# Symmetry-breaking transitions in quiescent and moving solitons in fractional
couplers

Dmiry V. Strunin1 and Boris A. Malomed2,3 1School of Mathematics, Physics and
Computing, University of Southern Queensland, Toowoomba, Queensland 4350,
Australia
2Department of Physical Electronics, School of Electrical Engineering, Faculty
of Engineering, and Center for Light-Matter Interaction, Tel Aviv University,
P.O.B. 39040, Tel Aviv, Israel
3Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile

###### Abstract

We consider phase transitions, in the form of spontaneous symmetry breaking
(SSB) bifurcations of solitons, in dual-core couplers with fractional
diffraction and cubic self-focusing acting in each core, characterized by Lévy
index $\alpha$. The system represents linearly-coupled optical waveguides with
the fractional paraxial diffraction or group-velocity dispersion (the latter
system was used in a recent experiment, which demonstrated the first
observation of the wave propagation in an effectively fractional setup). By
dint of numerical computations and variational approximation (VA), we identify
the SSB in the fractional coupler as the bifurcation of the subcritical type
(i.e., the symmetry-breaking phase transition of the first kind), whose
subcriticality becomes stronger with the increase of fractionality $2-\alpha$,
in comparison with very weak subcriticality in the case of the non-fractional
diffraction, $\alpha=2$. In the Cauchy limit of $\alpha\rightarrow 1$, it
carries over into the extreme subcritical bifurcation, manifesting backward-
going branches of asymmetric solitons which never turn forward. The analysis
of the SSB bifurcation is extended for moving (tilted) solitons, which is a
nontrivial problem because the fractional diffraction does not admit Galilean
invariance. Collisions between moving solitons are studied too, featuring a
two-soliton symmetry-breaking effect and merger of the solitons.

##  I Introduction

The Schrödinger equation with the kinetic-energy operator represented by a
fractional derivative was first derived by Laskin as a generalization of
canonical quantum mechanics [1, 2, 3]. The scaled form of the one-dimensional
fractional Schrödinger equation in the free space, with the Lévy index (LI)
$\alpha$ (so named by Mandelbrot [4]) is

$i\frac{\partial\psi}{\partial
t}=\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}\psi,$ (1)

where the fractional operator is defined as the Riesz derivative [5, 6],

$\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}\psi(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}dp|p|^{\alpha}\int_{-\infty}^{+\infty}dx^{\prime}e^{ip(x-x^{\prime})}\psi(x^{\prime}).$
(2)

It is built as a nonlocal operator, produced by the juxtaposition of the
direct and inverse Fourier transforms, with the fractional differentiation
represented by factor $|p|^{\alpha}$ in the Fourier space. In the Laskin’s
fractional quantum mechanics, $\alpha$ takes values

$1<\alpha\leq 2.$ (3)

The limit of $\alpha=2$ corresponds to the canonical Schrödinger equation,
with operator (2) reducing to the normal second derivative. The case of
$\alpha=1$ (the “square root of the Laplacian”) is known as the Cauchy driver
[7, 8], which occurs, in particular, in the fractional equation for a photonic
wave function [9]. Further, it was found that the original LI range (3) may be
expanded to $0<\alpha\leq 2$ [9].

It is relevant to mention that there are other definitions of the fractional-
order differentiation, such as the Caputo derivative [2], but the derivation
of the fractional Schrödinger equation by Laskin yields the model with the
Riesz derivative [1, 3]. The nonlocal character of the fractional-derivative
operators produces essentially nontrivial consequences in bounded domains,
with the respective boundary conditions [8, 10]. In particular, different
definitions of the fractional derivative produce essentially non-equivalent
results in the case of boundary conditions and under the action of external
potentials.

While experimental realization of the fractional quantum mechanics is still
missing, it was proposed to emulate it by means of the classical wave
propagation in photonics, utilizing the commonly known similarity of the
Schrödinger equation and the equation for the paraxial propagation of optical
beams [11, 12]. The theoretically elaborated setup for the emulation of the
fractional diffraction is based on the 4f optical configuration, which
performs the spatial Fourier decomposition of the beam by means of a lens,
then carries the beam through an appropriately shaped phase plate to impart
the local phase shift which emulates the action of the fractional diffraction
as per Eq. (2), and finally retrieves the beam from its Fourier decomposition,
using another lens [11]. Other emulations of the fractional quantum mechanics
were proposed in Lévy crystals [13] and polariton condensates [14].

Very recently, the first _experimental realization_ of the effective
fractional group-velocity dispersion (rather than transverse diffraction) in a
fiber-laser setup has been reported [15]. While the fractional dispersion acts
in the temporal domain, its experimental emulation was carried out by means of
a 4f setup similar to the one outlined above, with the difference that the
lenses performed the spectral decomposition of the optical signal and inverse
recombination. To avoid misunderstanding, it is relevant to stress that the
temporal variable plays the role of an efficient coordinate in optical fibers,
while the evolution variable is the propagation distance. Therefore, the
effective fractional dispersion reported in Ref. [15] emulates the temporal
counterpart of the fractional diffraction (cf. Ref. [16]), but not a
fractional derivative with respect to the evolution variable, which occurs in
various models of anomalous diffusion [17, 18]. Of course, a caveat is that
the proposed and reported realizations of the fractional diffraction and
dispersion in optics actually report the simulation of these effects by light,
but not their direct action.

Theoretical studies of models with the fractional diffraction were developed
by including external potentials [in particular, parity-time ($\mathcal{PT}$)
symmetric ones [19]-[24]] and studying the propagation of Airy waves in the
two-dimensional geometry, with the fractional-diffraction operator
$\left(-\partial^{2}/\partial x^{2}-\partial^{2}/\partial
y^{2}\right)^{\alpha/2}$ [25, 26]. The implementation of the fractional
diffraction/dispersion in optical waveguides makes it natural to include the
self-focusing Kerr nonlinearity of the material, which leads to the
consideration of fractional nonlinear Schrödinger equations (FNLSEs) [27].

The work based on FNLSEs has produced many theoretical results, including the
modulational instability of continuous waves [28], critical or supercritical
collapse [29], and diverse species of solitons [30]-[27]. These are quasi-
linear “accessible solitons” [33, 34], gap solitons maintained by lattice
potentials [38]-[42], solitary vortices [43, 44, 45], multi-peak modes
[46]-[49], clusters [50], discrete solitons [51], and dark modes [52].
Solitons produced by the interplay of the fractional diffraction and quadratic
nonlinearity were predicted too [53, 54], as well as dissipative solitons
produced by the fractional complex Ginzburg-Landau equation [55].

A generic effect produced by self-trapping nonlinearity is spontaneous
symmetry breaking (SSB) in double-well potentials [57]. SSB phenomenology in
such settings was studied in detail theoretically [58]-[63] and experimentally
[64]-[67] in diverse physical settings. Recently, the theoretical analysis was
extended for nonlinear systems combining the fractional diffraction and
symmetric potentials [68]-[71]. An appropriate platform for the realization of
SSB in the form of two-component solitons with broken symmetry between the
components is offered by dual waveguides, which are often represented by
double-core optical fibers [72, 73]. The transition from symmetric to
asymmetric two-component solitons, i.e., the SSB bifurcation in such fibers,
was studied in detail theoretically [74]-[79], and it was recently
demonstrated in an experiment [80]. In another recent work [81], some families
of symmetric and asymmetric solitons were found in the double-core system with
fractional diffraction.

The objective of the present work is to identify the SSB bifurcation of two-
component solitons in the fractional dual-core waveguide, i.e., the
corresponding symmetry-breaking phase transition. We do this analytically, by
means of the variational approximation (VA), and in a systematic numerical
form. An essential finding is that _deeper_ system’s fractionality [i.e.,
smaller LI $\alpha$ in Eq. (2)] enhances the _subcritical_ character [82] of
the bifurcation. In other words, the fractionality makes the SSB of the two-
component solitons a more strongly pronounced phase transition of the first
kind.

An essential peculiarity of the fractional diffraction is that it does not
admit the Galilean invariance. Therefore, the generation of moving solitons
and the consideration of the SSB bifurcation for them is a nontrivial problem,
which we address here too, and which was not considered previously. Collisions
between moving solitons are studied too

Thus, we address the system of linearly-coupled FNLSEs with the cubic self-
focusing nonlinearity,

$\displaystyle i\frac{\partial u_{1}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}u_{1}-|u_{1}|^{2}u_{1}-u_{2},$ $\displaystyle
i\frac{\partial u_{2}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}u_{2}-|u_{2}|^{2}u_{2}-u_{1},$ (4)

where the coupling coefficient in front of terms $-u_{2}$ and $-u_{1}$ is
fixed to be $1$ by means of scaling. The only irreducible control parameter of
the system is LI $\alpha$. In terms of optics, Eqs. (4) describe a pair of
parallel planar waveguides coupled by tunneling of light. In this case, $t$
and $x$ are, respectively, the propagation distance and transverse coordinate.
The same system realized in the temporal domain, i.e., with $x$ standing for
the reduced time [83], may be construed as a model of a laser built of dual-
core fibers with the effective fractional dispersion, following the
experimental method reported in Ref. [15]. The possibility of using dual-core
fibers in lasers is a well-known option [84].

There may also be a possibility to consider Eqs. (4) as a system of coupled
Gross-Pitaevskii equations for a Bose-Einstein condensate of quantum particles
governed by the fractional Schrödinger equations and filling a pair of
parallel cigar-shaped traps, which are linearly coupled by tunneling of the
particles. In that case, $t$ is time and $x$ is the coordinate along the
traps. However, the derivation of such a mean-field model for the condensate
requires accurate analysis, which should be a subject of a separate work. In
particular, it is necessary to check if collisional effects in this system may
be properly represented by the usual mean-field cubic terms. Experimental
realization of the system may be a challenging objective too.

The presentation is organized below as follows. The framework for the
construction of soliton solutions and analysis of their stability and dynamics
are presented in Section 2. The analytical approach, based on the VA, is
developed in Section 3. Numerical results for quiescent and moving solitons
are summarized in Section 4. The work is concluded by Section 5.

##  II The framework for soliton solutions

Stationary-state solutions to Eq. (4) with propagation constant $k$ (in the
application to BEC, $-k$ is the chemical potential) are looked for as

$u_{1,2}(x,t)=U_{1,2}(x)e^{ikt},$ (5)

with real functions $U_{1,2}(x)$ satisfying equations

$\displaystyle kU_{1}+\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}U_{1}-U_{1}^{3}-U_{2}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle kU_{2}+\frac{1}{2}\left(-\frac{\partial^{2}}{\partial
x^{2}}\right)^{\alpha/2}U_{2}-U_{2}^{3}-U_{1}$ $\displaystyle=$ $\displaystyle
0.$ (6)

Taking into regard that $U_{1,2}(x)$ are even functions of $x$ and using the
definition of the fractional derivative given by Eq. (2), the explicit form of
Eq. (6) can be written as

$\displaystyle
kU_{1}+\frac{1}{2\pi}\int_{0}^{+\infty}p^{\alpha}dp\int_{-\infty}^{+\infty}\cos(px)\cos(px^{\prime})U_{1}(x^{\prime})dx^{\prime}-U_{1}^{3}-U_{2}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
kU_{2}+\frac{1}{2\pi}\int_{0}^{+\infty}p^{\alpha}dp\int_{-\infty}^{+\infty}\cos(px)\cos(px^{\prime})U_{2}(x^{\prime})dx^{\prime}-U_{2}^{3}-U_{1}$
$\displaystyle=$ $\displaystyle 0.$ (7)

We consider values of LI in the interval of

$1<\alpha\leq 2,$ (8)

as it is well known that, at $\alpha\leq 1$, the FNLSE gives rise to the
collapse (critical collapse at $\alpha=1$, and supercritical at $\alpha<1$)
[29, 27].

In the case of the normal diffraction, $\alpha=2$, obvious solutions of Eq.
(6) in the form of symmetric solitons is

$U_{1}=U_{2}=\sqrt{2(k-1)}\mathrm{sech}\left(\sqrt{2(k-1)}x\right).$ (9)

The norm (power) of this solution is

$N=\int_{-\infty}^{+\infty}\left[\left(U_{1}(x)\right)^{2}+\left(U_{2}(x)\right)^{2}\right]dx=4\sqrt{2\left(k-1\right)}.$
(10)

With the increase of $N$, the symmetric states become unstable through SSB,
and stable asymmetric solitons appear. While there are no exact solutions for
the asymmetric solitons, the SSB point, at which they emerge, can be found
exactly for $\alpha=2$ [74]:

$\left(N_{\mathrm{SSB}}\right)_{\mathrm{exact}}(\alpha=2)=8/\sqrt{3}.$ (11)

On the other hand, the VA predicts this point at

$\left(N_{\mathrm{SSB}}\right)_{\mathrm{VA}}(\alpha=2)=2\sqrt{6}\approx 4.899$
(12)

[79, 72], the relative error of this result being $\simeq 6\%$.

Asymmetry of the solitons produced by the SSB bifurcation is defined by
parameter

$\Theta=N^{-1}\int_{-\infty}^{+\infty}\left[\left(U_{1}(x)\right)^{2}-\left(U_{2}(x)\right)^{2}\right]dx.$
(13)

The bifurcation is characterized by diagrams which displays $\Theta$ as
functions of $k$ or $N$ [see, in particular, Fig. 3 below].

Solutions of Eq. (4) for moving solitons, i.e., obliquely propagating light
beams with slope $c$ in the planar waveguide, are sought for as

$u_{1,2}=u_{1,2}\left(\xi\equiv x-ct,t\right).$ (14)

Accordingly, Eq. (4) is rewritten in terms of $\left(\xi,t\right)$ as

$\displaystyle i\frac{\partial u_{1}}{\partial t}-ic\frac{\partial
u_{1}}{\partial\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial\xi^{2}}\right)^{\alpha/2}u_{1}-|u_{1}|^{2}u_{1}-u_{2},$
$\displaystyle i\frac{\partial u_{2}}{\partial t}-ic\frac{\partial
u_{2}}{\partial\xi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\partial^{2}}{\partial\xi^{2}}\right)^{\alpha/2}u_{2}-|u_{2}|^{2}u_{2}-u_{1}.$
(15)

Solutions to Eq. (15) are further looked for as $u_{1,2}\left(\xi,t\right)=$
$U_{1,2}(\xi)e^{ikt}$ [cf. Eq. (5)], with complex functions $U_{1,2}(\xi)$
satisfying the following system of stationary equations:

$\displaystyle
kU_{1}+ic\frac{dU_{1}}{d\xi}+\frac{1}{2}\left(-\frac{d^{2}}{d\xi^{2}}\right)^{\alpha/2}U_{1}-\left|U_{1}\right|^{2}U_{1}-U_{2}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
kU_{2}+ic\frac{dU_{2}}{d\xi}+\frac{1}{2}\left(-\frac{d^{2}}{d\xi^{2}}\right)^{\alpha/2}U_{2}-\left|U_{2}\right|^{2}U_{2}-U_{1}$
$\displaystyle=$ $\displaystyle 0.$ (16)

The stability of solitons was addressed by considering solutions including
small perturbations $a_{1,2}$ and $b_{1,2}$,

$u_{1,2}(x,t)=\left[U_{1,2}(x)+a_{1,2}(x)e^{\lambda
t}+b_{1,2}^{\ast}(x)e^{\lambda^{\ast}t}\right]e^{ikt}\,,$ (17)

where $\lambda$ is the instability growth rate, and $\ast$ stands for the
complex conjugate. The linearization of Eq. (4) for the perturbations leads
the system of the corresponding Bogoliubov – de Gennes equations:

$\displaystyle\left[-\left(k-i\lambda\right)-\frac{1}{2}\left(-\frac{d^{2}}{d\xi^{2}}\right)^{\alpha/2}+2\left|U_{1,2}\right|^{2}\right]a_{1,2}+U_{1,2}^{2}b_{1,2}+a_{2,1}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left[-\left(k+i\lambda\right)-\frac{1}{2}\left(-\frac{d^{2}}{d\xi^{2}}\right)^{\alpha/2}+2\left|U_{1,2}\right|^{2}\right]b_{1,2}+\left(U_{1,2}^{\ast}\right)^{2}a_{1,2}+b_{2,1}$
$\displaystyle=$ $\displaystyle 0.$ (18)

The stability condition is that solutions of Eq. (18) must produce only
eigenvalues with Re$\left(\lambda\right)=0$ [89, 85]. Below, the prediction
for the stability of the solitons, provided by these equations, is
corroborated by direct simulations of the evolution of perturbed solitons.

##  III The variational approximation (VA)

To apply VA, we note that Eq. (7) can be derived from the Lagrangian,

$L=\int_{-\infty}^{+\infty}\left[\frac{k}{2}\left(U_{1}^{2}+U_{2}^{2}\right)\right]dx+H,$
(19)

with Hamiltonian

$\displaystyle
H=\int_{-\infty}^{+\infty}\left\\{-\frac{1}{4}\left[U_{1}^{4}(x)+U_{2}^{4}(x)\right]-U_{1}(x)U_{2}(x)\right\\}dx$
$\displaystyle+\frac{1}{4\pi}\int_{0}^{+\infty}p^{\alpha}dp\int_{-\infty}^{+\infty}dx\int_{-\infty}^{+\infty}dx^{\prime}\cos\left(p\left(x-x^{\prime}\right)\right)\left[U_{1}(x)U_{1}(x^{\prime})+U_{2}(x)U_{2}(x^{\prime})\right],$
(20)

cf. Ref. [45]. The ansatz for the asymmetric soliton can be adopted in the
simple form, which follows the pattern of the above-mentioned solution (9):

$U_{1}(x)=\sqrt{\frac{N}{2W}}\left(\cos\chi\right)\mathrm{sech}\left(\frac{x}{W}\right),U_{2}(x)=\sqrt{\frac{N}{2W}}\left(\sin\chi\right)\mathrm{sech}\left(\frac{x}{W}\right),$
(21)

where variational parameters are width $W$ and norm-distribution angle $\chi$,
while $N$ is considered as a given total norm. The asymmetry parameter (13)
corresponding to the ansatz is

$\displaystyle\Theta_{\mathrm{VA}}$ $\displaystyle=$
$\displaystyle\cos\left(2\chi\right)\equiv\sqrt{1-S^{2}},$ (22) $\displaystyle
S$ $\displaystyle\equiv$ $\displaystyle\sin\left(2\chi\right)$ (23)

(parameter $S$ is used below). The substitution of ansatz (21) in Lagrangian
defined by Eqs. (19) and (20) yields

$\displaystyle L_{\mathrm{VA}}$ $\displaystyle=$
$\displaystyle\frac{N}{2}k-\frac{N^{2}}{12W}\left(1-\frac{1}{2}\sin^{2}(2\chi)\right)-\frac{N}{2}\sin(2\chi)$
(24)
$\displaystyle+\left(1-2^{1-\alpha}\right)\Gamma(1+\alpha)\zeta(\alpha)\frac{N}{2(\pi
W)^{\alpha}},$

where $\Gamma$ and $\zeta$ are the Gamma- and zeta-functions. Parameters of
the asymmetric solitons are predicted by the Euler-Lagrange equations,

$\frac{\partial L_{\mathrm{VA}}}{\partial W}=\frac{\partial
L_{\mathrm{VA}}}{\partial\left(\sin(2\chi\right))}=0,$ (25)

which amount to relation $W=(N/6)\sin(2\chi)$, and an equation for
$S\equiv\sin\left(2\chi\right)$:

$S^{\alpha-1}\left(1-\frac{S^{2}}{2}\right)=\frac{\alpha}{\pi^{\alpha}}\left(1-2^{1-\alpha}\right)\Gamma(1+\alpha)\zeta(\alpha)\left(\frac{6}{N}\right)^{\alpha}.$
(26)

In particular, the threshold at which SSB takes place, giving rise to families
of asymmetric solutions which branch off from symmetric ones, that correspond
to $S=1$, is produced by the substitution of $S=1$ in Eq. (26):

$\left(N_{\mathrm{SSB}}\right)_{\mathrm{VA}}(\alpha)=\frac{6}{\pi}\left[2\alpha\left(1-2^{1-\alpha}\right)\Gamma(1+\alpha)\zeta(\alpha)\right]^{1/\alpha}.$
(27)

In the case of $\alpha=2$, expression (27) exactly reduces to Eq. (12). In the
opposite limit of $\alpha-1\rightarrow+0$ [see Eq. (8)], Eq. (27) yields

$\left(N_{\mathrm{SSB}}\right)_{\mathrm{VA}}\left(\alpha\rightarrow
1\right)=12\pi^{-1}\ln 2\approx 2.648.$ (28)

Finally, the VA prediction for the asymmetry dependence on the norm,
$\Theta(N)$, is obtained, in an implicit form, as a combination of Eqs. (22),
(23), and (26):

$\left(1-\Theta_{\mathrm{VA}}^{2}\right)^{(\alpha-1)/2}\left(1+\Theta_{\mathrm{VA}}^{2}\right)=\frac{2\alpha}{\pi^{\alpha}}\left(1-2^{1-\alpha}\right)\Gamma(1+\alpha)\zeta(\alpha)\left(\frac{6}{N}\right)^{\alpha}.$
(29)

This relation takes an explicit form in the limit of $\alpha\rightarrow 1$:

$\Theta_{\mathrm{VA}}(N;\alpha\rightarrow
1)=\sqrt{\frac{\left(N_{\mathrm{SSB}}\right)_{\mathrm{VA}}\left(\alpha\rightarrow
1\right)}{N}-1},$ (30)

where $\left(N_{\mathrm{SSB}}\right)_{\mathrm{VA}}\left(\alpha\rightarrow
1\right)$ is the value given by Eq. (28). Dependence (30) is displayed below
in Fig. 3, and the predictions produced by Eqs. (29) and (27) are compared
with numerical results in Fig. 3.

##  IV Numerical results

###  IV.1 The spontaneous-symmetry-breaking (SSB) bifurcation of stationary
states and their stability

Soliton solutions of Eq. (7) were produced by means of the squared-operator
iteration method [86, 85]. Then, the spectrum of stability eigenvalues
$\lambda$ was produced solving Eq. (18) by means of the Fourier collocation
method. Both algorithms were realized in the Matlab shell, as outlined in Ref.
[85]. Direct simulations of Eq. (4) were performed by means of the
pseudospectral method [85, 90, 91, 92],

A typical profile of an asymmetric soliton is presented in Fig. 1(a). Panels
(b) and (c) of the figure demonstrate that this soliton is unstable,
spontaneously transforming into a robust breather, which is a dynamical state
effectively symmetrized by persistent oscillations between its two components.
The shape of stable asymmetric soliton is similar to that in Fig. 1. As for
symmetric solitons, those ones which are unstable spontaneously turn into
stable asymmetric ones, with residual internal oscillations (not shown here in
detail).

Figure 1: The stationary profile of an unstable asymmetric soliton (a), its
spectrum of perturbation eigenvalues (b), and perturbed evolution (c), for
$\alpha=1.6$, $k=1.8$, and $N=3.729$. Shapes of stable asymmetric solitons are
similar to the one displayed here.

Families of numerically found stationary symmetric and asymmetric solitons,
and the SSB bifurcation which couples them, are displayed in Figs. 2 and 3 by
respective dependences $N(k)$ and $\Theta(N)$, in the interval of LI values
$1.1\leq\alpha\leq 2.0$, cf. Eq. (8) (the numerical solution is technically
challenging for $\alpha<1.1$ because of slow convergence). The latter figure
includes the comparison with the VA prediction, given above by Eq. (29). It is
seen that the VA is reasonably accurate, with the relative discrepancy in
terms of $N$ for fixed $\Theta$ being $\leq 6\%$ in Fig. 3(a). In particular,
the VA results are quite reliable for stable branches $\Theta(N)$. In the case
of $\alpha=2$, the findings are tantamount to the well-known results for the
usual coupler [72, 73].

Note that the (in)stability of all solution branches, as shown in Figs. 2 and
3, complies with the Vakhitov-Kolokolov criterion, $dN/dk>0$, which is the
necessary stability condition for self-trapped modes [87, 85]. In particular,
the asymmetric solitons belonging to the backward- and forward-going segments
of the respective solution branches are stable or not in agreement with the
criterion.

Figures 2 and 3(a) demonstrate that, as it might be expected, the increase of
the norm leads to destabilization of symmetric solitons and emergence of
asymmetric ones via the SSB bifurcation. Similar to the known feature of SSB
in the usual coupler ($\alpha=2$), Fig. 3 shows that the bifurcation is of the
subcritical type [82], thus representing a phase transition of the first kind,
which admits hysteresis and bistability. The subcritical bifurcation gives
rise to branches of asymmetric solitons that originally go in the backward
direction (which corresponds to the decrease of $N$), and then turn forward at
critical points. These points represent the minimum value of $N$ at which the
asymmetric solitons exist. An essential conclusion suggested by Fig. 3 is that
_deeper fractionality_ , i.e., larger $2-\alpha$, makes the subcritical
character of the SSB bifurcation stronger, while this feature is very weak in
the usual case, $\alpha=2$. As an extension of this trend, Fig. 3 shows that,
at $\alpha=1.2$, the branch of the asymmetric solitons admits almost no
extension past the critical turning point. Additional numerical results
demonstrate that, in the explored range of values of $k$, the $\Theta(N)$
curve does not reach the turning point for $\alpha=1.1$. A conjecture is that
the SSB bifurcation becomes an _extreme subcritical one_ , with no turning
points, in the limit of $\alpha\rightarrow 1$. Indeed, this feature is
explicitly demonstrated by the VA curve produced by Eq. (30), which is plotted
in Fig. 3(a).

Note that the first example of the extreme subcritical SSB bifurcation was
reported in the model based on the single NLS equation with the usual
diffraction and nonlinear double-well potential [88],

$i\frac{\partial u}{\partial t}=-\frac{1}{2}\frac{\partial^{2}u}{\partial
x^{2}}-\left[\delta\left(x-1\right)+\delta\left(x+1\right)\right]|u|^{2}u,$
(31)

where $\delta$ is the delta-function, with separation $2$ between the
potential wells fixed by scaling. An exact analytical solution of Eq. (31)
produces the extreme subcritical bifurcation, with branches of asymmetric
solitons going backward from the bifurcation point,
$N(\Theta=0)=\frac{2}{3}+\frac{8}{27}\left(\frac{3}{4}+\ln 2\right)\approx
1.09$, up to $N(\Theta=1)=1$, and never turning forward.

Figure 2: Norm $N$ of symmetric and asymmetric solitons plotted versus the
propagation constant, $k$, at different values of LI $\alpha$, as indicated by
labels. Solid and dashed lines mark stable and unstable branches,
respectively, according to the eigenvalue spectrum produced by Eq. (18).

Figure 3: (a) Symmetry parameter $\Theta$, defined as per Eq. (13), versus the
norm, for families of asymmetric solitons, at indicated values of LI $\alpha$.
Plotted are both the numerical results and their analytical counterparts,
predicted by VA as per Eq. (29). For $\alpha\rightarrow 1$, the VA curve is
plotted according to Eq. (30). Solid and dashed segments of the numerically
generated branches mark stable and unstable solutions, respectively. (b) The
value of the norm at the SSB bifurcation point versus LI $\alpha$, as produced
by the numerical solution and predicted by VA, see Eq. (27). At $\alpha=2$,
the red point shows the exact value (11), which is identical to the
corresponding numerically found one. At $\alpha=1$, the VA value is given by
Eq. (28).

The basic characteristic of the SSB is the value of the norm,
$N_{\mathrm{SSB}}$, at the bifurcation point, which is predicted by VA as per
Eq. (27).The numerically found value is displayed, as a function of $\alpha$,
along with its VA-predicted counterpart, in Fig. 3(b). The mismatch between
the numerical and variational values does not exceed $6\%$.

In addition to the symmetric and asymmetric solitons, solutions for
antisymmetric ones, with $U_{1}(x)=-U_{2}(x)$, were constructed too, but they
are completely unstable (similar to the known situation in the usual coupler,
with $\alpha=2$ [72, 73]). This conclusion is explained by the fact that the
linear-coupling term in Hamiltonian (20), with density $-U_{1}(x)U_{2}(x)$, is
negative for symmetric and asymmetric modes, but positive for the
antisymmetric ones, the higher energy (Hamiltonian) implying the instability.

###  IV.2 Spontaneous symmetry breaking (SSB) of moving (tilted) solitons

As mentioned above, the incompatibility of the Galilean invariance with the
fractional diffraction makes velocity (tilt) $c$ of the solitons a nontrivial
parameter in Eq. (16). First, we address effects of $c$ on the SSB
bifurcation, which was not addressed in previous works. For selected values of
$c$, bifurcation diagrams produced by the numerical solution of Eq. (16) are
displayed in Fig. 4(a), which demonstrates that the bifurcation keeps its
subcritical character. The shift of the $N(k)$ curves to larger $k$ is similar
to the effect of the Galilean boost in the case of the usual diffraction
($\alpha=2$): in that case, the removal of the velocity terms by means of the
boost produces a shift of the propagation constant $\Delta k=c^{2}/2$. For
values $c=0.4$ and $0.8$, which are presented in Fig. 4(a), this expression
yields $\Delta k(c=0.4)=0.08$ and $\Delta k(c=0.8)=0.32$, being close to the
shifts observed in Fig. 4(a). A new effect, which is absent in the case of the
usual diffraction, is the decrease of norm $N_{\mathrm{SSB}}$ at the
bifurcation point with the increase of $c$. It can be explained by the fact
that, for complex profile functions $U_{1,2}(x)$, in the case of $c\neq 0$,
the coupling Hamiltonian is
$H_{\mathrm{coupling}}=\int_{-\infty}^{+\infty}$Re$\left\\{U_{1}(x)U_{2}^{\ast}(x)\right\\}dx$,
cf. Eq. (20). Its value is reduced due to averaging of oscillations of the
complex integrand. In turn, the relative attenuation of the coupling naturally
leads to a decrease of value $N_{\mathrm{SSB}}$ at the point where the
nonlinearity becomes strong enough to initiate the SSB.

Figure 4: (a) SSB bifurcation diagrams at LI $\alpha=1.6$ for different values
of the soliton’s speed (tilt) $c$. (b) Power $N$ versus $c$ for fixed
$\alpha=1.6,k=2.4$.

Another manifestation of the effect of $c$ on the families of stable and
unstable symmetric and asymmetric solitons is displayed in Fig. 4(b) for fixed
values of LI $\alpha$ and propagation constant $k$. It is seen that tilt $c$
can be used to switch the optical beams between the asymmetric and symmetric
shapes, which may find applications in the design of photonic devices. The
trend towards the decrease of $N$ for large values of $c$, especially for the
symmetric solitons (for which the two-component structure is not essential,
while the fractionality remains a dominant factor), can be explained by noting
that rapid oscillations of the complex wave function $\psi(x)$ caused by large
$c$ make the value of the fractional derivative in Eq. (2) smaller. Therefore,
the strength of the self-focusing term (determined by the value of the norm),
which is necessary to balance the fractional diffraction in solitons, becomes
lower.

###  IV.3  Collisions between moving solitons

Once stable solitons are produced in the moving (tilted) state, they can be
used to explore collisions of soliton pairs in the coupler [93]. For this
purpose, two solitons were numerically constructed as solutions of Eq. (16),
$u_{1,2}^{\pm}$, with velocities $\pm c$. Then, direct simulations of Eq. (4)
were run, with the input in the form of the pair of solitons
$u_{1,2}^{\pm}(x)$ placed, respectively, at $x<0$ and $x>0$, with a
sufficiently large distance between them.

Here we focus on the following (most natural) settings for collisions between
mutually symmetric solitons, with equal values of the propagation constant,
$k$: (1) two stable symmetric solitons; (2) two stable asymmetric solitons,
with the same $k$, in the flipped configuration, where soliton $u_{1,2}^{+}$
has a larger component $u_{1}$ and a smaller one $u_{2}$, and vice versa for
$u_{1,2}^{-}$ (cf. Ref. [93]); and (3) two stable asymmetric solitons, which
are mirror images of each other.

Outcomes of collisions between stable symmetric solitons, at gradually
increasing speeds $\pm c$, are presented in Fig. 5. In all cases, the
colliding solitons bounce back – naturally, remaining far separated for
smallest speeds, and approaching closer to each other for larger $c$. Up to
the case of $c=0.06$, the entire picture remains fully symmetric, with respect
to both the two components in each soliton, and two colliding solitons as
well. Next, starting from $c=0.08$, the simulations demonstrate onset of
collision-induced SSB, which becomes obvious in the case of $c=0.10$. In this
case, the collision breaks the symmetry between the components, as well as
between the colliding solitons. In particular, it is worthy to note that the
post-collision amplitude of component $u_{2}$ in the left soliton is much
larger than before the collision. The collision-induced SSB effect is
explained by the instability of the transient state formed by the colliding
solitons when they are separated by a relatively small distance. A
qualitatively similar SSB mechanism was discovered by simulations of soliton-
soliton collisions in the single equation with the normal diffraction
($\alpha=2$) and cubic-quintic nonlinearity [94].

|
---|---
$c=0.03$ | $c=0.06$
|
$c=0.08$ | $c=0.09$
|
$c=0.1$ | $c=0.12$
Figure 5: The onset of SSB in collisions of slowly moving symmetric solitons,
with $\alpha=1.6$, $k=1.4$, and velocities $\pm c$. The norms of the solitons
are $N=3.420$ ($c=0.03)$, $N=3.412$ ($c=0.06$), $N=3.404$ ($c=0.08$),
$N=3.399$ ($c=0.09$), $N=3.393$ ($c=\pm 0.10$), $N=3.380$ ($c=\pm 0.12$).

With subsequent growth of $c$, the collision picture remains approximately the
same as shown in Fig. 5 for $c=0.12$ up to $c=0.2$ (not shown here in detail).
At still larger speeds, the symmetry is gradually recovered, leading,
eventually, to practically elastic collisions at $c\geq 0.34$, which is a
natural outcome of fast collisions [95].

Results of collisions between stable asymmetric solitons in the mutually
flipped states, as defined above, are demonstrated in Fig. 6. The general
picture is similar to that outlined above for the collisions between symmetric
solitons. Namely, at low speeds, $c\leq 0.04$, the solitons bounce back,
without breaking the symmetry between the colliding ones. In fact, in this
case each soliton switches from the intrinsic asymmetric shape into a nearly
symmetric one, as concerns the relation between its two components. Then,
starting from $c=0.06$, the collision-induced SSB effect sets in, leading to
strong symmetry breaking at $c=0.1$, with a dominant $u_{1}$ component of the
left soliton in the post-collision state. Approximately the same inelastic
outcome of the collision persists up to $c\simeq 0.40$ (not shown here in
detail), while the further increase of the speed gradually leads to a
transition to quasi-elastic collisions.

|
---|---
$c=0.04$ | $c=0.06$
|
$c=0.08$ | $c=0.10$
Figure 6: The gradual onset of SSB in collisions of slowly moving mutually
flipped asymmetric solitons, with $\alpha=1.6$, $k=2.6$, and velocities $\pm
c$. The norms of the solitons are $N=3.749$ ($c=0.04$), $N=3.748$ ($c=0.06$),
$N=3.746$ ($c=0.08$), $N=3.744$ ($c=0.1$), cf. Fig. 5.

The situation when a fully inelastic collision of the mutually flipped
asymmetric solitons gives rise to very strong symmetry breaking is
additionally illustrated by Fig. 7(a), for the same values of $\alpha$ and $k$
as in Fig. 6, and $c=0.42$. It is observed that, in the post-collision state,
component $u_{1}$ almost vanishes, while nearly all the initial norm is
trapped in component $u_{2}$ in the form of a quasi-soliton, which performs
slow erratic motion. Actually, the dominant component and direction of motion
of the emerging mode are selected by the system randomly (as confirmed by
additional numerical results), as a result of the above-mentioned instability
in the transient state created by the collision.

Figure 7: Examples of strongly inelastic collisions of solitons with $k=2.6$
at $\alpha=1.6$. (a)The post-collision dynamics of originally flipped
asymmetric solitons for $c=0.42$ and $N=3.627$. (b) Merger of asymmetric
solitons colliding in the unflipped state, for $c=0.4$ and $N=3.638$.

Lastly, in Fig. 7(b) we present an example of a fully inelastic collision
between identical stable asymmetric solitons, i.e., ones in the unflipped
configuration. In this case, the solitons merge into a single strongly
asymmetric quiescent one, with the same dominant component $u_{1}$ as in the
original solitons.

##  V Conclusion

As a contribution to the quickly developing studies of solitons in systems
with fractional diffraction, we have addressed the phenomenology of the SSB
(spontaneous symmetry breaking) in the one-dimensional dual-core system, with
the Riesz fractional derivative and cubic self-focusing acting in the cores,
and linear coupling between the cores. The corresponding system of FNLSEs
(fractional nonlinear Schrödinger equations) models tunnel-coupled planar
optical waveguides with the fractional diffraction, as well as coupled
waveguides with the fractional group-velocity dispersion in the temporal
domain (the latter setting was recently realized in the experiment [15], and
may be appropriate for the realization of the results predicted in the present
work).

By means of systematic numerical computations and the analytical method based
on the VA (variational approximation), we have identified the SSB in the
system as the bifurcation of the subcritical type, i.e., the phase transition
of the first kind. With the increase of the fractionality, i.e., parameter
$2-\alpha$, where $\alpha$ is the LI (Lévy index), the subcritical character
of the SSB bifurcations gets stronger pronounced, in comparison with very weak
subcriticality in the case of the usual (non-fractional) diffraction,
$\alpha=2$. In the limit of $\alpha\rightarrow 1$, the bifurcation becomes the
extreme subcritical one, i.e., with the backward-going branches of asymmetric
solitons which never turn forward. The VA produces reasonably accurate results
for the SSB, in spite of the complex structure of the system. The
(in)stability of families of asymmetric solitons exactly follows the Vakhitov-
Kolokolov criterion. By means of the numerical method, the study of the SSB
bifurcation and families of symmetric and asymmetric solitons has been
extended for moving (tilted) ones, which is a nontrivial issue for the
fractional system, as it breaks the Galilean invariance. Collisions between
moving solitons are systematically studied too, demonstrating another SSB
effect, as well as merger of the colliding solitons.

As an extension of the analysis, it may be relevant to consider a similar
system of linearly coupled fractional complex Ginzburg-Landau equations, with
the objective to predict symmetric and asymmetric dissipative solitons, as
well as bound states of such solitons in the respective laser systems, cf.
Ref. [96] where a similar analysis was developed in the case of the non-
fractional diffraction.

## Acknowledgment

We appreciate a useful discussion with W. B. Cardoso. The work of B.A.M. was
supported, in part, by grant No. 1695/22 of the Israel Science Foundation.

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