# Drift instabilities in localised Faraday patterns

©Juan F. Marín<EMAIL_ADDRESS>Departamento de Física, Universidad de
Santiago de Chile, Usach, Av. Víctor Jara 3493, Estación Central, Santiago,
Chile.  Rafael Riveros Ávila  Instituto de Física, Pontificia Universidad
Católica de Valparaíso, Casilla 4059, Chile  Saliya Coulibaly  Université de
Lille, CNRS, UMR 8523, Laboratoire de Physique des Lasers Atomes et Molécules,
F-59000 Lille, France  Majid Taki  Université de Lille, CNRS, UMR 8523,
Laboratoire de Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Mónica A. García-Ñustes  Instituto de Física, Pontificia Universidad Católica
de Valparaíso, Casilla 4059, Chile

###### Abstract

Nature is intrinsically heterogeneous, and remarkable phenomena can only be
observed in the presence of intrinsically nonlinear heterogeneities.
Spontaneous pattern formation in nature has fascinated humankind for
centuries, and the understanding of the underlying symmetry-breaking
instabilities has been of longstanding scientific interest. In this article,
we provide theoretical and experimental evidence that heterogeneities can
generate convection (drift instabilities) in the amplitude of localised
patterns. We derive a minimal theoretical model describing the growth of
localised Faraday patterns under heterogeneous parametric drive, unveiling the
presence of symmetry-breaking nonlinear gradients. The model reveals new
dynamics in the phase of the underlying patterns, exhibiting convective
instabilities when the system crosses a secondary bifurcation point. We
discuss the impact of our results in the understanding of convective
instabilities induced by heterogeneities in generic nonlinear extended systems
far from equilibrium.

###### pacs:

05.45.Yv 05.45.-a, 89.75.Kd

##  I Introduction

Patterns are ubiquitous in non-equilibrium systems Cross and Greenside (2009).
They appear in very different systems, such as fluids, chemical reactions,
magnetic textures, nonlinear optics, and granular layers Pismen (2006); Miles
(1984); Kenig  _et al._ (2009); Périnet  _et al._ (2009); Clerc  _et al._
(2010). A paradigmatic toy-model for the study of pattern formation in
extended non-equilibrium systems is the wave-envelope description in the
continuum limit of a chain of coupled nonlinear dissipative oscillators, such
as the torsion pendula depicted in Fig. 1(a). Besides its simplicity, such
system has proven to give profound insights into the generic phenomenon of
pattern formation. A vertically-oscillating drive $f(t)=f_{0}\cos\Omega t$
forcing the common axis of the pendula has the effect of modulating the
effective gravity– a parameter of the system –, and when the amplitude of such
driving is above certain threshold value the system undergoes the celebrated
parametrical instability Landau and Lifshitz (2013). Above the threshold of
instability, the homogeneous envelope of a plane wave turns unstable and
become modulated due to the development of weakly nonlinear harmonics– see the
insets below Fig. 1(a). One observes in the envelope the spontaneous formation
of a regular pattern oscillating with a well defined wavelength and a sub-
harmonic response, i. e. oscillating with frequency $\omega=\Omega/2$. These
patterns are known as Faraday patterns, and were first reported by Faraday in
his seminal experimental work on vertically vibrated fluids and granular
layers Faraday (1831a, b). Just above the threshold of instability, the
amplitude of the pattern grows with a slow characteristic time until
saturation. Saturation is provided by a double dynamical compensation: one
between nonlinearity and dispersion, and another between dissipation and the
parametric drive. In such an envelope description, taking the continuum limit
in the weakly nonlinear analysis of the system of coupled pendula, one obtains
that the amplitude is governed by the following parametrically driven
nonlinear Schrödinger (pdnlS) equationAlexeeva  _et al._ (2000),

$\partial_{t}\psi=-(\mu+i\nu)\psi-i\alpha\partial_{xx}\psi-i|\psi|^{2}\psi+\gamma(x)\psi^{*},$
(1)

which is written in dimensionless form. In this model, pattern dynamics is
described through a complex order parameter $\psi$ that depends on space and
time. Parameter $\mu$ is the dissipation, $\alpha$ is the diffraction length,
$\nu:=\Omega/2-\omega_{o}$ is the detuning between the sub-harmonic frequency
of the generated Faraday wave and the natural frequency $\omega_{o}$ of the
system, and $\gamma_{o}:=f_{o}/4\omega_{o}$ is the normalised amplitude of the
drive (pump).

Figure 1: (a) Schematic of a system of coupled pendula under parametrical
drive as a model to study pattern formation phenomena. The envelope of linear
plane waves becomes modulated as nonlinear harmonics grows in time, generating
a pattern with a characteristic wavelength and frequency. (b) Normalised
amplitudes $|\bar{\psi}|$ and $|\bar{C}|$ obtained numerically from Eq. (1)
and Eq. (4), respectively, with $\mu=0.45$, $\nu=1$, $\alpha=1$, and
$\sigma_{i}=16$. (c) Normalised maximum amplitude of the saturated Faraday
wave, $|\bar{C}_{\max}|$, as a function of $\gamma_{o}$, according to Eq. (9)
(solid line) and the numerical solutions of the pdnlS Eq. (1) (stars). Drift
is observed above a critical pump $\gamma_{D}\simeq 0.837$ indicated with a
vertical dashed line.

If $\gamma=\gamma_{o}=\hbox{constant}$, spatial instabilities occur if
$\gamma>\gamma^{h}_{c}:=\mu$ Miles (1984). In such scenario, the instability
spontaneously grows into a pattern formation for positive values of $\nu$
through a supercritical bifurcation Coullet  _et al._ (1994); Clerc  _et al._
(2009) with leading eigenmodes of wave-number
$\kappa_{c}=\pm\sqrt{\nu/\alpha}$. The effect of a heterogeneous pump in this
system has been recently studied Urra  _et al._ (2019) by introducing the
space-dependent Gaussian profile
$\gamma(x)=\gamma_{o}\exp\left({-x^{2}/2\sigma_{i}^{2}}\right)$, where
$\gamma_{o}$ is the pump strength and $\sigma_{i}$ its extension. The linear
stability analysis under a slightly non-uniform Gaussian drive, i.e.
$\sigma_{i}\ll\sqrt{\alpha}$, shows that the envelope $C_{0}$ of the pattern
has a discrete response given by the Gauss-Hermite polynomials Urra  _et al._
(2019). Urra et al. Urra  _et al._ (2019) showed that the threshold of the
instability of the $m$-th Gauss-Hermite mode is given by

$\gamma_{o}^{(m)}=\mu+(2m+1)\frac{1}{\sigma_{i}}\sqrt{\frac{\nu}{\alpha}}.$
(2)

Thus, localised Faraday patterns appear if $\gamma_{o}>\gamma_{o}^{(0)}$. Near
the fundamental threshold of instability, Faraday patterns are rather regular
and highly ordered both in the real and imaginary parts of $\psi$ as well as
on its phase, $\arg(\psi)$. The amplitude of the pattern increases as we
increase the value of $\gamma_{o}$. However, at some point in the interval
$(\gamma_{o}^{(2)},\gamma_{o}^{(3)})$, a secondary instability emerges and
pattern dynamics become complex both in amplitude and phase. In many cases, a
secondary instability is a precursor to spatiotemporal chaos and transition to
turbulence Zhang and Viñals (1995); Schmid and Henningson (2001).

In this article, we use the normal form theory to formally derive an amplitude
equation for localised Faraday patterns induced by heterogeneous parametric
drive. As will be shown, the evolution of the instabilities at the first order
of nonlinearity is described by a quintic Complex Ginzburg-Landau equation
with Webber-like and nonlocal self-phase modulation terms in the form of
nonlinear gradients. Such nonlinear gradients triggers convection, i.e. _drift
instabilities_ , through a spontaneous nonlinear symmetry breaking above a
secondary bifurcation point. Importantly, such drift is entirely induced by
the heterogeneity of the pump and cannot be observed under homogeneous drive.
Theoretical conditions for drift appearance are reproduced in a fluid
experiment where Faraday waves are induced by a localised bottom motion in a
water cell, providing experimental confirmation of the predicted drift.

##  II Results

###  II.1 Theoretical prediction of convection in localised Faraday waves

We have characterised the dynamics of localised Faraday waves near the
fundamental threshold of instability through Eq. (1) using the method of
normal forms [see Section V: Methods]. It is a widely used formalism in which
a nonlinear system takes the “simplest”– or so called _normal_ – form
preserving the essential features of the system, such as its behaviour near a
bifurcation point Haragus and Iooss (2010). To characterise the dynamics close
to the spatial instability, we introduce a bifurcation parameter
$\delta:=\gamma_{o}-\gamma_{o}^{(0)}$ and a small dimensionless parameter
$\epsilon:=\sqrt{\alpha}/\sigma_{i}\ll 1$ characterising the length of the
injection zone. With the condition $\epsilon\ll 1$ we are assuming that the
injection zone is large compared with the diffraction length, i.e. a weakly
heterogeneous system. We study the behaviour of solutions around a Faraday
wave with critical wavenumber $\kappa_{c}$ introducing a modulating complex
amplitude function $C(x,t)$ – an _order parameter_ –, i.e.

$\psi(x,t)=C(x,t)e^{i\kappa_{c}x}+c.c.+h.o.h.,$ (3)

where $h.o.h$ denotes the high-order harmonics. Below the bifurcation point,
$\gamma_{o}<\gamma_{o}^{(0)}$, one obtains the stable homogeneous solution
$C=0$ (disordered phase). Such homogeneous solution becomes unstable just
above the bifurcation point, and $C(x,t)$ has its own spatiotemporal dynamics
modulating the underlying Faraday pattern (self-organised phase). At the
leading order of non-linearity, one obtains the following governing equation
for the amplitude,

$\partial_{t}C=\frac{2\nu}{\alpha\mu}\partial_{xx}C+\left(\delta-\frac{\mu}{2\sigma_{i}^{2}}x^{2}\right)C-i\frac{2}{\mu}\sqrt{\frac{\nu}{\alpha}}\left[3\partial_{x}(|C|^{2}C)-2C\partial_{x}(|C|^{2})\right]-\frac{9}{2\mu}|C|^{4}C.$
(4)

For non-extended systems, the homogeneous limit of Eq. (4) turns into the
well-known amplitude equation previously derived by Coullet et al. Coullet
_et al._ (1994). The first two terms in the right-hand side of Eq. (4) are
Weber-like terms that give the known linear limit of the heterogeneous system
Urra  _et al._ (2019). At the modulational instability, the amplitude of the
unstable Gauss-Hermite modes is saturated by the quintic nonlinearity given by
the last term in Eq. (4). Importantly, the terms proportional to
$\partial_{x}(|C|^{2}C)$ and $C\partial_{x}(|C|^{2})$ in Eq. (4) are new non-
local terms unveiling self-phase modulation (SPM) effects in the slowly
evolving modes of the localised Faraday patterns. Such terms are commonly
introduced in the nonlinear Schrödinger equation to account for self-
steepening effects and self-frequency shifting via stimulated Raman scattering
Kumar (1990); Porsezian and Nakkeeran (1996); Gedalin  _et al._ (1997);
Nakkeeran  _et al._ (1998); Liu  _et al._ (2009). To our knowledge, this is
the first time a complex quintic Ginzburg-Landau equation with Webber-like
terms has been derived with these non-local terms appearing naturally from the
weakly nonlinear analysis. The appearance of such terms lead us to discover
absolute and drift instabilities in the localised Faraday waves, as we show
below.

In Fig. 1(b) and Fig. 1(c) we compare the predictions of our amplitude
equation and the pdnlS equation. There is a remarkable agreement between both
models near the fundamental threshold of instability. In Fig. 1(b), we solve
numerically Eqs. (1) and (4) with no-flux boundary conditions. The initial
conditions are given by small-amplitude random distributions of values in
space for the real and imaginary parts of $\psi$ and $C$. Figure 1(b) confirms
that the extension $\sigma$ and the envelope profile of the Faraday pattern
are well described by the amplitude equation. As one would expect, predictions
from the amplitude equation show some mismatch from the outcome of the pdnlS
equation away from the fundamental threshold. Figure 1(c) demonstrate that the
amplitude equation (4) well describes the nature of the bifurcation of the
Faraday pattern, as we will see later.

Figure 2: Drift instabilities in the amplitude equation (4). Numerical results
were obtained with $\mu=0.45$, $\nu=1$, $\alpha=1$, and $\sigma_{i}=16$. (a,
b, c) A wave exhibiting absolute stability for $\gamma_{o}=0.80$. (d, e, f)
Convective instability for $\gamma_{o}=1.2$. (g, h, i) A strongly nonlinear
convective instability for $\gamma_{o}=1.62$.

The SPM terms in Eq. (4) are induced by nonlinear gradients, which reveal
underlying complex dynamics in the phase. The gradient terms break the
transverse reflection symmetry ($x\to-x$) and dramatically affect the pattern
selection in the system Louvergneaux  _et al._ (2004); Coulibaly  _et al._
(2008). Thus, the nonlinear gradients in Eq. (4) give a _nonlinear symmetry
breaking_ where convection is characterised by an amplitude-dependence in the
group velocity. In Fig. 2, we demonstrate drift instabilities in the system,
which has been obtained from the numerical simulation of Eq. (4) for the given
values of parameters.

In Fig. 2(a), we show a scenario where amplification of the wave-amplitude
dominates, and the system exhibits absolute stability. In the absence of SPM
terms, the localised Faraday wave spontaneously grows centred around the
Gaussian pump with width $\sigma_{w}=(\sqrt{\nu}\sigma_{i}/\mu)^{1/2}$ Urra
_et al._ (2019). In Fig. 2(a), we indicate with dashed lines the with
$\sigma_{w}$ centred at the Gaussian pump for reference, evidencing a clear
shift in $\mbox{Re}(C)$ due to the SPM terms. Figure 2(b) shows a zoom-in at
the beginning of the simulation, showing that there is indeed a transient
drift towards the left as the pattern grows. However, convection is too weak
to overcome saturation, and the system rapidly becomes absolutely stable.
Although the imaginary part of $C$ exhibits a similar shift as the real part,
the amplitude $|C|$ exhibit no shift and the phase shows no spatiotemporal
dynamics in this case, as shown in Fig. 2(c).

Increasing the value of $\gamma_{o}$, we found for some value
$\gamma_{\text{D}}\in(\gamma_{o}^{(2)},\gamma_{o}^{(3)})$ that the propagation
overcomes amplification and the system exhibits a nonlinear transition to
convective instability. This scenario is shown in Fig. 2(d), evidencing a
drift in the real part of the envelope of Faraday waves. The real and
imaginary parts of $C$ exhibit the same drift, excepting a phase factor.
However, the amplitude $|C|$ of the wave has no drift, as we show in Fig.
2(e). Thus, convection in the system is entirely due to the phase dynamics, as
evidenced in Fig. 2(f).

A further increase in $\gamma_{o}$ increases the drift speed, as shown in Fig.
2(g). The drifting wave also experiences a change in both frequency and
wavelength. These observations confirm the nonlinear nature of the drift
instability. Notice that the drifting pattern becomes non-propagating near the
tails of the Gaussian pump, as highlighted in the zoom-in of Fig. 2(h). This
pinning effect stems from the spatial heterogeneity of the pump, which imposes
a smooth amplitude variation at the tails of the Gaussian inducing a boundary
condition-like. Such kind of boundary conditions induces spatial variations
for the pattern envelope that can become comparable to its modulation
amplitude. This results in the coupling between the slow scale of the envelope
to the fast scale of the modulation of the underlying Faraday pattern, leading
to the pinning of the drifting wave Clerc  _et al._ (2012). Moreover, the
phase dynamics exhibit travelling waves inside the injection zone defined by
the Gaussian parametrical drive, as evidenced in figures 2(f) and 2(i). The
mean speed of such travelling waves decreases as $\gamma_{o}$ decreases.

###  II.2 Drift in the model system

Given that the amplitude equation (4) predicts drift above a secondary
threshold of instability, we consider again the pdnlS model, Eq. (1), to find
out if it exhibits such drift in the underlying Faraday patterns. In the upper
panel of Fig. 3, we show the results from the numerical simulation of the
pdnlS model for the given values of the parameters.

In Fig. 3(a), we have fixed $\gamma_{o}$ slightly below the secondary
instability onset $\gamma_{D}$ and the localised pattern is absolutely stable,
as expected. Notice that the maximum of the wave amplitude is shifted towards
the left of the origin, in accordance with the behaviour observed in Fig.
2(a). Remarkably, if $\gamma_{o}$ is slightly above the threshold
$\gamma_{D}$, the unidirectional drift becomes evident, as shown in Fig. 3(b).
These results evidence the nonlinear symmetry breaking in the system and
confirm the predictions from our normal form (4). Further increasing the pump
$\gamma_{o}$ above the threshold of convective instability, the system evolves
towards a more complex convective state with a superposition of oscillations
(suggesting an underlying Hopf instability) and drift towards both directions
as shown in Fig. 3(c). Remarkably, as we will discuss later in this article,
we have found in a fluid experiment the same drift within the range of
parameters predicted by our normal form, as shown in Fig. 3(d-f).

Figure 3: (Upper panel) Drift instability found in the numerical simulations
of the pdnlS equation (1) with $\mu=0.45$, $\alpha=1$, $\nu=1.0$, and
$\sigma_{i}=16$ for (a) $\gamma_{o}=0.55$, (b) $\gamma_{o}=0.8375$, and (c)
$\gamma_{o}=0.9000$. (Lower panel) Experimental evidence of the drift
instability found in a fluid subject to vertical and localised vibrations. The
amplitude of the vertical acceleration is (d) $\Gamma_{0}=0.56$, (e)
$\Gamma_{0}=0.70$, and (f) $\Gamma_{0}=0.84$. The detected height of the
surface of the fluid is shown in the colour scale.

To gain further insight into the dynamical behaviour of the drifting localised
patterns, we derive an explicit expression for the pattern speed as a function
of parameter $\gamma_{o}$ following a similar procedure as in Ref. Clerc  _et
al._ (2012). We write our amplitude equation Eq. (4) in the polar form by
setting $C=\rho\exp(i\theta)$. After removing the phase factor, the imaginary
part reads

$\partial_{\tau}\theta=4\partial_{XX}\theta+\frac{8}{\rho}\partial_{X}\rho\partial_{X}\theta-20\rho^{2}\partial_{X}\rho,$
(5)

with $X:=x\sqrt{\nu/\alpha}$ and $\tau:=2\mu t$. As depicted in Figs. 2(f) and
2(i), the phase $\theta(x,t)=\arg(C)$ is composed of two different
superimposed dynamical behaviours: (i) a travelling wave with some phase speed
$v=\omega/k$ and (ii) a monotonically increasing one modulo $2\pi$, such that
$\arg(C)\in[-\pi,\,\pi]$ (following the conventional restriction of the
inverse trigonometric functions to the principal branch). Based on this
observation, we propose the ansatz
$\theta(X,\,\tau)=\theta_{v}(X-{\color[rgb]{0,0,1}v}\tau)+\chi(\tau)\mod(2\pi)-\pi$,
where $\theta_{v}$ and $\chi$ account for the travelling and the increasing
dynamics, respectively. Next, let us take the space and time average
$\langle\theta(X,\tau)\rangle_{X}=\sigma_{w}^{-1}\int_{-\sigma_{w}/2}^{\sigma_{w}/2}dX\,\theta(X,\tau)$
and
$\langle\theta(X,\tau^{\prime})\rangle_{\tau}=(\omega/T)\int_{\tau}^{\tau+T}d\tau^{\prime}\,\theta(X,\tau^{\prime})$,
respectively. Close to the drift instability threshold,
$d\langle\theta\rangle_{X}/dt$ has slow dynamics on time, and therefore
$\chi(\tau)\simeq\langle\chi(\tau^{\prime})\rangle_{\tau}$. Introducing a new
bifurcation parameter $\delta_{D}:=\gamma_{o}-\gamma_{D}$ characterising the
distance from the point of drift instability, expanding $\chi(\tau)$ as
$\chi=\langle\chi(\tau^{\prime})\rangle_{\tau}+\delta_{D}\langle\chi(\tau^{\prime})\rangle_{\tau}-\delta_{D}^{2}\langle\chi(\tau^{\prime})\rangle_{\tau}^{2}+\ldots$,
and averaging Eq. (5) on space and time, one obtains

$\frac{d\chi}{d\tau}=\frac{8\omega C_{0}}{\langle
v\rangle}\left(1+2\delta_{D}\chi-3\delta_{D}^{2}\chi^{2}\right),$ (6)

where $C_{0}:=\langle(\rho
k)^{-1}\partial_{X}\theta_{v}\partial_{X}\rho\rangle_{X,\tau}$ and $\langle
v\rangle:=d\langle\theta(X,\tau)\rangle_{X}/dt$ is the average phase speed of
the drifting pattern. Solving analytically Eq. (6), and assuming that near the
drift instability the frequency of the drifting waves increase linearly with
$\delta_{D}$ as
$\omega(\delta_{D})=\Omega_{0}\delta_{D}+\mathcal{O}(\delta_{D}^{2})$, we get
the following maximum average speed

$\langle v\rangle_{\max}=\sqrt{\beta\left(\gamma_{o}-\gamma_{D}\right)},$ (7)

with $\beta:=32\Omega_{0}C_{0}/3$. For $\gamma_{o}$ close to $\gamma_{D}$, the
pattern velocity increases as the square root of $\gamma_{o}$.

###  II.3 Experimental evidence of the drift instabilities in a fluid
experiment

Guided by the above theoretical study of drift instabilities, we have
performed experiments of localised Faraday waves in a fluid subject to
vertical and localised vibrations. The experimental setup is schematised in
Fig. 4(a) and described in detail in Section V. Figure 4(b) shows a typical
detection of the free-surface elevation from the images obtained from the
high-speed camera. It is known that the pdnlS equation (1) governs the
envelope of the free surface of a vertically vibrated fluid in the vicinity of
Faraday instability Urra  _et al._ (2019). In particular, the pdnlS equation
captures the water surface displacement of the transverse mode if the
viscosity is small Miles (1984). The parameters in the dimensional pdnlS
equation describing the envelope are determined by the characteristics of the
fluid system, such as the injection frequency $\omega$ and the displacement
amplitude of forcing $a_{o}$. These values are needed to compute
$\gamma_{o}=\Gamma_{o}/4g$ and $\nu=(\omega^{2}/\omega_{o}^{2}-1)/2$, where
$\Gamma_{o}=4\omega^{2}a_{o}$ and $g$ is the acceleration of gravity.

Figure 4: (a) Experimental setup for generating Faraday patterns in the
surface of fluid under localised vertical vibrations. (b) Typical detection of
the free surface from the images received from the high-speed camera. (c)
Experimental measurements of the mean speed $\langle v\rangle$ of the drifting
patterns as a function of the acceleration amplitude $\Gamma_{0}$. The fit
using theoretical predictions with and without noise effects is shown in solid
and dashed red lines, respectively. (d) Typical behaviour of the drifting
patterns in experiments under different acceleration amplitudes.

We looked for the predicted drift instabilities in the fluid experiment shown
in Fig. 4(a) starting from a fixed forcing frequency. We gradually increase
the forcing amplitude by raising the displacement amplitude only. In Fig.
3(d), we show an experimental spatiotemporal diagram were the Faraday pattern
exhibit absolute stability. Notice that the maximum wave amplitude exhibits a
shift towards the left, in accordance with our previous observations in Fig.
3(a) and Fig. 2(a). Further increasing $\Gamma_{0}$, we were able to identify
the unidirectional drift shown in Fig. 3(e). We notice by direct comparison
between figures 3(b), 3(e), 2(d), and 2(g) the correspondence between theory,
pdnlS simulations, and experiments. Remarkably, the drift phenomenon is robust
enough to be captured by the experiment despite the slight difference in scale
given by the parameters of the equation. We also remark the appearance of a
fast zig-zag kind of motion, whose wavelength seems to depend on the forcing
amplitude. Such zig-zag movement appears above a certain instability threshold
and will be studied elsewhere.

We identify a forcing amplitude dependence on the drift velocity. Figure 4(c)
shows the experimental measurements of the mean drift velocity $\langle
v\rangle$ for different values of $\Gamma_{o}$, whereas Fig. 4(d) summarises
the different drifting states observed. The fit of the experimental speed
measurements using Eq. (7) is shown with dashed red line in Fig. 4(c). We
notice that such a curve fits well the experimental observations for
$\Gamma_{0}\gtrsim 0.676$. However, considering the effects of additive noise,
which are usually small and unavoidably present in experiments, the most
probable value of the mean drift velocity follows the law Agez  _et al._
(2013)

$\langle
v\rangle=A\sqrt{\frac{1}{2}\left(\Delta_{D}+\sqrt{\Delta_{D}^{2}+2\xi}\right)},$
(8)

where $\Delta_{D}:=\Gamma_{0}-\Gamma_{c}$ and $\xi$ is the noise intensity.
Figure 4(c) shows the fitting curve using Eq. (8) in solid red line. The
outcome parameter values from the fitting procedure were $A=2.463$,
$\Gamma_{c}=0.683$, and $\eta=0.01$. We observe a remarkable agreement between
the experimental measurements and the fit using Eq. (8). Neglecting noise
effects on the bifurcation, we have $\xi\to 0$ in Eq. (8), and we recover our
theoretical prediction on Eq. (7), which is valid for the deterministic case.

Further increasing $\Gamma_{o}$ above the drift instability threshold, we
observe the complex convective state shown in Fig. 3(f). In this case, pattern
dynamics exhibit branching phenomena, intermittence, and convection towards
both possible directions. Simulations of the pdnlS equation (1) exhibit
spatiotemporal complexity near this combination of parameters, as shown in
Fig. 3(c). The pdnlS approximation of the hydrodynamic equations describing
the Faraday instability captures qualitatively well the observed phenomenon.
However, the normal form of Eq. (4) is valid for
$\gamma_{o}\sim\gamma_{o}^{(0)}$ and cannot capture the complex convective
states observed far above the instability threshold. Higher-order corrections
would be needed to capture such spatiotemporal complexities in the amplitude
equation.

###  II.4 Impact of self-phase modulation terms on the nonlinear saturation of
localised Faraday patterns

Given the above theoretical and experimental evidence of drift instabilities
in localised Faraday patterns, we wondered on the impact of such convection in
the nonlinear saturation of the pattern. Following our previous calculations
Urra  _et al._ (2019), we perform a multiple-scale analysis in Eq. (4) to
investigate the role of the new SPM terms in the nonlinear saturation
mechanism. We have obtained that SPM has no effects in the saturation of the
fundamental mode close to its threshold of instability. Assuming that the
spatial and temporal scale of perturbations on the wave amplitude is large,
one obtains for $\gamma_{o}\sim\gamma_{o}^{(0)}$ that the evolution of the
amplitude $D$ of the fundamental mode is governed by

$\frac{d}{dt}D=\delta D-\frac{9}{2\sqrt{3}\mu}D^{5},$ (9)

which is in full agreement with our previous results Urra  _et al._ (2019)
obtained without SPM terms. Indeed, Eq. (9) predicts the scaling law
$D\propto\delta^{1/4}$, which has been previously verified in experiments Urra
_et al._ (2019) and gives the signature of a supercritical bifurcation. Figure
1(c) shows the maximum stationary amplitude $|C_{\max}|$ of the Faraday wave
as a function of the pump strength $\gamma_{o}$. We show in solid line the
prediction from Eq. (9) and with stars the values obtained from the numerical
simulations of the pdnlS Eq. (1). We conclude that the results of the
multiple-scale analysis on the amplitude equation (4) are in good agreement
with the numerical simulations of the pdnlS equation (1).

##  III Discussion

In a previous work Urra  _et al._ (2019) we have derived an amplitude equation
describing the growth of nonlinear instabilities around the fundamental
threshold $\gamma_{o}^{(0)}$, performing an appropriate expansion on the
dispersion relation obtained from the WKBJ analysis. The resulting amplitude
equation gives the first insight into how the nonlinear instability of the
Gauss-Hermite modes gives rise to a saturated solution. However, until now
such amplitude equation has not been formally derived using the standard tools
from the weakly nonlinear analysis.

The drift instabilities reported in this article are purely induced by the
heterogeneity of the drive and weak asymmetries in the system, which are
always present in real-life situations. Indeed, under a homogeneous drive, the
amplitude of the Faraday pattern will be space-independent and the SPM terms
causing the drift in Eq. (4) vanishes. Notice that only the heterogeneous
Weber-like term in Eq. (4) vanishes after taking the limit
$\sigma_{i}\to\infty$ (i.e., $\gamma(x)\to\gamma_{o}$). However, although the
SPM terms are not directly coupled to the Gaussian width $\sigma_{i}$, they
are _indirectly_ linked through the heterogeneous profile of $C(x,t)$ induced
by the Gaussian. We believe this is the reason why such drift has not been
observed in previous studies, which have been mainly focused on homogeneously
driven systems. The role of heterogeneities is increasingly becoming an
important factor in many nonlinear systems in different areas of physics Urra
_et al._ (2019); Edri  _et al._ (2020a, b); García-Ñustes  _et al._ (2021).
Our findings validate the use of non-local terms, such as those naturally
appearing in Eq. (4), in other phenomenological models to capture
heterogeneity-induced drift instabilities observed in simulations and
experiments.

##  IV Conclusions

In summary, we have shown that heterogeneity can trigger drift instabilities
in the evolving amplitude of localised Faraday patterns. Our analysis started
from a heterogeneous version of the pdnlS equation as a model for
parametrically induced waves. Under the assumption of a slowly varying
Gaussian pump, we have derived the evolution equation governing the slow
dynamics of the amplitude of the pattern. The resulting amplitude equation
includes new non-local SPM terms that reveal complex dynamics in the
underlying patterns. We have shown that above a secondary threshold of
instability, such new terms trigger convection through a nonlinear symmetry
breaking induced by nonlinear gradients. We performed a thorough search in an
experiment of localised Faraday patterns in the free surface of fluid under
localised parametric drive, and found the drift instability around the
parameter values predicted by the amplitude equation. Numerical simulations of
the pdnlS equation were also in remarkable agreement with theory and
experiments. These results extends our general understanding of the dynamical
effects of heterogeneities in nonlinear extended parametrical oscillators, and
how pattern selection can be dramatically affected by the nature of the
heterogeneous pump.

##  V Methods

### Normal forms

The theory of normal forms gives a powerful method for the systematic
construction of local, near-identity, and non-linear transformations to
simplify the equations describing the dynamics of complex nonlinear problems
near a bifurcation point Haragus and Iooss (2010). We start writing Eq. (1) in
the form $\partial_{t}\boldsymbol{\Psi}=L\boldsymbol{\Psi}+\mathbf{N}$, where
$\boldsymbol{\Psi}:=(\psi_{R},\psi_{I})$, $\psi:=\psi_{R}+i\psi_{I}$, $L$ is a
linear operator, and $\mathbf{N}$ is a nonlinear vector. We perform our
transformations around a Faraday wave solution with critical wavenumber
$\kappa_{c}$ according to

$\Psi=C(\eta,t)\left(\begin{array}[]{c}1\\\
0\end{array}\right)e^{i\kappa_{c}x}+\sum_{n=1}^{\infty}\mathbf{W}^{[n]}+c.c.,$
(10)

where $C(\eta,t)$ is a complex amplitude that varies slowly as a function of
$\eta:=\epsilon^{1/2}x$, as described in Section II, and the vector functions
$\mathbf{W}^{[n]}\in\mathbb{C}^{2},\,n\geq 1$, are higher order corrections to
be calculated. Assuming in Eq. (1) the scaling laws $C\sim\epsilon^{1/4}$,
$\delta\sim\epsilon$, $\partial_{t}\sim\epsilon$ and
$\partial_{x}\sim\epsilon^{1/2}$, one can expand the pdnlS equation at
different orders of $\epsilon^{1/4}$ and obtain a hierarchy of linear problems
of the form

$\mathcal{L}\mathbf{W}^{[n]}=\mathbf{b}_{n},$ (11)

which can be solved for $\mathbf{W}^{[n]}$ only if $\mathbf{b}_{n}$ is in the
image of the linear operator $\mathcal{L}$. According to the Fredholm
alternative Pismen (2006), at least one solution for $\mathbf{W}^{[n]}$ exists
if $\exists!|\mathbf{v}\rangle\in\ker\mathcal{L}^{\dagger}$ such that
$\langle\mathbf{v}|\mathbf{b}_{n}\rangle=0$. Defining the inner product in the
vector function space as

$\langle\mathbf{f}|\mathbf{g}\rangle=\kappa_{c}\int_{x_{o}}^{x_{o}+1/\kappa_{c}}dx\,(\mathbf{f}^{*})^{T}\mathbf{g},\quad|\mathbf{f}\rangle,|\mathbf{g}\rangle\in\mathbb{C}^{2},$
(12)

the Fredholm alternative allows to obtain a solvability condition to calculate
$\mathbf{W}^{[n]}$ at each order of $\varepsilon^{1/4}$. Our amplitude
equation (4) is the solvability condition obtained at order
$\varepsilon^{5/4}$.

### The method of multiple scales

We have used the method of multiple scales to study the effects of SPM in the
growth of the Faraday pattern near the fundamental threshold of instability.
Such perturbation method allows to separate the time-scale dynamics of the
different Gauss-Hermite modes controlling divergences in the perturbative
developments Peyrard and Dauxois (2004).

Just above the fundamental threshold of instability, the fundamental Gauss-
Hermite mode, $\mathcal{G}_{0}$, dominates the amplitude dynamics of the
Faraday pattern; higher order modes are decaying modes Urra  _et al._ (2019).
As the bifurcation parameter is increased, more Gauss-Hermite modes are
excited. Each mode grows at their own growth rate/characteristic time.
Assuming that the spatial and temporal scale of perturbations on the wave
amplitude is large, the spatial derivative is re-scaled by a small parameter
$\varepsilon$: $\partial_{x}\to\varepsilon^{1/2}\partial_{x}$, and we define
slow time scales $T_{i}:=\varepsilon^{i}t$ (with $i=1,\,2,\,\ldots$). Writing
$C$ as a perturbative development of the different time-scales

$C(x,t):=\sum_{k=1}^{\infty}\varepsilon^{k/4}A_{k}(x,T_{1},T_{2},\ldots),$
(13)

and using a similar expansion in the bifurcation parameter,
$\delta=\delta_{o}+\varepsilon\delta_{1}+\varepsilon^{2}\delta_{2}+\ldots$,
one obtains at each order of $\varepsilon^{1/4}$ a hierarchy of problems
similar to Eq. (11) for $A_{k}$, $k\geq 1$.

At order $\varepsilon^{1/4}$, one obtains that $A_{1}$ is governed by the
(time-independent) Weber equation Urra  _et al._ (2019), whose general
solution is given by a linear combination of the Gauss-Hermite modes with
time-dependent coefficients $D_{n}(T_{1},\ldots)$, $n\geq 0$. In particular,
the saturation of the fundamental Gauss-Hermite mode will be given by the time
evolution of the first of such time-dependent coefficients,
$D_{o}(T_{1},\ldots)$.

At order $\varepsilon^{5/4}$, in the analogous hierarchical equation (11) one
obtains a vector $\mathbf{b}_{5}$ that contains SPM terms. Given that the set
of all Gauss-Hermite modes $\\{\mathcal{G}_{n}\\}_{n=0}^{\infty}$ is a basis
of the space, we write the functions $A_{2}$ and $A_{3}$ as the linear
expansion

$A_{k}=D_{o}^{(k)}(T_{1})\mathcal{G}_{0}+\sum_{n=1}^{\infty}D_{n}^{(k)}(T_{1})\mathcal{G}_{n}\quad,\quad
k=2,3.$ (14)

However, near the threshold of instability of the fundamental mode, the modes
$\mathcal{G}_{n}$ with $n\geq 2$ are stable. Thus, the coefficients
$D_{n}^{(k)}(T_{1})$ for $n\geq 2$ decays in time. Once the Faraday pattern is
completely evolved, the maximum amplitude of the pattern is $C_{\max}\simeq
D_{o}:=D$ and the functions $A_{k}$ ($k=1,2,3$) are even in space. Thus, using
the Fredholm alternative, from symmetry arguments and the inner product (12)
one obtains that the contribution of the self-phase modulation terms vanish,
and the solvability conditions leads to Eq. (9).

### Numerical methods

Numerical simulations of the pdnlS equation (1) and our amplitude equation (4)
where performed with no-flux boundary conditions, using a 401-point spatial
grid with resolution $dx=0.25$. We used finite differences of second order of
accuracy for the spatial derivatives, and the time integration was achieved
using a fourth-order Runge-Kutta scheme with time-step $dt=0.0001$. To obtain
the amplitude of the drifting localised Faraday patterns shown in Fig. 3,
first we obtain numerically the pattern profile $\psi(x,t)$ from the numerical
simulation of Eq. (1). Then, the complex amplitude $C$ is obtained by means of
the Hilbert transform in space,
$C(x,t)=(1/\pi)\int_{-L/2}^{L/2}\,d\chi\psi(\chi,t)/(x-\chi)$, which is
computed at each time step of the numerical integration.

### Experimental set-up

The experimental setup is depicted in Fig. 4(a). We used a rectangular water
channel of 15 mm long, 490 mm wide, and 100 mm deep whose bottom has a central
rectangular soft silicone elastomer, type Smooth-on Dragon Skin 10 Medium. The
Gaussian injection $\gamma(x)$ is done through this soft zone. For controlling
the amplitude and frequency of the injection, we used a brushless servomotor
with feedback (Model No. BLM-N23-50-1000-B) controlled with a DMC-40x0 motion
controller from Galil Inc. The servomotor is coupled to an axis for
transmitting the rotational motion to an evenly spaced 13-piston array
($\Delta x=16$ mm), placed right underneath the soft injection zone. The
pistons are placed in the axis with a set of rotatory cams for keeping them
locked during operation. Over the pistons, there are aluminium bars connected
to the soft zone. Each one of them has a tiny roller in contact with the
piston, making the transmission of rotational motion to a linear one possible.
The bars are pushed back towards the piston with springs. The setup resembles
the mechanical transmission system of a music box, as shown in Fig. 4(a), and
allows one to deform the bottom of the channel with spatial distribution.

For the experiment, the basin was filled up to 20 mm in height with water
mixed with white ink for computational detection purposes. We used a Phantom
VEO 440L camera controlled by the PCC 3.5 software provided along with the
camera for the data acquisition. The resolution of the captures was 1280 x 252
pixels, and for every amplitude, the acquisition time was around 94 s at 400
frames per second. Images were saved in grayscale for easing the contour
detection process.

Due to fluid inertia, a pre-processing of the images is necessary for greater
amplitude experiments. A thin water layer is formed in the front of the basin,
making the detection of the water-air interface more diffuse. To keep the
focus only on the studied physical system, the brightness and gain of the
images have been changed with the PCC software tools. The image processing has
been made with a Python code using the OpenCV package for image manipulation.
For localising the water-air interface, we used a Canny edge detector over the
images previously processed with a Gaussian blur and Sobel filter. Iterating
this process over time, we obtain the spatio-temporal water profiles
$\eta(x,t)$ shown in Fig. 3(d-f) and 4(d). As a post-process meant for direct
comparison with simulations, a Hilbert transform was made along the time axis
for every pixel in Matlab, obtaining the envelope of Faraday waves described
by $\left|\psi(x,t)\right|$ in the pdnlS equation.

###### Acknowledgements.

The authors thank Milena Páez-Silva for improvements in the experimental
setup. R.R.A., S.C., and M.A.G-Ñ. thanks the financial support of ECOS grant
No. 200006. J.F.M. thanks ECOS-Sud project No. C15E06 for supporting a
collaboration visit at Université de Lille, France, where part of this work
was conceived. J.F.M. also thanks the financial support of ANID FONDECYT
POSTDOCTORADO grant No. 3200499. M.A.G-Ñ. acknowledges the financial support
of ANID FONDECYT grant No. 1201434.

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# Characters of prime power degree in principal blocks

J. Miquel Martínez  Departament de Matemàtiques, Universitat de València,
46100 Burjassot, València, Spain<EMAIL_ADDRESS>
###### Abstract.

We describe finite groups whose principal block contains only characters of
prime power degree.

###### Key words and phrases:

Character degrees, principal block, prime powers

###### 2010 Mathematics Subject Classification:

20C20, 20C15

This research is partially supported by Ministerio de Ciencia e Innovación
PID2019-103854GB-I00, Generalitat Valenciana CIAICO/2021-163, as well as a
fellowship UV-INV-PREDOC20-1356056 from Universitat de València and a travel
grant associated to the same fellowship

##  1\. Introduction

Let $G$ be a finite group and let $\mathrm{cd}(G)$ denote the set of degrees
of its irreducible ordinary characters. The properties of $G$ that can be seen
in $\mathrm{cd}(G)$ have been a subject of study for over half a century. Some
fundamental results in this topic are the Isaacs–Passman theorems, Thompson’s
theorem and the Itô–Michler theorem, which was one of the first applications
of the classification of finite simple groups to character theory. The recent
survey [Mor23] gives a good overview of the open problems and the techniques
used to explore them.

In [Man85a] and [Man85b], O. Manz described the structure of finite groups all
whose characters have prime power degrees. He proved that these groups are
either solvable or isomorphic to a direct product $A\times S$ where $A$ is an
abelian group and $S$ is isomorphic to
${\mathrm{SL}}_{2}(4)\cong\mathfrak{A}_{5}$ or ${\mathrm{SL}}_{2}(8)$
(although this last part depends on Brauer’s height zero conjecture, which has
been proved recently in [MNST22]). A version of Manz’s results for Brauer
characters was obtained in [Ton14] and for real character degrees in [Bon22]
and [CT22].

Let $p$ be a prime and let $B_{0}(G)$ be the principal $p$-block of $G$. If
every nonlinear character in $B_{0}(G)$ has degree divisible by $p$ then it is
well known that $G$ has a normal $p$-complement [IS76] (this is in fact a
principal block version of Thompson’s theorem). If $\mathrm{cd}(B_{0}(G))$
consists only of powers of a different prime $q\neq p$ then it was proved in
[Mar21, Theorem 5.3] that $G/{\bf O}_{p^{\prime}}(G)$ has a normal abelian
$q$-complement. In view of these results, A. Moretó asked the author if it is
possible to describe finite groups $G$ such that $\mathrm{cd}(B_{0}(G))$
consists of powers of possibly different primes, providing a principal block
version to Manz’s results. Of course, in this case $p$-solvability is no
longer guaranteed (take $G=\mathfrak{A}_{5}$ and $p$ any prime dividing
$|G|$), but it is possible to accurately describe the structure of $G$ and
this description is the purpose of this note.

Since this property is inherited by factor groups and normal subgroups (see
Lemma 3.1), we inevitably run into the problem of determining which finite
simple groups satisfy our hypothesis. The following completely describes these
groups.

######  Theorem A.

Let $S$ be a nonabelian finite simple group, and let $p$ be a prime dividing
$|S|$. Then $\mathrm{cd}(B_{0}(S))$ contains only prime powers if and only if
$(S,p)$ is one of the following

  1. (i)

$S=\operatorname{PSL}_{2}(q)$ for $q$ a Fermat or Mersenne prime and
$p\not\in\\{2,q\\}$,

  2. (ii)

$S={\mathrm{SL}}_{2}(2^{n})$ where $q=2^{n}\pm 1$ is a prime,
$p\not\in\\{2,q\\}$,

  3. (iii)

$S={\mathrm{SL}}_{2}(8)$ and $p\in\\{2,3,5,7\\}$,

  4. (iv)

$S={\mathrm{SL}}_{2}(4)\cong\mathfrak{A}_{5}$ and $p\in\\{2,3,5\\}$,

  5. (v)

$S=\operatorname{PSL}_{2}(9)\cong\mathfrak{A}_{6}$ and $p=5$.

and in all cases there are exactly two primes dividing the degrees in
$\mathrm{cd}(B_{0}(S))$.

Most of the work towards the proof of Theorem A follows from the results of
[MZ01] and [BBOO01], where the prime power degree characters of finite
(quasi-)simple groups were determined. In fact, Theorem A is fairly simple to
obtain using these results and the work done in [RSV20] and [GRSS20] on simple
groups of Lie type.

For general finite groups we have the following description.

######  Theorem B.

Let $G$ be a finite group, $p$ a prime dividing $|G|$ and assume
$\mathrm{cd}(B_{0}(G))$ consists only of prime powers. Then one of the
following happens:

  1. (i)

$G/{\bf O}_{p^{\prime}}(G)$ is a solvable group described in [Man85a],

  2. (ii)

there is a normal subgroup ${\bf O}_{p^{\prime}}(G)\subseteq
M\triangleleft\,G$ such that

$M/{\bf O}_{p^{\prime}}(G)=H\times S$

where $H$ is an abelian $p$-group and $(S,p)$ is one of the pairs from Theorem
A. Further, $G/M$ is isomorphic to a subgroup of ${\mathrm{Out}}(S)$.

We remark that the only case where $G/M$ is not cyclic is when
$S\cong\mathfrak{A}_{6}$. Indeed, the characters in the principal $5$-block of
$\operatorname{Aut}(\mathfrak{A}_{6})$ have degrees $1,9$ and $16$. In all
other cases, ${\mathrm{Out}}(S)$ is cyclic.

The following is an immediate corollary of Theorem B and the main result of
[Man85a].

######  Corollary C.

Let $G$ be a finite group, and $p$ a prime such that $\mathrm{cd}(B_{0}(G))$
consists only of prime powers. Then there are at most $3$ primes dividing the
degrees in $\mathrm{cd}(B_{0}(G))$.

Problems on character degrees of finite groups have led to the study of the so
called character degree graph $\Gamma(G)$ whose vertices are primes dividing
the degree of some character of $G$, and two vertices $p,q$ are connected if
there is $\chi\in\operatorname{Irr}(G)$ with $pq\mid\chi(1)$. Shortly after
Manz’s work, it was proved that $\Gamma(G)$ has at most three connected
components, and if $G$ is solvable then it has at most two (see [Lew08,
Theorems 4.2 and 6.4]). In [MS05] an analogous graph $\Gamma(B)$ was
introduced for a $p$-block $B$, where the degrees considered are only those of
characters that lie in $\operatorname{Irr}(B)$. In [MS05, Corollary C] the
authors show that if $G$ is $p$-solvable then $\Gamma(B)$ has at most three
connected components and if $G$ is solvable then it has two, so it seems that
$\Gamma(B)$ behaves somewhat similarly to $\Gamma(G)$. As pointed out by
Moretó, it is interesting to speculate whether $\Gamma(B_{0}(G))$ has at most
$3$ connected components in general, mimicking the situation in $\Gamma(G)$.

We prove Theorem A in Section 2 and we prove Theorem B and Corollary C in
Section 3.

### Acknowledgements

The results in this note were obtained while the author visited the Department
of Mathematics of the Rheinland-Pfälzische Technische Universität (formerly TU
Kaiserslautern). He thanks Gunter Malle for supervising his visit and for a
thorough read of this manuscript, and the entire department for their warm
hospitality. Furthermore, he would like to thank Alexander Moretó for his
question and very useful conversations on the topic, and Annika Bartelt for
clarifying some formulas for unipotent characters.

##  2\. Simple groups

The aim of this section is to prove Theorem A. We start by recalling some
classical results in number theory.

######  Lemma 2.1 (Zsigmondy’s theorem).

Let $q$ be a prime and $n>1$ an integer. Then

  1. (i)

there is a prime dividing $q^{n}-1$ that does not divide $q^{m}-1$ for all
$m<n$ unless $q=2$ and $n=6$ or $n=2$ and $q+1$ is a power of $2$,

  2. (ii)

there is a prime dividing $q^{n}+1$ that does not divide $q^{m}+1$ for all
$m<n$ unless $q=2$ and $n=3$.

######  Lemma 2.2.

Suppose that $q$ is an odd prime and $q^{n}+1=2^{s}$ for some positive
integers $n$ and $s$. Then $n=1$.

###### Proof.

See [HB82, Chapter IX, Lemma 2.7]. ∎

The following immediately follows from the previous lemmas.

######  Lemma 2.3.

Assume $q$ is a power of $2$ such that $q-1$ and $q+1$ are prime powers. Then
$q\in\\{4,8\\}$.

###### Proof.

Assume $q>8$ and that both $q-1$ and $q+1$ are prime powers. By Lemma 2.2,
$q-1$ is a prime, and therefore $q+1$ is a power of $3$. By Lemma 2.1 there is
a prime dividing $q+1$ that does not divide $2+1=3$, a contradiction. ∎

The next result is one of our main tools for discarding simple groups for
Theorem A.

######  Lemma 2.4.

If $G$ is not a $p$-solvable group then $|\mathrm{cd}(B_{0}(G))|\geq 3$, and
if $p\geq 5$ there are at least $3$ character degrees in
$\mathrm{cd}(B_{0}(G))$ not divisible by $p$.

###### Proof.

This follows from the main results of [Mar21] and [GRSS20]. ∎

Next, we exclude most families of finite simple groups as candidates for
Theorem A.

######  Proposition 2.5.

Assume $S$ is not one of
$\operatorname{PSL}_{n}(q),\operatorname{PSU}_{n}(q),{\mathrm{PSp}}_{2n}(q)$.
Then there is some $\chi\in\operatorname{Irr}(B_{0}(S))$ whose degree is not a
prime power.

###### Proof.

Assume first that $S$ is an alternating group $\mathfrak{A}_{n}$ for $n\geq
7$. If $n\leq 9$ this is easily checked in [GAP]. If $n\geq 10$ then by the
main result of [BBOO01] there is at most one nonlinear representation of
$\mathfrak{A}_{n}$ of prime power degree $q$. By Lemma 2.4 there is a
character $\chi\in\operatorname{Irr}(B_{0}(G))$ of degree $\chi(1)\neq q$ so
we are done in this case.

If $S$ is a simple group appearing in cases (7)–(27) of [MZ01, Theorem 1.1]
then this is also easily checked in [GAP]. Finally, if $S$ is a simple group
not appearing in any of the cases (2)–(27) of [MZ01, Theorem 1.1] then this
implies that $S$ is a simple group of Lie type and the nonlinear character of
$S$ with prime power degree is the Steinberg character $\mathsf{St}_{S}$. Then
by Lemma 2.4 there is a nonlinear character
$\chi\in\operatorname{Irr}(B_{0}(S))$ with $\chi(1)\neq\mathsf{St}_{S}(1)$ so
we are done. ∎

Thus we are left to deal with the groups
$\operatorname{PSL}_{n}(q),\operatorname{PSU}_{n}(q)$ and
${\mathrm{PSp}}_{2n}(q)$. We begin with an easy observation.

######  Proposition 2.6.

Let $S$ be a simple group of Lie type in characteristic $p$. Then Theorem A
holds for $(S,p)$.

###### Proof.

Assume $S$ is not one of the groups in Proposition 2.5. The group
$S={\mathrm{PSp}}_{4}(2)^{\prime}\cong\operatorname{PSL}_{2}(9)$ for $p=2$ can
be checked in [GAP]. Otherwise, by [Cab18, Theorem 3.3],
$\operatorname{Irr}(B_{0}(S))=\operatorname{Irr}(S)\setminus\\{\mathsf{St}_{S}\\}$
where $\mathsf{St}_{S}$ denotes the Steinberg character of $S$. Since
$\mathsf{St}_{S}(1)$ is a prime power, $\mathrm{cd}(B_{0}(S))$ consists only
of prime powers if and only if every character degree of $S$ is a prime power.
By the main result of [Man85b] we have that
$S\in\\{{\mathrm{SL}}_{2}(4),{\mathrm{SL}}_{2}(8)\\}$. ∎

To deal with the remaining groups we will need to use so-called unipotent
characters, introduced by G. Lusztig. In the case of
$S=\operatorname{PSL}_{n}(q)$ or $\operatorname{PSU}_{n}(q)$, these are
characters of ${\mathrm{SL}}_{n}(q)$ or respectively ${\mathrm{SU}}_{n}(q)$
but as argued in, for example, the first paragraph of [GRSS20, Proposition
4.4], these characters contain ${\bf Z}({\mathrm{SL}}_{n}(q))$ or resp. ${\bf
Z}({\mathrm{SU}}_{n}(q))$ in their kernels, and so they can be seen as
characters of $S$, and furthermore they are contained in the principal block
of $G$ if and only if they are in the principal block of $S$ (using [CE04,
Lemma 17.2]).

In this case, they are parametrized by partitions $n$ (see [GM20, Section
4.3]). Let $p$ be a prime dividing $|S|$ and let $e$ denote the order of $q$
modulo $p$ if $S=\operatorname{PSL}_{n}(q)$ and the order of $-q$ modulo $p$
if $S=\operatorname{PSU}_{n}(q)$. Further, let $r$ denote the remainder of $n$
divided by $m$. Following [FS82] we have that a unipotent character of $S$
parametrized by the partition $\alpha$ of $n$ belongs to the principal
$p$-block if its $e$-core is $(r)$.

If $S={\mathrm{PSp}}_{2n}(q)$ then the unipotent characters are characters of
${\mathrm{Sp}}_{2n}(q)$ parametrized by certain symbols (see [GM20, Section
4.4]). Exactly as before, they can be seen as characters of $S$ and a
unipotent character of ${\mathrm{Sp}}_{2n}(q)$ belongs to
$B_{0}({\mathrm{Sp}}_{2n}(q))$ if and only if it belongs to $B_{0}(S)$.

######  Lemma 2.7.

Let $G={\mathrm{SL}}_{n}(q)$ or ${\mathrm{SU}}_{n}(q)$ and let $\chi$ be a
unipotent character of $G$. Then $\chi(1)$ is not a prime power unless
$\chi=1_{G}$ or $\chi=\mathsf{St}_{G}$.

###### Proof.

Let $\chi$ be the unipotent character parametrized by the partition $\alpha$
of $n$. It is easy to see in [GM20, Propositions 4.3.1 and 4.3.5] that
$\chi(1)$ has nontrivial $q$-part and nontrivial $q^{\prime}$-part unless
$\alpha=(n)$ or $\alpha=(1^{n})$, which correspond to $1_{G}$ and
$\mathsf{St}_{G}$ respectively. ∎

######  Proposition 2.8.

Let $S=\operatorname{PSL}_{n}(q)$ or $\operatorname{PSU}_{n}(q)$ with $n\geq
3$ and let $p\nmid q$ be a prime dividing $|S|$. Then there is
$\chi\in\operatorname{Irr}(B_{0}(S))$ with $\chi(1)$ not a prime power.

###### Proof.

. Let $\epsilon=1$ if $S=\operatorname{PSL}_{n}(q)$ and $\epsilon=-1$ if
$S=\operatorname{PSU}_{n}(q)$. By Lemma 2.4, $S$ has to be one of the groups
in cases (3) or (4) of [MZ01, Theorem 1.1], so $n$ is an odd prime and the
prime power degrees are $\mathsf{St}_{S}(1)=|S|_{q}$ and
$(q^{n}-\epsilon)/(q-\epsilon)$. By Lemma 2.1, $(q^{n}-\epsilon)/(q-\epsilon)$
can not be a power of $2$.

If $q$ is odd then both prime power degrees are odd. If $p=2$ then the order
of $\epsilon q$ modulo $p$ is necessarily $1$, and therefore all unipotent
characters belong to $B_{0}(S)$, and we are done by Lemma 2.7. If $p$ is odd
then [GMV19, Theorem B] guarantees the existence of a character of even degree
in $B_{0}(S)$. Thus we assume $q$ is a power of $2$.

Assume first $n\geq 5$. If $p\geq 5$ then [GRSS20, Table 1] produces a
unipotent character in $B_{0}(S)$ different from $1_{G}$ and
$\mathsf{St}_{S}$, which can not have prime power degree by Lemma 2.7. If
$p=3$ then we argue identically with [RSV20, Table 3].

We are left with the groups $\operatorname{PSL}_{3}(q)$ and
$\operatorname{PSU}_{3}(q)$ with $q$ a power of $2$. If $p=3$ or
$p\mid(q+\epsilon)$ then the last paragraph of the proof of [RSV20,
Proposition 3.10] and the first paragraph of the proof of [GRSS20, Proposition
4.5] produces a character in $B_{0}(S)$ of degree $q^{3}-\epsilon$, which is
not a prime power. If $p\geq 5$ then if $p\nmid(q+\epsilon)$ then by the order
formula for $\operatorname{PSL}_{3}(q)$ and $\operatorname{PSU}_{3}(q)$ we
have that either $p\mid(q-\epsilon)$ or $p\mid(q^{2}+\epsilon
q+1)=(q^{3}-\epsilon)/(q-\epsilon)$.

In the first case, the order of $\epsilon q$ modulo $p$ is $1$, and it follows
that the unipotent character defined by the partition $(1,2)$ belongs to
$B_{0}(S)$ and has degree $q(q+\epsilon)$ by [GM20, Propositions 4.3.1 and
4.3.5].

In the second case, we have that the prime power degrees of $S$ are
$\mathsf{St}_{S}(1)=|S|_{q}$ and $(q^{3}-\epsilon)/(q-\epsilon)$ which must be
a power of $p$. Since $p\geq 5$, Lemma 2.4 guarantees the existence of a
character $\chi\in\operatorname{Irr}(B_{0}(S))$ of $p^{\prime}$-degree
different from $\mathsf{St}_{S}(1)$, so $\chi(1)$ is not a prime power. ∎

######  Proposition 2.9.

Let $S=\operatorname{PSL}_{2}(q)$ and let $p\nmid q$ be a prime dividing $|S|$
and assume first that every character degree in $B_{0}(S)$ is a prime power.
Then $(S,p)$ is one of the pairs in Theorem A.

###### Proof.

If $q$ is odd then it is well known that $\mathrm{cd}(S)=\\{1,q,q+1,q-1,(q\pm
1)/2\\}$ where the last sign depends on the congruence of $q$ modulo $4$. By
Lemma 2.4 the set $\mathrm{cd}(B_{0}(S))$ has size at least 3, of which forces
at least one of $q+1$ or $q-1$ to be a power of $2$. If $p=2$ then let
$\chi\in\operatorname{Irr}(S)$ be a character of such degree. Notice that by
the order formula for $\operatorname{PSL}_{2}(q)$, $\chi$ has $2$-defect zero
and so it can not belong to the principal $2$-block of $S$. This forces $p\neq
2$. If $q+1$ is a power of $2$ the same argument works.

If $q$ is a power of $2$ then $\mathrm{cd}(S)=\\{1,q,q+1,q-1\\}$ which again
forces $q+1$ or $q-1$ to be a prime power (and they both are prime powers only
if $q\in\\{4,8\\}$ by Lemma 2.3). Assume that $q>8$. If $r=q-1$ is a prime
power then by Lemma 2.2 in fact $r$ is a prime. A character of degree $r$ has
$r$-defect zero and so it can not belong to the principal $r$-block, forcing
$p\neq r$. If $r=q+1$ is a prime power then again by Lemma 2.1 we have that
$r$ is a prime and we can mimick the previous argument to reach the desired
conclusion. ∎

######  Proposition 2.10.

Let $S={\mathrm{PSp}}_{2n}(q)$ and let $p\nmid q$ be a prime dividing $|S|$.
Then there is $\chi\in\operatorname{Irr}(B_{0}(S))$ with $\chi(1)$ not a prime
power.

###### Proof.

By Lemma 2.4, we may assume that $S$ is one of the groups in cases (5) or (6)
of [MZ01, Theorem 1.1]. Assume first that we are in case (5), so that the
irreducible characters of $S$ whose degree is a prime power have degrees
$\mathsf{St}_{S}(1)=|S|_{q}$ and $(q^{n}+1)/2$. Notice that $q^{n}+1$ can not
be a power of $2$ by Lemma 2.1.

If $p$ is odd then there must exist a character
$\chi\in\operatorname{Irr}(B_{0}(S))$ of even degree by [GMV19, Theorem B]. If
$p=2$ then by [CE04, Theorem 21.14], all unipotent characters belong to
$B_{0}(S)$. Consider the unipotent character $\chi$ parametrized by the symbol
$\binom{0\,n}{1}$. By [GM20, Proposition 4.4.15] and using the order formula
for ${\mathrm{Sp}}_{2n}(q)$, it is easy to see that $\chi(1)_{q}>1$ and
$\chi(1)_{q^{\prime}}>1$ and so $\chi(1)$ is not a prime power.

If we are in case (6) of [MZ01, Theorem 1.1], the characters of prime power
degree have degrees $3^{l}$ and $(3^{n}-1)/2$ which is only a power of $2$ if
$n=2$ so $S={\mathrm{PSp}}_{4}(3)$, which is checked in [GAP]. If $n>2$ then
the same argument as before applies. ∎

Proof of Theorem A. The only if direction is done in Propositions 2.5, 2.6,
2.8, 2.9, 2.10.

For the if direction the cases (iii)–(v) can be checked in [GAP]. Assume first
that $S=\operatorname{PSL}_{2}(q)$ where $q+1$ is a power of $2$ and let
$p\neq\\{2,q\\}$. Then the characters of degree $(q-1)$ and $(q-1)/2$ (if they
exist) have $p$-defect zero by the order formula for
$\operatorname{PSL}_{2}(q)$, so they can not belong to the principal
$p$-block. This forces

$\mathrm{cd}(B_{0}(S))\subseteq\\{1,q,q+1,(q+1)/2\\}.$

The case where $q$ or $q-1$ is a power of $2$ is done analogously. ∎

##  3\. Theorem B and Corollary C

Our notation for this section follows [Nav98] and [Nav18].

######  Lemma 3.1.

Assume $\mathrm{cd}(B_{0}(G))$ consists only of prime powers. If
$N\triangleleft\,G$ then $B_{0}(N)$ and $B_{0}(G/N)$ consist only of prime
powers.

###### Proof.

For all $\theta\in\operatorname{Irr}(B_{0}(N))$ there is some
$\chi\in\operatorname{Irr}(B_{0}(G))$ such that $\chi_{N}$ contains $\theta$
by [Nav98, Theorem 9.4], and $\theta(1)$ divides $\chi(1)$ by standard
Clifford theory, and the first claim follows. For the second claim recall that
$\operatorname{Irr}(B_{0}(G/N))\subseteq\operatorname{Irr}(B_{0}(G))$. ∎

######  Theorem 3.2.

Assume that $G$ is not $p$-solvable and that $\mathrm{cd}(B_{0}(G))$ consists
only of prime powers. Then there is some ${\bf O}_{p^{\prime}}(G)\subseteq
M\triangleleft\,G$ with $M/{\bf O}_{p^{\prime}}(G)=H\times S$ where $H$ is an
abelian $p$-group and $(S,p)$ is one of the pairs in Theorem A. Also, $G/M$ is
isomorphic to a subgroup of ${\mathrm{Out}}(S)$.

###### Proof.

Arguing by induction on $|G|$, we may assume ${\bf O}_{p^{\prime}}(G)=1$. Let
$E$ be the layer of $G$. We claim that $E$ is quasisimple. Indeed, write
$E=K_{1}\cdots K_{t}$ for components $K_{1},\dots,K_{t}$, assume $t>1$ and let
$Z={\bf Z}(E)$. Then $Z\triangleleft\,G$ and by [KS03, 6.5.6]
$E/Z=E_{1}\times\dots\times E_{t}$ where $E_{i}=K_{i}Z/Z$. By Lemma 3.1,
$B_{0}(E/Z)$ consists only of prime powers, and so does $B_{0}(E_{i})$. Now
since the $E_{i}$’s are not $p$-solvable, by Lemma 2.4 and [Mar21, Lemma 5.2],
for each $E_{i}$ there are characters
$\theta,\eta\in\operatorname{Irr}(B_{0}(E_{i}))$ of coprime degree. Let
$\theta\in\operatorname{Irr}(B_{0}(E_{1}))$ and
$\eta\in\operatorname{Irr}(B_{0}(E_{2}))$ be nonlinear and such that
$\theta(1)$ and $\eta(1)$ are coprime and consider
$\psi=\theta\times\eta\times 1_{E_{3}}\times\dots\times
1_{E_{t}}\in\operatorname{Irr}(B_{0}(E/Z))$. Then $\psi(1)$ is not a prime
power, a contradiction. This forces $E$ to be quasisimple and again by Lemma
3.1, $(E/{\bf Z}(E),p)$ is one of the pairs from Theorem A. Furthermore, the
$p$-complement of ${\bf Z}(E)$ is a normal $p^{\prime}$-subgroup of $G$, and
since ${\bf O}_{p^{\prime}}(G)=1$ we have that ${\bf Z}(E)$ is a $p$-group.

Now let $C={\bf C}_{G}(E)$ so that $G/C$ is almost simple with socle $S=E/{\bf
Z}(E)$ (arguing as in Step 7 of [GMS22, Theorem 2.10]). Now
$C\triangleleft\,G$ and $M=CE\triangleleft\,G$ is a central product because
$[C,E]=1$. Arguing as before, it is easy to see that $B_{0}(C)$ can not
contain nonlinear characters, so $C/{\bf O}_{p^{\prime}}(C)$ is an abelian
$p$-group by [Nav98, Theorem 6.10]. Now since ${\bf
O}_{p^{\prime}}(C)\triangleleft\,G$ we get that $C$ is an abelian $p$-group
and ${\bf Z}(E)=C\cap E$. Now if the Schur multiplier of $S$ has order not
divisible by $p$ then ${\bf Z}(E)=1$ and $M=C\times S$. By [MT11, Tables 24.2
and 24.3], the only pair $(S,p)$ for which the Schur multiplier of $S$ has
order divisible by $p$ is $(\mathfrak{A}_{5},2)$, and it is easily checked in
[GAP] that the universal central extension $2.\mathfrak{A}_{5}$ has a
character of degree $6$ in its principal $2$-block, so in all cases $M=C\times
S$. Finally, since $G/C$ is almost simple with socle $S$ we have that $G/M$ is
isomorphic to a subgroup of ${\mathrm{Out}}(S)$. ∎

Proof of Theorem B. If $G$ is $p$-solvable then by [Nav98, Theorem 10.20] we
have that $\operatorname{Irr}(B_{0}(G))=\operatorname{Irr}(G/{\bf
O}_{p^{\prime}}(G))$ and therefore $G/{\bf O}_{p^{\prime}}(G)$ is one of the
groups from [Man85a]. Otherwise, apply Theorem 3.2.∎

If $2^{n}+1$ is a prime, then it is well known that $n$ is a power of $2$. It
is also well known that if $2^{n}-1$ is a prime then $n$ is a prime. Since the
automorphism group of $S={\mathrm{SL}}_{2}(2^{n})$ is a cyclic group of order
$n$, if we are in case (ii) of Theorem A then ${\mathrm{Out}}(S)$ has prime
power order.

Proof of Corollary C. We may assume $G$ is as in case (ii) of Theorem B. If
$\chi\in\operatorname{Irr}(B_{0}(G))$ has degree $\chi(1)=r^{t}$ for some
prime $r$ and $\chi_{S}\neq 1_{S}$ then $\chi_{S}$ contains characters of
degree a power of $r$ in $B_{0}(S)$ by [Nav98, Theorem 9.4]. Otherwise
$\chi_{S}=1_{S}$ and $\chi_{M}$ contains only linear characters so $\chi(1)$
divides $|G/M|$ by [Nav18, Theorem 5.12]. Now, $G/M$ is isomorphic to a
subgroup of ${\mathrm{Out}}(S)$ which has prime power order in all cases from
Theorem A, so we are done.∎

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