Color online) Evolution of intensities ($|u|^{2}$, $|w|^{2}$) and
the squared central amplitudes ($A^{2}_{u}$, $A^{2}_{w}$) for the stationary
solution [Fig. 1 a)] in the unstable region for $\chi_{0}=8$. Frames a) and b)
demonstrate the unstable dynamics of the fundamental- and second-harmonics
waves and their amplitudes, respectively, when the mismatch parameter $q$ is
not modulated ($q_{1}=0$). Frames c) and d) correspond to stabilization of
these solitons when the mismatch parameter $q(z)$ is modulated ($q_{1}=96.2$,
$\omega=20$).

Figure 3: (Color online) Evolution of intensities ($|u|^{2}$, $|w|^{2}$) and
the squared central amplitudes ($A^{2}_{u}$, $A^{2}_{w}$) for the stationary
solution [Fig. 1 b)] in the unstable region for $\chi_{0}=5$. Frames a) and b)
demonstrate the unstable dynamics of the fundamental- and second-harmonics
waves and their amplitudes, respectively, when the mismatch parameter $q$ is
not modulated ($q_{1}=0$). Frames c) and d) correspond to stabilization of
these solitons when the mismatch parameter $q(z)$ is modulated ($q_{1}=96.2$,
$\omega=20$).

Figure 4: (Color online) Evolution of width and the squared central amplitude
for stationary solutions with and without modulation of the mismatch parameter
$q$. Frame a) demonstrate the unstable dynamics of the $V$ \- type soliton in
the case of $\chi_{0}=2$, solid (black) and dotted (blue) curves correspond to
soliton’s full width at half maximum (fwhm) and the squared central amplitude
($|w(0,z)|^{2}$), respectively, of the second-harmonic; and for $\chi_{0}=5$
dash-dotted (red) and dashed (green) curves correspond to the soliton’s full
width at half maximum (fwhm) and the squared central amplitude
($|w(0,z)|^{2}$), respectively, when the mismatch parameter is not modulated
($q_{1}=0$). Frame b) corresponds to stabilization of the solitons for the
same stationary solutions when the mismatch parameter $q(z)$ of Eq. (2) is
modulated with the frequency $\omega=10$ ($q_{1}=32$: black and blue curves,
and $q_{1}=25$: red and green curves). In both frames, $q_{0}=2$.

For the case of a large mismatch parameter we carried out numerical
calculations of the original equations (2), for the parameter $q_{0}=22$ and
the maximum propagation distance, $z$, that the simulations are performed for
is $z_{max}=500$. In this case, according to our numerical analysis, $V$ \-
type solitons with power $P_{v}=4\sqrt{22}$ are unstable in the region
$0<\chi_{0}\lessapprox 2.8$. However, according to Eqs. (3), this unstable
region can be shifted to the stable region by modulating the mismatch
parameter. Fig. 5 shows the stability of solitons for the cases of
$\chi_{0}=1$, and $\chi_{0}=2.5$, when the mismatch parameter are modulated
with amplitudes $q_{1}=46$, and $q_{1}=36$, respectively.

Figure 5: (Color online) Evolution of width and the squared central amplitude
for stationary solutions with and without modulation of the mismatch parameter
$q$ in the case of large mismatch parameter, $q_{0}=22$. Frame a) demonstrate
the unstable dynamics of the $V$ \- type soliton in the case of $\chi_{0}=1$,
dash-dotted (red) and dotted (blue) curves correspond to the soliton’s full
width at half maximum (fwhm) and the squared central amplitude
($|w(0,z)|^{2}$), respectively, of the second-harmonic; and for
$\chi_{0}=2.5$, dashed (black) and solid (green) curves correspond to
soliton’s full width at half maximum (fwhm) and the squared central amplitude
($|w(0,z)|^{2}$), respectively, when the mismatch parameter is not modulated
($q_{1}=0$). Frame b) corresponds to stabilization of the solitons for the
same stationary solutions when the mismatch parameter $q(z)$ of Eq. (2) is
modulated with the frequency $\omega=9$ ($q_{1}=46$: red and blue curves, and
$q_{1}=36$: black and green).

##  4 Modulation of the Kerr nonlinearity

In this section, we deal with the system of a constant mismatch parameter
($q=q_{0}$) and a longitudinally modulated cubic nonlinearity,
$\chi\rightarrow\chi(z)$. To derive the corresponding average over the rapid
modulation of the original system (2), we will use the following
transformations to new fields for the FH and SH

$u=\bar{u}e^{i\varGamma(z)(\frac{1}{2\sigma}\lvert\bar{u}\rvert^{2}+\rho\lvert\bar{v}\rvert^{2})},\>w=\bar{v}e^{i\frac{\varGamma(z)}{\sigma}(2\sigma\lvert\bar{v}\rvert^{2}+\rho\lvert\bar{u}\rvert^{2})},$
(13)

where $\varGamma(z)$ is the anti-derivative of $\chi_{1}(z)$, i.e.,
$\varGamma_{z}(z)=\chi_{1}(z)$. For convenience, we will drop the bar sign in
the subsequent calculations. Inserting the transformations (13) into the
original equations (2) allows us to exclude the strongly and rapidly varying
terms from the original system and derive an averaged system [14]. Then we can
replace the original equations (2) with the equivalent equations:

$\begin{split}iu_{z}&-\varGamma(z)\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{z}u+\frac{\partial^{2}u}{\partial
x^{2}}+i\varGamma(z)\left(\frac{1}{2\sigma}\lvert u\rvert^{2}+\rho\lvert
v\rvert^{2}\right)_{xx}u\\\ &+2i\varGamma(z)\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert
v\rvert^{2}\right)_{x}u_{x}-\varGamma^{2}(z)\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\right]^{2}u-u\\\
&+u^{*}ve^{i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert
u\rvert^{2}\right]}+\chi_{0}\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)u=0,\end{split}$ (14)

$\begin{split}iv_{z}&-\varGamma(z)\left(2\sigma\lvert v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{z}v+\frac{\partial^{2}v}{\partial
x^{2}}+i\frac{\varGamma(z)}{\sigma}\left(2\sigma\lvert v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{xx}v\\\ &+i\frac{2\varGamma(z)}{\sigma}\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{x}v_{x}-\frac{\varGamma^{2}(z)}{\sigma^{2}}\left[\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)_{x}\right]^{2}v-qv\\\
&+\frac{1}{2}u^{2}e^{-i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert u\rvert^{2}\right]}+\chi_{0}\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)v=0.\end{split}$ (15)

It follows from Eqs. (14) and (15) that

$\begin{split}&\left(\frac{1}{2\sigma}\lvert u\rvert^{2}+\rho\lvert
v\rvert^{2}\right)_{z}=\frac{-i}{2\sigma}uu_{xx}^{*}+\frac{i}{2\sigma}u^{*}u_{xx}-\frac{\varGamma(z)}{\sigma}\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\lvert u\rvert^{2}\right]_{x}\\\
&-\frac{i(\rho-1)}{2\sigma}u^{*2}ve^{i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert
u\rvert^{2}\right]}+\frac{i(\rho-1)}{2\sigma}u^{2}v^{*}e^{-i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert u\rvert^{2}\right]}\\\
&-\frac{i\rho}{\sigma}vv_{xx}^{*}+\frac{i\rho}{\sigma}v^{*}v_{xx}-\frac{2\rho}{\sigma}\frac{\varGamma(z)}{\sigma}\left[\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)_{x}\lvert
v\rvert^{2}\right]_{x},\end{split}$ (16)

and

$\begin{split}&\left(2\sigma\lvert v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{z}=-i\rho uu_{xx}^{*}+i\rho
u^{*}u_{xx}-2\rho\varGamma(z)\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\lvert u\rvert^{2}\right]_{x}\\\
&+i(\rho-1)u^{*2}ve^{i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert
u\rvert^{2}\right]}-i(\rho-1)u^{2}v^{*}e^{-i\frac{\varGamma(z)}{\sigma}\left[2\sigma(1-\rho)\lvert
v\rvert^{2}+(\rho-1)\lvert u\rvert^{2}\right]}\\\
&-i2vv_{xx}^{*}+i2v^{*}v_{xx}-\frac{4\varGamma(z)}{\sigma}\left[\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)_{x}\lvert
v\rvert^{2}\right]_{x}.\end{split}$ (17)

We insert Eq. (16) and Eq. (17) into Eqs. (14) and (15), then we average over
the period $\Lambda=2\pi/\omega$ for the rapid oscillations, using the
relations

$\displaystyle\langle\varGamma(z)\rangle=0,\,\,\,\,\langle\varGamma^{2}(z)\rangle=\beta^{2},$
$\displaystyle\langle e^{\pm
i\frac{\varGamma(z)}{\sigma}\Theta}\rangle=\frac{1}{\Lambda}\int_{0}^{\Lambda}e^{\pm
i\frac{\varGamma(z)}{\sigma}\Theta}dz=J_{0}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right),$
$\displaystyle\langle\varGamma(z)e^{\pm
i\frac{\varGamma(z)}{\sigma}\Theta}\rangle=\frac{1}{\Lambda}\int_{0}^{\Lambda}\varGamma(z)e^{\pm
i\frac{\varGamma(z)}{\sigma}\Theta}dz$ $\displaystyle=\pm
i\frac{\chi_{1}}{\omega}J_{1}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right),$

where $\Theta=\left[(1-\rho)(2\sigma\lvert v\rvert^{2}-\lvert
u\rvert^{2}\right)]$ and $J_{i}(\cdot),i=0,1$ are the zero- and first-order
Bessel functions. In the averaging process fast oscillating terms (terms with
$\varGamma(z)$) vanish. We obtain the following system of averaged coupled
equations for the FH:

$\begin{split}&iu_{z}+\beta^{2}\left[\frac{1}{\sigma}\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\lvert
u\rvert^{2}\right]_{x}+\frac{2\rho}{\sigma^{2}}\left[\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)_{x}\lvert
v\rvert^{2}\right]_{x}\right]u\\\
&-\beta^{2}\left[\left(\frac{1}{2\sigma}\lvert u\rvert^{2}+\rho\lvert
v\rvert^{2}\right)_{x}\right]^{2}u+\frac{\partial^{2}u}{\partial
x^{2}}-\frac{\chi_{1}}{\omega}\frac{\rho-1}{2\sigma}J_{1}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)u\left(u^{2}v^{*}+u^{*2}v\right)\\\
&-u+u^{*}vJ_{0}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)+\chi_{0}\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)u=0,\\\ \end{split}$ (18)

and SH:

$\begin{split}&i\sigma
v_{z}+\beta^{2}\left[2\rho\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\lvert
u\rvert^{2}\right]_{x}+\frac{4}{\sigma}\left[\left(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2}\right)_{x}\lvert
v\rvert^{2}\right]_{x}\right]v\\\
&-\frac{\beta^{2}}{\sigma^{2}}\left[\left(2\sigma\lvert v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{x}\right]^{2}v+\frac{\partial^{2}v}{\partial
x^{2}}+\frac{\chi_{1}}{\omega}(\rho-1)J_{1}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)v\left(u^{2}v^{*}+u^{*2}v\right)\\\
&-qv+\frac{1}{2}u^{2}J_{0}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)+\chi_{0}(2\sigma\lvert
v\rvert^{2}+\rho\lvert u\rvert^{2})v=0.\end{split}$ (19)

The Hamiltonian of the above averaged system, i.e.

$iu_{z}=\frac{\delta\langle H\rangle}{\delta u^{*}},\,\,\,\ i\sigma
v_{z}=\frac{\delta\langle H\rangle}{\delta v^{*}},$

is then

$\begin{split}\langle H\rangle=&\int_{-\infty}^{+\infty}\bigg{[}\lvert
u_{x}\rvert^{2}+\lvert
v_{x}\rvert^{2}+\beta^{2}\left[\left(\frac{1}{2\sigma}\lvert
u\rvert^{2}+\rho\lvert v\rvert^{2}\right)_{x}\right]^{2}\lvert u\rvert^{2}\\\
&+\frac{\beta^{2}}{\sigma^{2}}\left[\left(2\sigma\lvert v\rvert^{2}+\rho\lvert
u\rvert^{2}\right)_{x}\right]^{2}\lvert
v\rvert^{2}-\frac{\chi_{0}}{4\sigma}\lvert u\rvert^{4}-\chi_{0}\sigma\lvert
v\rvert^{4}+q\lvert v\rvert^{2}+\lvert u\rvert^{2}\\\ &-\chi_{0}\rho\lvert
v\rvert^{2}\lvert
u\rvert^{2}-\frac{1}{2}J_{0}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)(u^{2}v^{*}+u^{*2}v)\bigg{]}dx.\end{split}$
(20)

The effective $\chi^{(2)}_{\textnormal{eff}}$ non-linearity parameter now
become
$\chi^{(2)}_{\textnormal{eff}}=\chi^{(2)}J_{0}\left(\frac{\chi_{1}}{\sigma\omega}\Theta\right)$,
i.e. it depends also on the power imbalance of FH and SH. In the case of
AMBEC, it corresponds to the dependence of $\chi^{(2)}$ on the population
imbalance between the atoms and the molecular BEC [15].

To check the validity of this averaged model, we perform numerical simulation
of the original system (2), and the averaged equations, Eqs. (18) and (19),
with the same initial condition obtained by solving numerically for the
stationary states of Eqs. (2) with the parameters $q=2$, $\chi_{0}=5$. This
comparison is reported in Fig. 6 for evolution of the stationary solutions.
With the modulation parameters $\omega=30$, $\chi_{1}=6$ and $\chi_{1}=10$, we
have confirmed agreement between the results from the original and the
averaged models.

Figure 6: (Color online) Evolution of stationary solutions in the cases of two
different modulation parameters for the Kerr nonlinearity. In the left-hand
side, frames a) and b), we demonstrate the evolution of the squared central
amplitude for the fundamental- and second-harmonics ($|u(0,z)|^{2}$ and
$|v(0,z)|^{2}$, respectively) in the case of $\chi_{1}=6$. On the right-hand
side, frames c) and d), corresponds to the case of $\chi_{1}=10$. Dashed (red)
and solid (blue) curves correspond to the original model, Eqs. (2), and the
averaged equations, Eqs. (18) and (19), respectively.

##  5 Conclusion

In conclusion, we suggest the method of dynamical stabilization of solitons in
media with competing quadratic and cubic nonlinearities, based on the rapid
modulations of the longitudinal variable of the mismatch parameter. The
criteria for the dynamical stabilization of solitons for mismatch management
are obtained from analysis of averaged equations. In addition, in the case of
the management of the cubic nonlinearity, we have shown that stable optical
solitons can exist in medium supported by competing quadratic and varying Kerr
non-linearities. It is shown in the analysis of an averaged system, that the
Kerr non-linearity management is described by the same non-modulated
Hamiltonian but with an effective $\chi^{(2)}_{\textnormal{eff}}$ parameter
that depends on the intensity imbalance between the FH and SH fields. Finally,
we have confirmed that the predictions of the average model corroborate with
the full numerical simulations.

## Acknowledgments

This work has been supported by the state budget of the Republic of
Uzbekistan, and for M. Ö. through ORU-RR-2021/2022. We also thank the authors
of the XMDS software [16], which was used here for the dynamical simulations.

## References

  * [1] A. V. Buryak, P. Di Trapani, D. V. Skryabin, and S. Trillo, Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications, Phys. Rep. 370, 63 – 235 (2002).
  * [2] B. Oleś and K. Sacha, Solitons in coupled atomic–molecular Bose–Einstein condensates, J. Phys. B: At. Mol. Opt. Phys. 40, 1103 (2007).
  * [3] A. V. Buryak, Y. S. Kivshar, and S. Trillo, Optical solitons supported by competing nonlinearities, Optics Letters 20, 1961 (1995).
  * [4] H. Saito and M. Ueda, Dynamically Stabilized Bright Solitons in a Two-Dimensional Bose-Einstein Condensate, Phys. Rev. Lett. 90, 040403 (2003); F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, Adiabatic Dynamics of Periodic Waves in Bose-Einstein Condensates with Time Dependent Atomic Scattering Length, ibid. 90, 230402 (2003); F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel and B. A. Malomed, Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length, Phys. Rev. A 67, 013605 (2003). G. D. Montesinos, V. M. Perez-Garcia and P. J. Torres, Stabilization of solitons of the multidimensional nonlinear Schrödinger equation: matter-wave breathers, Physica D 191, 193 (2004).
  * [5] B. A. Malomed, Soliton Management in Periodic Systems (Springer US), 2006.
  * [6] A. Biswas, D. Milovic and M. Edwards, Mathematical Theory of Dispersion-Managed Optical Solitons (Springer US), 2010.
  * [7] E. Kengne, W.-M. Liu, and B. A. Malomed, Spatiotemporal engineering of matter-wave solitons in Bose–Einstein condensates, Physics Reports 899, 1-62 (2021).
  * [8] O. Y. Driben, B. A. Malomed, A. Gubeskys, and V. A. Yurovsky, Mismatch management for optical and matter-wave quadratic solitons, Phys. Rev. E 75, 026612 (2007).
  * [9] O. V. Matusevich, V. A. Trofimov, E. A. Yudina, and B. A. Malomed, The evolution of two-frequency solitons in an optical fiber with a longitudinally nonuniform nonlinearity, Optics and Spectroscopy 106, 99 (2009).
  * [10] J. F. Corney and O. Bang, Solitons in quadratic nonlinear photonic crystals, Phys. Rev. E 64, 047601 (2001).
  * [11] M. Conforti, Exact cascading nonlinearity in quasi-phase-matched quadratic media, Opt. Lett. 39, 2427 (2014).
  * [12] F. Kh. Abdullaev, M. Ögren, and J. S. Yuldashev, Matter waves in atomic-molecular condensates with Feshbach resonance management, Phys. Rev. E 104, 024222 (2021).
  * [13] M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, Quasi-phase-matched second harmonic generation: tuning and tolerances, IEEE J. Quant. Electron. 28, 2631 (1992).
  * [14] D. E. Pelinovsky, P. G. Kevrekidis, and D. J. Frantzeskakis, Averaging for Solitons with Nonlinearity Management, Phys. Rev. Lett. 91, 240201, (2003); V. Zharnitsky and D. E. Pelinovsky, Averaging of nonlinearity-managed pulses, Chaos 15 037105 (2005).
  * [15] M. Ögren and K. V. Kheruntsyan, Role of spatial inhomogeneity in dissociation of trapped molecular condensates, Phys. Rev. A 82, 013641 (2010).
  * [16] http://www.xmds.org