# Noise-induced, ac-stabilized sine-Gordon breathers: Emergence and statistics

Duilio De Santis<EMAIL_ADDRESS>Claudio Guarcello  Bernardo
Spagnolo  Angelo Carollo  Davide Valenti  Dipartimento di Fisica e Chimica “E.
Segrè”, Group of Interdisciplinary Theoretical Physics, Università degli Studi
di Palermo, I-90128 Palermo, Italy  Dipartimento di Fisica “E. R. Caianiello”,
Università degli Studi di Salerno, I-84084 Fisciano, Salerno, Italy  INFN,
Sezione di Napoli, Gruppo Collegato di Salerno - Complesso Universitario di
Monte S. Angelo, I-80126 Napoli, Italy  Radiophysics Department, Lobachevsky
State University, 603950 Nizhniy Novgorod, Russia

###### Abstract

Noisy and ac forcing can cooperatively lead to the emergence of sine-Gordon
breathers robust to dissipation. This phenomenon is studied, for both Neumann
and periodic boundary conditions (NBC and PBC, respectively), at different
values of the main system parameters, such as the noise intensity and the ac
frequency-amplitude pair. In all the considered cases, nonmonotonicities of
the probability of generating only breathers versus the noise strength are
observed, implying that optimal noise ranges for the breather formation
process exist. Within the latter scenarios, the statistics of the breathers’
number, position, and amplitude are analyzed. The number of breathers is found
to grow, on average, with the noise amplitude. The breathers’ spatial
distribution is sharply peaked at the system’s edges for NBC, whereas it is
essentially uniform for PBC. The average breather amplitude is dictated by the
ac frequency-amplitude pair. Finally, a size analysis shows that the minimum
system length for the generation mechanism is given by the typical breather
half-width (width) in NBC (PBC).

###### keywords:

Perturbed sine-Gordon equation , Solitons , Breathers , Noise , Long Josephson
junctions

##  1 Introduction

The soliton is a cornerstone of nonlinear science, and it is nowadays
ubiquitous in physics and beyond [1, 2]. Two core reasons behind the
widespread success of this concept are: (i) the soliton displays remarkable
particle-like properties which are both very robust to perturbations and
compelling for a plethora of applications; (ii) solitons mathematically emerge
as solutions of partial differential equations involving fundamental types of
nonlinearity that can be adopted for the description of many kinds of
phenomena [3, 4, 5, 6, 7]. One of the most celebrated playgrounds for soliton
dynamics is the one-dimensional sine-Gordon (SG) equation, which constitutes a
topic of never-ending interest for a broad spectrum of researchers [8]. Such a
model, whose mechanical equivalent is a chain of pendula linked with linear
springs [9], is indeed encountered in all sorts of scientific domains,
including fundamental and applied superconductivity, crystal dislocations,
ultra-short optical pulses, elementary particles, biology, geology, and
gravity [10, 11, 12, 8].

In the SG landscape, two basic building blocks of the solitonic spectrum can
be identified: kinks/antikinks and breathers [5, 6]. Kinks (antikinks) are
step-like structures with positive (negative) topological charge. Interactions
among pairs of kinks are possible and, understandably, they are repulsive in
the kink-kink case and attractive in the kink-antikink one. The latter fact is
responsible for the emergence of topologically neutral bound states, known as
breathers, between kinks and antikinks. Breathers are exponentially localized
in space and they ‘breathe’ (i.e., oscillate) in time. Both theoretical and
experimental issues related to the latter excitations, which represent the
main concern of the below discussions, are of great importance in a
significant number of scientific works. In particular, extensive
investigations have been performed on SG breathers in various perturbed
frameworks [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], interacting-breather
scenarios [24, 25, 26, 27], biological systems [11, 28], cuprate
superconductors [29, 30], geological media [31, 32], graphene-based structures
[33], magnetic systems [34], liquid crystals [35, 36], network models [37,
38], self-induced transparency media [39], and long Josephson junctions (LJJs)
[40, 41, 42, 43, 44, 45, 46]. Note that SG breathers are also found in two
dimensions [47, 48, 49] and that long-lived breather excitations can exist
even in modified SG models [50, 51, 52, 53].

Here, a study on the phenomenon of breather formation is presented in the
noisy and ac-driven SG system with loss. Stochastic and sinusoidal forcing can
indeed cooperatively lead to the emergence of long-time stable breathers in
random locations [45]. The manifestation of such events is investigated, for
both Neumann and periodic boundary conditions (NBC and PBC, respectively), at
different values of the main system parameters, such as the noise strength and
the ac frequency-amplitude combination. Interestingly, nonmonotonic behaviors
of the probability of generating exclusively breathers are obtained as a
function of the noise intensity. This fact highlights the existence of optimal
noise strength ranges for the above mechanism to occur. Once the most
favorable scenarios for exciting breathers are identified, the statistics of
their number, position, and amplitude are analyzed, and the following overall
features are observed. The number of breathers grows, on average, with the
noise strength. The breathers’ spatial distribution displays sharp peaks at
the system’s edges for NBC, whereas it is essentially uniform for PBC. This
qualitative difference is due to the existence of ‘edge-breather’ modes [54,
55, 14], peculiar to the former boundary condition (BC) class, providing an
energetic advantage with respect to their ‘bulk’ counterparts. The average
amplitude of the breather oscillations is dictated by the ac frequency-
amplitude combination. More specifically, the typical breather amplitude is
not quite that predicted by the unperturbed SG theory, but its value is
readily understood within the damped-driven scenario. Lastly, the effect of
the system size on the generation mechanism is analyzed, and a minimal length
for its occurrence is found. This ‘critical’ length is roughly given by the
characteristic breather half-width (width) in NBC (PBC). The response’s
sensitivity to the BC type is, once again, explained by edge-breathers, which
require half of the space than bulk excitations.

Direct applications of these findings include, but are not limited to, the
notorious open problem of the breather’s experimental observation in LJJs [41,
43, 44, 45, 46]. The considered generation technique was indeed recently
employed for devising two detection schemes: (i) a ‘destructive’ protocol in
which the breather’s properties are embedded into the resistive switching
statistics of the device [45]; (ii) a ‘non-destructive’ approach exploiting
the fact that a breather enhances the heat transport in a thermally biased LJJ
[46].

This work is structured as follows. Section 2 describes the SG model, along
with both its fundamental solitonic solutions in the unperturbed case and its
physical interpretation in the context of LJJs. The results are illustrated in
Sec. 3 and, finally, conclusions are drawn in Sec. 4.

##  2 Materials and methods

The analysis is concerned with the dissipative, harmonically-driven SG system
in the presence of thermal noise. In dimensionless units, the equation for the
field ${\varphi(x,t)}$ reads [5, 6]

$\varphi_{xx}-\varphi_{tt}-\alpha\varphi_{t}=\sin\varphi-\eta\sin(\omega
t)-\gamma_{T}(x,t),\;\;x,t\in\left[-\frac{l}{2},\frac{l}{2}\right]\times\left[0,t_{\rm{sim}}\right].$
(1)

Here, the notation ${\partial\varphi/\partial x=\varphi_{x}}$ is adopted for
the partial derivatives, ${\alpha}$ is the dissipation coefficient, ${\omega}$
and ${\eta}$ are, respectively, the frequency and the amplitude of the
external force, ${\gamma_{T}(x,t)}$ is the zero-average Gaussian noise source,
delta-correlated both in ${x}$ and ${t}$, with intensity ${2\alpha\Gamma}$,
${l}$ is the system length, and ${t_{\rm{sim}}}$ is the simulation time.

Flat initial conditions are considered throughout the work for Eq. (1)

$\varphi(x,0)=\varphi_{t}(x,0)=0,$ (2)

along with either NBC

$\varphi_{x}(-l/2,t)=\varphi_{x}(l/2,t)=0$ (3)

or PBC

$\varphi(-l/2,t)=\varphi(l/2,t).$ (4)

To properly set the stage for the discussions below, it is perhaps useful
recalling that the perturbation-free version of Eq. (1) (i.e., with
${\alpha=\eta=\Gamma=0}$) is satisfied, for ${l=\infty}$, by the elementary
topological solitons [5, 6]

$\varphi_{\pm}(x,t)=4\arctan\left[\exp\left(\pm\frac{x-vt}{\sqrt{1-v^{2}}}\right)\right]$
(5)

commonly labelled as kinks (${\varphi_{+}}$) and antikinks (${\varphi_{-}}$).
The nonlinear waves described by Eq. (5) travel with constant velocity
${v<1}$, and also display relativistic behavior. Within this context, the
breather arises as a localized, oscillating bound state between a kink and an
antikink, and its waveform [5, 6]

$\varphi_{b}(x,t)=4\arctan\left\\{\frac{\sqrt{1-\omega_{b}^{2}}}{\omega_{b}}\frac{\sin\left[\frac{\omega_{b}\left(t-v_{e}x\right)}{\sqrt{1-v_{e}^{2}}}\right]}{\cosh\left[\frac{\sqrt{1-\omega_{b}^{2}}\left(x-v_{e}t\right)}{\sqrt{1-v_{e}^{2}}}\right]}\right\\}$
(6)

can indeed be understood as the analytic continuation of the profile that
features a colliding kink-antikink pair. In Eq. (6), ${0<\omega_{b}<1}$ and
${v_{e}<1}$ are, respectively, the excitation’s proper frequency and the
envelope velocity. Note that the parameter ${\omega_{b}}$ yields, for
${v_{e}=0}$, the oscillation period ${T_{b}=2\pi/\omega_{b}}$, the energy
${E_{b}=16\sqrt{1-\omega_{b}^{2}}}$ 111The energy is obtained by integrating,
over the whole spatial domain, the energy density
${\varepsilon(x,t)=(\varphi_{t}^{2}+\varphi_{x}^{2})/2+1-\cos\varphi}$
associated to the unperturbed SG equation, for ${\varphi=\varphi_{b}}$., the
amplitude ${A_{b}=4\arctan\left(\sqrt{1-\omega_{b}^{2}}/\omega_{b}\right)}$,
and the characteristic length ${\lambda_{b}\sim 1/\sqrt{1-\omega_{b}^{2}}}$ of
the nonlinear mode. Thus, low (high) frequencies and high (low) oscillation
amplitudes correspond to high (low) energy breathers.

As mentioned in Sec. 1, the SG framework is successfully employed in a wide
range of scientific contexts. The LJJ stands out among the latter as a
prototypal physical system in condensed matter for exploring the rich SG
phenomenology. In this regard, Eq. (1) accounts for quasiparticle dissipation,
a spatially-uniform alternating (ac) current, and thermal fluctuations, and it
accurately reproduces the behavior of the driven superconducting device. The
field ${\varphi(x,t)}$ here represents the Josephson phase, i.e., the phase
difference between the macroscopic wave functions relative to the two
superconductors. The characteristic scales are given by the Josephson
penetration depth ${\lambda_{J}=\sqrt{\Phi_{0}/\left(2\pi J_{c}L_{P}\right)}}$
(${l=L/\lambda_{J}}$ is the normalized junction length) and the Josephson
plasma frequency ${\omega_{p}=\sqrt{2\pi J_{c}/\left(\Phi_{0}C\right)}}$ [56,
57], where ${\Phi_{0}}$ is the magnetic flux quantum, ${J_{c}}$ is the
critical current per unit length, ${L_{P}}$ is the inductance per unit length,
and ${C}$ is the capacitance per unit length. The dissipation constant reads
${\alpha=G/\left(\omega_{p}C\right)}$ ($G$ is the effective normal
conductance), ${\omega}$ (${\eta}$) is given in units of ${\omega_{p}}$
($J_{c}$), and the noise strength is ${\Gamma=2ek_{B}T/\left(\hbar
J_{c}\lambda_{J}\right)}$, in which ${e}$ is the electron charge, ${k_{B}}$ is
the Boltzmann constant, ${T}$ is the absolute temperature, and ${\hbar}$ is
the reduced Planck constant. Physically, Eqs. (5)-(6) (more precisely, their
perturbed counterparts) describe magnetic energy packets in the junction [56,
57], and thus the above initial condition [Eq. (2)] corresponds to an
excitation-free initial state, whereas the BC specifies the device’s geometry
[8]: overlap for NBC [Eq. (3)] and annular for PBC [Eq. (4)], see Fig. 1.
Various LJJ geometries have been studied over the years [58, 59, 8], and the
two considered here are, arguably, the most popular.

Figure 1: Pictorial view of the LJJ geometries considered in the work. Panel
(a), i.e., the overlap case, corresponds to NBC [Eq. (3)], while panel (b),
i.e., the annular case, to PBC [Eq. (4)]. Single magnetic flux quanta
${\Phi_{0}}$, mathematically described by kinks, are sketched as red arrows.
The symbols ${L}$ and ${W}$ indicate, respectively, the length and the width
of the device.

Equation (1) is numerically solved by means of an implicit finite-difference
algorithm, see Refs. [23, 45] for a detailed account on the matter. Throughout
the paper, the discretization steps are ${\Delta x=\Delta t=0.01}$, the system
length is ${l=50}$ unless otherwise specified, the simulation time is
${t_{\rm{sim}}=500}$, the damping parameter is ${\alpha=0.2}$ [45], and the
forcing frequency is below-plasma, i.e., ${\omega<1}$, see Eq. (6).

##  3 Results and discussion

### Noise-induced, ac-locked breathers

As first discussed in Ref. [45] for the PBC case, the combined action of
thermal fluctuations and ac forcing can lead to the emergence, in random
locations, of breathers surviving way beyond their radiative decay lifetime of
${\sim 1/\alpha=5}$ [23]. More specifically, the excited modes are resonant
with the external current source, their morphology closely resembles that of
analytical breathers, e.g., frequency-dependent width and amplitude are
exhibited in analogy with Eq. (6), and they also present a noteworthy
stability with respect to position. Various questions on the statistics
underlying this interesting phenomenon naturally arise and have yet to be
quantitatively addressed: at given ${\omega}$, ${\eta}$, ${\Gamma}$, and BC
type, how likely is one to observe ${n}$ breathers in a particular
realization? How likely is a breather to be found at location ${x}$? What is
the average breather amplitude? Do any of these features depend on the noise
strength ${\Gamma}$? Moreover, Ref. [45] mainly focuses on a single ac
frequency-amplitude combination and PBC. Thus one may wonder how the latter
choices influence the overall picture. A better understanding of these topics
can surely be useful, especially in view of the long-sought experimental
breather detection in LJJs, see Sec. 1.

Ideally, one would address the above queries by exploring a comprehensive set
of physical scenarios, but this is rather impractical from a computational
viewpoint. For each ac frequency-amplitude pair and BC type, a large enough
set of runs must indeed be performed at different noise intensities for the
results to be statistically relevant. Equation (1) also has to be simulated
for a sufficiently long time to let the breather generation events to unfold.
In view of the previous facts, the choice of focusing on the medium-to-high
breather frequency range is made by taking the combinations ${\omega=0.6}$,
${\eta=0.59}$ and ${\omega=0.7}$, ${\eta=0.44}$ 222The amplitude ${\eta}$
paired to each ${\omega}$ value is the lowest at which the phenomenon of
interest is observed. The exclusive generation of breathers is usually
possible for a range of ac amplitudes above the analyzed ones, see Fig. 3 in
Ref. [45], but it is reasonable to first consider the cases closer to the
unperturbed SG limit.. For those frequency-amplitude pairs, both in NBC and
PBC, ensembles of ${N=5000}$ numerical experiments are examined at different
${\Gamma}$ values.

Figure 2: Four different numerical realizations. In each panel, the main plot
shows the spatio-temporal evolution of ${\varphi}$, whereas the inset
corresponds to the phase snapshot ${\varphi(x,t^{\star})}$ (the instant
${t=t^{\star}}$ is indicated with a dashed green horizontal line in the
contour graph), which is used to analyze the number, position, and amplitude
of the excited breathers, see the circles/squares and dashed vertical lines in
black. The results on the left are obtained with ${\omega=0.6}$,
${\eta=0.59}$, and ${\Gamma=0.003}$, whereas those on the right with
${\omega=0.7}$, ${\eta=0.44}$, and ${\Gamma=0.01}$. Furthermore, each panel
features a sketch of the LJJ geometry relative to the BC imposed in the
simulation. Shared parameter values: ${\Delta x=\Delta t=0.01}$, ${l=50}$,
${t_{\rm{sim}}=500}$, and ${\alpha=0.2}$.

To operatively illustrate how the analysis works, it is worth starting from
some examples. Figure 2 displays four runs: the two on the left are obtained
with ${\omega=0.6}$, ${\eta=0.59}$, and ${\Gamma=0.003}$, whereas those on the
right with ${\omega=0.7}$, ${\eta=0.44}$, and ${\Gamma=0.01}$. Each panel also
features a sketch of the LJJ geometry relative to the BC imposed in that
specific simulation. The distinctive traits of the phenomenon, summarized
above, can be identified in all the contour plots, which present the spatio-
temporal evolution of the phase ${\varphi}$. In particular, one or more
breathers are seen to emerge in random spots within a few driving cycles, and
they persist up to ${t_{\rm{sim}}=500}$ 333As one appreciates from Fig. 1 in
Ref. [45], the nonlinear modes can survive even for ${t>500}$.. A peculiar
feature of NBC, already evident in Fig. 2(a), is that single kinks/antikinks
can form bound states with virtual companions at the boundaries. These edge-
breathers, which can also be encountered in the earlier SG literature [54, 55,
14], are intuitively expected to be more likely than the bulk counterparts due
to their reduced energy cost.

The focus of the present study is exclusively on the cases where, similarly to
Fig. 2(a)-(c), breathers are the only emerging excitations. The devised
strategy for analyzing these runs is to focus on a suitable instant
${t=t^{\star}\lesssim t_{\rm{sim}}}$ and apply a peak detection algorithm
444Scipy’s signal.find_peaks is employed here. Via extensitve preliminary
testing, the routine’s prominence parameter is tuned to capture breather-like
modes very reliably. to the profile ${\varphi(x,t^{\star})}$, see the insets
in Fig. 2. Such an approach takes advantage of the fact that the excited
breathers share the same oscillation cycle, locked to the ac force, in every
simulation. If this were not true, it clearly would not be possible, in
general, to measure all the breather amplitudes [see, e.g., the black
circles/squares in Fig. 2] through a single ${\varphi}$ snapshot. Moreover, an
underlying assumption is that the configuration at ${t^{\star}\lesssim
t_{\rm{sim}}}$ is somewhat representative of the realization. This is fairly
true, especially for the moderate noise amplitude values considered for the
statistical analysis. The observed features are indeed mostly unaltered, after
the occurrence of the generation events over a rather short transient, for
hundreds of breathing periods. For example, the black dashed vertical lines in
Fig. 2 indicate the detected breather positions at ${t=t^{\star}}$, and they
are consistent with the entire simulated evolution. One last remark concerns
the time ${t^{\star}}$, which is operatively chosen, within the final two
driving cycles of each run, as the instant characterized by the maximal phase
excursion, see the dashed green horizontal lines in the contour plots in Fig.
2.

Figure 3: Probabilities of having no excitations (${\mathcal{P}_{a}}$,
purple), breathers only (${\mathcal{P}_{b}}$, blue), and at least one kink-
type excitation (${\mathcal{P}_{c}}$, green) versus ${\Gamma}$. The results on
the left (circles) are obtained with ${\omega=0.6}$ and ${\eta=0.59}$, whereas
those on the right (squares) with ${\omega=0.7}$ and ${\eta=0.44}$. Each panel
features a sketch of the LJJ geometry relative to the BC imposed in the
simulations. Also, the gray shaded areas denote the ranges of noise strengths
selected for the statistical analysis: ${\Gamma\in[0.001,0.01]}$ for panels
(a) and (c); ${\Gamma\in[0.007,0.04]}$ for panels (b) and (d). Shared
parameter values: ${\Delta x=\Delta t=0.01}$, ${l=50}$, ${t_{\rm{sim}}=500}$,
${\alpha=0.2}$, and ${N=5000}$.

Reasonable ranges of the ${\Gamma}$ parameter for the statistical analysis
should involve a high fraction of ‘breathers only’ cases. To this end, the
probability ${\mathcal{P}_{b}}$ of generating exclusively breathers is
evaluated as a function of the noise strength, along with the probability of
having no excitations ${\mathcal{P}_{a}}$, and that of exciting at least one
kink-type structure ${\mathcal{P}_{c}}$. As illustrated in Fig. 3, whose left
panels [(a) and (c), see the circles] are obtained with ${\omega=0.6}$ and
${\eta=0.59}$, whereas those on the right [(b) and (d), see the squares] with
${\omega=0.7}$ and ${\eta=0.44}$, the behavior is quite similar for NBC and
PBC. For the smallest noise intensities, the probability ${\mathcal{P}_{a}}$
of having no excitations is ${1}$, see the purple curves. For intermediate
${\Gamma}$ values, ${\mathcal{P}_{b}}$ peaks at ${\gtrsim 0.9}$, see the blue
curves. A further increase of the thermal fluctuations leads to less coherent
breathers, and kink-type structures begin to take over, as indicated by the
${\mathcal{P}_{c}}$ (green) curves. It may be worth observing that, in both
the left and right panels of Fig. 3, for NBC the probability
${\mathcal{P}_{b}}$ rises (falls) a bit faster than for PBC, at lower (higher)
noise intensities, a fact which is likely due to the existence of the easier-
to-create (easier-to-destroy) edge-breathers in the former BC type. Another
relevant aspect of Fig. 3 is that the ${\mathcal{P}_{b}(\Gamma)}$ curves at
${\omega=0.6}$ and ${\eta=0.59}$, see the blue circles, reach their peak
earlier than their counterparts at ${\omega=0.7}$ and ${\eta=0.44}$, see the
blue squares. Based on these findings, the below investigation takes place in
the gray shaded ${\Gamma}$ regions in Fig. 3(a)-(c), which extend roughly over
one order of magnitude around the ${\mathcal{P}_{b}(\Gamma)}$ plateaus,
leaving out the noise intensities associated with the transition towards the
regime of significant thermal agitation (signaled by ${\mathcal{P}_{c}}$’s
growth). More quantitatively, for ${\omega=0.6}$ and ${\eta=0.59}$
(${\omega=0.7}$ and ${\eta=0.44}$) the highlighted area is
${\Gamma\in[0.001,0.01]}$ (${\Gamma\in[0.007,0.04]}$).

Figure 4: Main plots: PDFs of the number of breathers ${n}$ observed at
different noise strengths. Insets: Averages of the breather number,
${\left\langle n\right\rangle}$, as a function of ${\Gamma}$ (which takes
values in the same sets as in the corresponding main panels). The results on
the left (circles) are obtained with ${\omega=0.6}$ and ${\eta=0.59}$, whereas
those on the right (squares) with ${\omega=0.7}$ and ${\eta=0.44}$. Each panel
also features a sketch of the LJJ geometry relative to the BC imposed in the
simulations. Shared parameter values: ${\Delta x=\Delta t=0.01}$, ${l=50}$,
${t_{\rm{sim}}=500}$, ${\alpha=0.2}$, and ${N=5000}$.

### Breather statistics

Figure 4’s main plots present the probability density functions (PDFs) of the
breather number ${n}$ for different noise strengths, while the insets show the
average breather numbers ${\left\langle n(\Gamma)\right\rangle}$. Like before,
panels (a) and (c) [(b) and (d)] are obtained with ${\omega=0.6}$ and
${\eta=0.59}$ [${\omega=0.7}$ and ${\eta=0.44}$]. Upon increase of the noise
intensity, all the PDF peaks are seen to shift from ${n=1}$ towards higher
${n}$ values, i.e., it gradually becomes more probable to observe several
breathers in each run, see the circles/squares. Generally speaking, PBC seem
characterized by less breathers than NBC, as one appreciates by comparing
panels (a) and (b) with panels (c) and (d), respectively. This information can
also be deduced from the insets, where monotonically growing profiles of the
average breather number are displayed. Overall, it should be clear from Fig. 4
that the noise amplitude represents an useful control parameter, since it
allows to tune the quantity ${\left\langle n\right\rangle}$.

Figure 5: Breather counts as a function of the position ${x}$, for different
noise strengths ${\Gamma}$. The results on the left (circles) are obtained
with ${\omega=0.6}$ and ${\eta=0.59}$, whereas those on the right (squares)
with ${\omega=0.7}$ and ${\eta=0.44}$. The legends in panels (a) and (b) also
apply to panels (c) and (d), respectively. The gray shaded areas indicate the
ranges ${[100,450]}$ [panels (a) and (c)] and ${[100,700]}$ [panels (b) and
(d)]. Furthermore, each plot features a sketch of the LJJ geometry relative to
the BC imposed in the simulations. Shared parameter values: ${\Delta x=\Delta
t=0.01}$, ${l=50}$, ${t_{\rm{sim}}=500}$, ${\alpha=0.2}$, and ${N=5000}$.

Another topic of interest, that is, the likelihood of a breather being found
at a given position, is addressed in Fig. 5. More precisely, the plots
illustrate the breather counts versus ${x}$, for different noise strengths,
with ${\omega=0.6}$ and ${\eta=0.59}$ (left panels, circles) and
${\omega=0.7}$ and ${\eta=0.44}$ (right panels, squares). For both frequency-
amplitude pairs, a striking difference between the upper (NBC) and lower (PBC)
results emerges. The NBC case indeed features distributions that are sharply
peaked at the boundaries, whereas the PBC counterparts are essentially
uniform. These outcomes are well understood: being essentially single-soliton
modes, edge-breathers can be excited more easily for NBC than their bulk
analogues, while the absence of preferred locations for the generation events
is expected for PBC. Concerning the NBC panels, note also that, for
${|x|\lesssim l/2}$, regions with near-zero breather counts extend over the
typical modes’ width due to boundary interactions; on the other hand, a
behavior similar to PBC is recovered for ${|x|\ll l/2}$, as seen by direct
comparison with the visual aid of the gray shaded areas. Figure 5 allows one
to conclude that the BC type significatively impacts on the spatial
distribution of breathers 555Furthermore, Fig. 5 explains the systematic
difference in the averages of the breather number ${n}$ between NBC and PBC,
see Fig. 4.. Instead, changing the value of the noise strength ${\Gamma}$
seems to play a role mainly in shifting the distributions’ average.

Figure 6: Averages of the breather amplitude, ${\left\langle A\right\rangle}$,
and their standard deviations (shaded areas), as a function of ${\Gamma}$. The
results on the left (circles) are obtained with ${\omega=0.6}$ and
${\eta=0.59}$, whereas those on the right (squares) with ${\omega=0.7}$ and
${\eta=0.44}$. The dashed green horizontal lines denote the analytical
breather amplitudes, see Eq. (6), at ${\omega=0.6}$ [panels (a) and (c)] and
at ${\omega=0.7}$ [panels (b) and (d)]. The dashed red horizontal lines
indicate, instead, the amplitudes observed by simulating the noiseless damped-
driven system [${\omega=0.6}$ and ${\eta=0.59}$ for panels (a) and (c),
${\omega=0.7}$ and ${\eta=0.44}$ for panels (b) and (d)], with ad-hoc
analytical breather initial conditions. Each plot also features a sketch of
the LJJ geometry relative to the BC imposed in the numerical experiments.
Shared parameter values: ${\Delta x=\Delta t=0.01}$, ${l=50}$,
${t_{\rm{sim}}=500}$, ${\alpha=0.2}$, and ${N=5000}$.

Consider now the averages of the breather amplitude ${\left\langle
A\right\rangle}$, which are plotted in Fig. 6 (circles and squares), along
with their standard deviations (shaded areas), as a function of the noise
intensity. Here, the same qualitative trends are obtained in all cases. The
${\left\langle A(\Gamma)\right\rangle}$ curves are indeed almost constant, and
the corresponding standard deviations gradually increase with ${\Gamma}$. The
fact that panels (a) and (b) are practically identical, respectively, to
panels (c) and (d) suggests that the BC are not very influential on the
average breather amplitude. The latter quantity turns out to be dictated by
the frequency-amplitude pair, as shown by the systematic difference between
the simulation results on the left and those on the right 666This feature can
be appreciated also in the plots of Fig. 2, see the insets.. Note that the
observed ${\left\langle A\right\rangle}$ values are higher than their
unperturbed counterparts, i.e., the analytical breather amplitudes calculated
via Eq. (6) and denoted by the dashed green horizontal lines, see Fig. 6. A
very close match for the ensemble averages is found with the amplitudes
extracted from separate numerical experiments, with ad-hoc analytical breather
initial conditions, in the noise-free (${\Gamma=0}$) damped-driven scenario,
see the dashed red horizontal lines. In other words, the typical amplitudes
exhibited by noise-induced breathers are nearly equal to those at which exact
breathers spontaneously relax in the presence of dissipative and ac
perturbations [${\omega=0.6}$ and ${\eta=0.59}$ for panels (a) and (c),
${\omega=0.7}$ and ${\eta=0.44}$ for panels (b) and (d)]. Finally, in regard
to the standard deviations displayed in Fig. 6, more scattered breather
amplitude values constitute a reasonable outcome in the presence of a
progressively larger amount of noise.

### Size analysis

One last aspect that we wish to highlight is the effect of the system size
${l}$ on the breathers’ emergence mechanism. In particular, one might wonder
whether a minimum junction length exists for the generation events to occur.
To this end, parameter sets corresponding to peaks of the breather-only
probability ${\mathcal{P}_{b}}$, see Fig. 3, are now chosen: (i)
${\omega=0.6}$, ${\eta=0.59}$, and ${\Gamma=0.003}$; (ii) ${\omega=0.7}$,
${\eta=0.44}$, and ${\Gamma=0.01}$. For each of the latter, both in NBC and
PBC, ensembles of ${N=5000}$ runs are performed at different sizes
${l\in\left[1,32\right]}$.

Figure 7: Ensemble averages of the spatial standard deviation
${\bar{\sigma}_{\varphi}}$, defined in Eq. (7), as a function of the size
${l}$. The results on the left (circles) are obtained with ${\omega=0.6}$,
${\eta=0.59}$, and ${\Gamma=0.003}$, whereas those on the right (squares) with
${\omega=0.7}$, ${\eta=0.44}$, and ${\Gamma=0.01}$. The dashed red vertical
lines denote the breather half-widths, computed via Eq. (6), at ${\omega=0.6}$
[panel (a)] and at ${\omega=0.7}$ [panel (b)]. The dashed blue vertical lines
indicate the breather widths at ${\omega=0.6}$ [panel (c)] and at
${\omega=0.7}$ [panel (d)]. Each plot also features a sketch of the LJJ
geometry relative to the BC imposed in the numerical experiments. Shared
parameter values: ${\Delta x=\Delta t=0.01}$, ${t_{\rm{sim}}=500}$,
${\alpha=0.2}$, ${N=5000}$, and ${t_{0}=250}$.

To efficiently characterize the response at different ${l}$ values, the
following time-averaged spatial standard deviation of the phase is introduced

$\bar{\sigma}_{\varphi}=\frac{1}{t_{\rm{sim}}-t_{0}}\int_{t_{0}}^{t_{\rm{sim}}}dt\;\sigma_{\varphi}(t),$
(7)

with ${t_{0}=250}$ being a fixed time instant 777Different values have been
extensively tested, but the results presented in Fig. 7 show essentially no
dependence on the choice of ${t_{0}<t_{\rm{sim}}}$ within reasonable limits.,
and ${\sigma_{\varphi}(t)}$ the standard deviation of the instantaneous curve
${\varphi(x,t)}$. The quantity considered in Eq. (7) is intuitively near-zero
in situations free of solitonic excitations, i.e., for phase profiles close to
spatially flat. By contrast, whenever ${\varphi}$ is inhomogeneous in space
due to the presence of persistent breathers, ${\bar{\sigma}_{\varphi}}$
assumes larger values.

Ensemble averages of ${\bar{\sigma}_{\varphi}}$, over ${N=5000}$ simulations,
are shown versus the size ${l}$ in Fig. 7. The results on the left (circles)
are obtained with ${\omega=0.6}$, ${\eta=0.59}$, and ${\Gamma=0.003}$, while
those on the right (squares) with ${\omega=0.7}$, ${\eta=0.44}$, and
${\Gamma=0.01}$. Notably, all the
${\left\langle\bar{\sigma}_{\varphi}(l)\right\rangle}$ curves display rather
sharp transitions at precise system lengths. By comparing panels (a) and (b)
with, respectively, panels (c) and (d), one immediately sees that the critical
${l}$ value for NBC is roughly half of that in the corresponding PBC case.
This behavior is ultimately due to the fact that a junction must be long
enough to fit a breather for its emergence to be possible. For PBC the
transition occurs when ${l}$ is roughly equal to the characteristic breather
width, which can be evaluated via Eq. (6), see dashed blue vertical lines in
panels (c) and (d). On the other hand, edge-breathers are allowed in NBC, and
these modes require half of the space compared to their bulk counterparts. The
NBC critical length is then, quite reasonably, very close to the
characteristic breather half-width, see the dashed red vertical lines in
panels (a) and (b).

##  4 Conclusions

This study is concerned with the emergence of breathers, for both NBC and PBC,
in the damped and ac-driven SG equation in the presence of thermal noise.
Different ac frequency-amplitude pairs are examined, and the probability of
breather-only generation is seen to behave nonmonotonically versus the noise
intensity. Within the most favorable scenarios for exciting breathers, the
statistics of their number, position, and amplitude are addressed. In
particular, increasing the noise strength makes the number of breathers grow
on average. The breather distribution in space presents, for NBC, pronounced
peaks at the boundaries, while, for PBC, it is approximately flat. The reason
behind such a difference is the edge-breather mode, peculiar to NBC, that is
easier to excite than its bulk counterpart. Away from the edges, a similar
behavior for NBC and PBC is recovered, as one may imagine. The average
amplitude at which the breather oscillations occur is determined by the ac
frequency-amplitude pair, a fact that is well-understood in terms of the
damped-driven SG scenario. Finally, the analysis looks at the influence of the
system length on the overall picture. The existence of minimal length for
generating breathers, which is roughly equal to the breather half-width
(width) in NBC (PBC), is highlighted. Again, edge-breathers, which occupy half
of the space compared to bulk modes, explain the results’ sensitivity to the
different BC.

The above findings are clearly useful for, e.g., devising an experimental
configuration suitable for a breather detection scheme in LJJs [41, 43, 44,
45, 46]. The provided information on the effects of the system’s main
parameters, geometry, and size on the generation mechanism allows for
tailoring a source of remarkably stable breathers. Quantities such as the
average number of excited breathers are indeed shown to be controllable in
conceptually simple ways, for example, by changing the noise strength.

## Acknowledgments

The majority of the above numerical runs were performed on CINECA’s machine
Galileo100, Project: IscrC_NDJB. The authors acknowledge the support of the
Italian Ministry of University and Research (MUR).

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