aainstitutetext: Department of Mathematics and Applied Mathematics, University
of Cape Town, Rondebosch 7701, South Africabbinstitutetext: Centre for
Theoretical and Mathematical Physics, University of Cape Town, South Africa

# The energy-frequency diagram of the (1+1)-dimensional $\Phi^{4}$ oscillon

N. V. Alexeeva b,1 I. V. Barashenkov, 111Corresponding author. a Alain Dika a
and Raphael De Sousa<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Two different methods are used to study the existence and stability of the
(1+1)-dimensional $\Phi^{4}$ oscillon. The variational technique approximates
it by a periodic function with a set of adiabatically changing parameters. An
alternative approach treats oscillons as standing waves in a finite-size box;
these are sought as solutions of a boundary-value problem on a two-dimensional
domain. The numerical analysis reveals that the standing wave’s energy-
frequency diagram is fragmented into disjoint segments with
$\omega_{n-1}<\omega<\omega_{n-2}$, where $\omega_{n}=\omega_{0}/(n+1)$,
$n=2,3,...$, and $\omega_{0}$ is the endpoint of the continuous spectrum. All
standing waves are practically stable: perturbations may result in the
deformation of the wave’s radiation wings but do not affect its core. The
variational approximation involving the first, zeroth and second harmonic
components provides an accurate description of the oscillon with the frequency
in $(\omega_{1},\omega_{0})$, but breaks down as $\omega$ falls out of that
interval.

##  1 Introduction

Oscillons (also known as pulsons) were introduced as localised long-lived
pulsating structures in three-dimensional classical field theories Voronov1 ;
BM1 ; BM2 ; G1 ; CGM . The original motivation Zeldovich  was to model the
vacuum domain formation in theories with spontaneous symmetry breaking and its
cosmological implications. Oscillons have now been recognised to have a role
in the dynamics of inflationary reheating, symmetry-breaking phase
transitions, and false vacuum decay CGM ; Riotto ; GInt ; 11cosmo ; Dymnikova
; Broadhead ; bubbling ; Amin1 ; Stamatopoulos ; Zhou ; Amin2 ; Adshead ; GG ;
Bond ; Antusch ; Hong ; Cyn ; LozAm ; Takhistov ; Aurek . They arise as
natural ingredients in the bosonic sector of the standard model Farhi2 ;
Graham ; Gleiser4 ; Sfakianakis ; Piani  and axion-based models of fuzzy dark
matter Kolb ; Vaquero ; Kawa_axion ; Olle ; Arvanitaki ; Miyazaki ; string ;
Kasu ; Sang ; Imagawa . The Einstein-Klein-Gordon equations have also been
shown to exhibit oscillon solutions Maslov ; Zhang2 ; Nazari ; Kou1 ; Hira ;
Kou2 ; Naka .

For some time, the studies of three-dimensional oscillons were disconnected
from research into oscillatory solitons in 1D. The reason was that the latter
objects — interpreted as the kink-antikink bound states Kudryavtsev ; DHN ;
Getmanov ; Sugiyama ; Geicke ; CP  — were believed to persist over long
periods of time (or even indefinitely), emitting little or no radiation. By
contrast, the three-dimensional oscillons were thought to have a fairly short
lifespan BM1 ; BM2 ; G1 ; CGM .

More accurate mathematical and numerical analysis indicated, however, that the
oscillons in three dimensions and one-dimensional bound states share their
basic properties Honda ; Fodor1 ; Fodor3 . (The only exception, of course, is
the sine-Gordon breather — an exactly periodic solution which emits strictly
no radiation.) It is for this reason that we are using the oscillon
nomenclature for what would otherwise be called “bion” Getmanov ; BK ,
“approximate breather” SK  or “breather-like state on the line” Vachaspati .

In this paper, we employ a new variational approach to study localised
oscillations in the one-dimensional $\Phi^{4}$ equation — the one-dimensional
version of the system that bore the originally discovered pulsed states DHN ;
Kudryavtsev ; Voronov1 ; BM1 ; BM2 . We consider our present attack on the
one-dimensional $\Phi^{4}$ oscillon a step towards the consistent variational
description of its three-dimensional counterpart.

The 1+1 dimensional $\Phi^{4}$ theory is probably the simplest model with
spontaneous symmetry breaking exploited in the studies of quantum fields
Rychkov ; Bajnok ; Serone ; Bordag ; Graham-Weigel ; Tokacs ; Martin ; Ito
and phase transitions PT1 ; PT2 . The model is defined by the Lagrangian

$L=\frac{1}{2}\int\left[\Phi_{t}^{2}-\Phi_{x}^{2}-(\Phi^{2}-1)^{2}\right]\,dx$
(1)

and equation of motion

$\Phi_{tt}-\Phi_{xx}-2\Phi(1-\Phi^{2})=0.$ (2)

Its oscillon solution was discovered by Dashen, Hasslacher and Neveu DHN  who
constructed it as an asymptotic series in powers of the amplitude of the
oscillation. The first few terms in the expansion of DHN  (with typos
corrected) are

$\displaystyle\Phi=1+\frac{\epsilon}{\sqrt{3}}\,\mathrm{sech}(\epsilon
x)\cos(\omega t)+\frac{\epsilon^{2}}{12}\mathrm{sech}^{2}(\epsilon
x)\left[\cos(2\omega t)-3\right]$
$\displaystyle+\frac{\epsilon^{3}}{144\sqrt{3}}[98\,\mathrm{sech}(\epsilon
x)-103\,\mathrm{sech^{3}}(\epsilon x)]\cos(\omega
t)+\frac{\epsilon^{3}}{48\sqrt{3}}\mathrm{sech^{3}}(\epsilon x)\cos(3\omega
t)+O(\epsilon^{4}).$ (3)

Here $\omega^{2}=\omega_{0}^{2}-\epsilon^{2}$, $\epsilon\to 0$ and
$\omega_{0}$ denotes the endpoint of the continuous spectrum:

$\omega_{0}=2.$

Segur and Kruskal SK  proved that the series (3) does not converge and that
the true oscillon expansion includes terms that are nonanalytic in $\epsilon$.
(For the system-dynamic perspective on this argument, see Eleonskii et al
Eleonskii .) The nonanalytic terms lie beyond all orders of the perturbation
theory and make negligible contributions to the core of the oscillon; yet they
do not vanish as $|x|\to\infty$ and account for the oscillon-emitted
radiation. Boyd Boyd2  noted a close relationship between radiating (hence
decaying) oscillons of small amplitude and nanopterons: standing waves with
“wings” extending to the infinities. (See also Fodor2 .) The amplitude of the
wings is exponentially small in $\epsilon$ but does not decrease as
$x\to\pm\infty$. The sum of the standing wave and a solution to the linearised
equation with an exponentially small amplitude gives a highly accurate
approximation of the radiating oscillon Boyd2 .

Regarding oscillons of finite amplitude, numerical simulations have been the
primary source of information. The earliest observations of oscillons with
finite $\epsilon$ are due to Kudryavtsev Kudryavtsev , while more detailed and
accurate sets of simulations can be found in Getmanov ; Geicke ; Boyd2 ;
Fodor2 . For reviews, see BK ; Fodor3 ; Boyd1 .

The present study is motivated by the need to have an analytic tool capable of
providing insights into the structure and properties of the finite-amplitude
oscillons, similar to the collective coordinate technique used in the
nonlinear Schrödinger domain Malomed ; BAZ . Modelling on the multiscale
variational method developed for the theory with the symmetric vacuum BA , we
formulate a variational approach to the one-dimensional $\Phi^{4}$ oscillon.
The symmetry-breaking nature of the vacuum in (1) forces us to expand the set
of collective coordinates that was employed in BA . However, similar to BA ,
the expanded set does not include any radiation degrees of freedom. We
approximate the oscillon by a strictly periodic, nonradiating, state.

To validate our variational approximation, we carry out a numerical study of
the time-periodic solutions of equation (2) on a finite interval. The earlier
studies determined standing waves with frequencies
$\omega>\omega_{0}/\sqrt{2}$ Boyd1 ; we will reach below that record.

For each $\omega$, there is a continuous family of standing waves with
different amplitudes of the radiation wings and, consequently, different
energies. We focus on the waves with the lowest energy as these nanopterons
have the smallest wing amplitude Fodor2 ; Fodor3 ; Fodor1 . The energy-
frequency diagram of the lowest-energy standing waves is found to be in good
agreement with the diagram constructed variationally in a wide range of
frequencies.

The paper is organised into five sections. The variational approximation for
the $\Phi^{4}$ oscillons is presented in Section 2, with some technical
details relegated to the Appendices. In Section 3, we determine numerical
solutions describing standing waves in a finite-size box, and in Section 4,
the results of the variational and numerical approaches are compared. Section
5 summarises the conclusions of this work.

##  2 Multiscale variational approach

###  2.1 Method

The variational method is arguably the main analytical approach to solitons in
nonintegrable systems outside of perturbation expansions. In the context of
oscillons, the method was pioneered in Ref CGM , where the three-dimensional
$\Phi^{4}$ oscillon was approximated by a localised waveform

$\Phi=1+A(t)e^{-(r/b)^{2}}.$ (4)

Here $A(t)$ is an unknown oscillatory function describing the trajectory of
the structure’s central point and $b$ is an arbitrarily chosen value of its
width. (Ref Kev  followed a similar strategy when dealing with the two-
dimensional sine-Gordon equation.) Once the ansatz (4) has been substituted in
the lagrangian and the $r$-dependence integrated away, the variation of action
produces a second-order equation for $A(t)$.

The regular variational method does not suggest any optimisation strategies
for the choice of $b$. Making $b(t)$ another collective coordinate — as it is
done in the studies of the nonlinear Schrödinger solitons Malomed ; BAZ  —
gives rise to an ill-posed dynamical system not amenable to numerical
simulations CF ; BA . This difficulty is circumvented in the multiscale
approach, which considers the oscillon as a rapidly oscillating structure with
adiabatically changing parameters BA .

Following BA , we consider $\Phi$ to be a function of two time variables,
${\mathcal{T}_{0}}=t$ and ${\mathcal{T}_{1}}=\epsilon t$. The rate of change
is assumed to be $O(1)$ on either scale:
$\partial\Phi/\partial\mathcal{T}_{0},\partial\Phi/\partial\mathcal{T}_{1}\sim
1$. We require $\Phi$ to be periodic in $\mathcal{T}_{0}$, with a period of
$T$:

$\Phi(\mathcal{T}_{0}+T;\,\mathcal{T}_{1})=\Phi(\mathcal{T}_{0};\,\mathcal{T}_{1}).$

As $\epsilon\to 0$, the variables $\mathcal{T}_{0}$ and $\mathcal{T}_{1}$
become independent and the Lagrangian (1) transforms into

$\displaystyle
L=\frac{1}{2}\int\left[\left(\frac{\partial\phi}{\partial{\mathcal{T}_{0}}}+\epsilon\frac{\partial\phi}{\partial{\mathcal{T}_{1}}}\right)^{2}-\phi_{x}^{2}-4\phi^{2}-4\phi^{3}-\phi^{4}\right]dx,$
(5)

where we let $\Phi=1+\phi$. The action $\int Ldt$ is replaced with

$S=\int_{0}^{T}d{\mathcal{T}_{0}}\int
d{\mathcal{T}_{1}}\,L\left(\phi,\frac{\partial\phi}{\partial{\mathcal{T}_{0}}},\frac{\partial\phi}{\partial{\mathcal{T}_{1}}}\right).$
(6)

Modelling on the asymptotic expansion (3), we choose the trial function in the
form

$\displaystyle\phi=A\,\cos(\omega{\mathcal{T}_{0}}+\theta)\,\mathrm{sech}\left(\frac{x}{b}\right)+\left[B+C\cos
2(\omega{\mathcal{T}_{0}}+\theta)\right]\,\mathrm{sech}^{2}\left(\frac{x}{b}\right),$
(7)

where $A,B,C,b$ and $\theta$ are functions of the “slow” time variable
$\mathcal{T}_{1}$ while $\omega=2\pi/T$ ($\omega>0$).

Once the explicit dependence on $x$ and $\mathcal{T}_{0}$ has been integrated
away, equations (5) and (6) give

$S=T\int d{\mathcal{T}_{1}}\,\mathcal{L}_{\omega},$ (8)

where the effective Lagrangian $\mathcal{L}_{\omega}$ is given by

$\mathcal{L}_{\omega}=K+\frac{b}{2}\left(A^{2}+\frac{8C^{2}}{3}\right)\left(\dot{\theta}+\omega\right)^{2}-U,$
(9)

with

$\displaystyle
K=\frac{1}{2}\dot{A}(bA\dot{)}+\frac{2}{3}\dot{B}(bB\dot{)}+\frac{1}{3}\dot{C}(bC\dot{)}+\frac{{\dot{b}}^{2}}{2b}\left[\left(\frac{\pi^{2}}{36}+\frac{1}{3}\right)A^{2}+\frac{2\pi^{2}}{45}(2B^{2}+C^{2})\right]$
(10)

and

$\displaystyle
U=\left(\frac{1}{6b}+2b\right)A^{2}+\frac{4}{15}\left(\frac{1}{b}+5b\right)\left(2B^{2}+C^{2}\right)$
$\displaystyle+2b\left[\frac{8}{15}B(2B^{2}+3C^{2})+A^{2}(2B+C)\right]$
$\displaystyle+b\left[\frac{A^{4}}{4}+\frac{16}{35}\left(B^{4}+\frac{3}{8}C^{4}+3B^{2}C^{2}\right)+\frac{4}{5}A^{2}\left(2B^{2}+C^{2}+2BC\right)\right].$
(11)

In (10), we introduced a short-hand notation $\dot{f}\equiv\epsilon\partial
f/\partial\mathcal{T}_{1}$.

The variation of the lagrangian (9) with respect to the collective coordinates
$A,B,C,b$ and $\theta$ produces five equations of motion. The variable
$\theta$ is cyclic; the corresponding Euler-Lagrange equation gives rise to
the conservation law

$b\left(A^{2}+\frac{8}{3}C^{2}\right)(\dot{\theta}+\omega)=\ell,$ (12)

where $\ell=\mathrm{const}$. Making use of (12), $\dot{\theta}$ can be
eliminated from the remaining four equations which become a system of four
equations for four unknowns ($A,B,C$ and $b$):

$\displaystyle\ddot{A}+\frac{\ddot{b}}{2b}A+\frac{\dot{b}}{b}\dot{A}-\left(\frac{\pi^{2}}{36}+\frac{1}{3}\right)\left(\frac{\dot{b}}{b}\right)^{2}A+\left(4+\frac{1}{3b^{2}}\right)A-\frac{9\ell^{2}}{\left(3A^{2}+8C^{2}\right)^{2}}\frac{A}{b^{2}}$
$\displaystyle+4A(2B+C)+A^{3}+\frac{8}{5}A(2B^{2}+2BC+C^{2})=0,$ (13a)
$\displaystyle\ddot{B}+\frac{\ddot{b}}{2b}B+\frac{\dot{b}}{b}\dot{B}-\frac{\pi^{2}}{15}\left(\frac{\dot{b}}{b}\right)^{2}B+\frac{4}{5}\left(5+\frac{1}{b^{2}}\right)B$
$\displaystyle+\frac{3}{5}(5A^{2}+8B^{2}+4C^{2})+\frac{6}{5}A^{2}(2B+C)+\frac{24}{35}B(2B^{2}+3C^{2})=0,$
(13b)
$\displaystyle\ddot{C}+\frac{\ddot{b}}{2b}C+\frac{\dot{b}}{b}\dot{C}-\frac{\pi^{2}}{15}\left(\frac{\dot{b}}{b}\right)^{2}C+\frac{4}{5}\left(5+\frac{1}{b^{2}}\right)C-\frac{36\ell^{2}}{(3A^{2}+8C^{2})^{2}}\frac{C}{b^{2}}$
$\displaystyle+3\left(A^{2}+\frac{16}{5}BC\right)+\frac{12}{5}A^{2}(B+C)+\frac{36}{35}(4B^{2}+C^{2})C=0,$
(13c) $\displaystyle
A\ddot{A}+\frac{4}{3}B\ddot{B}+\frac{2}{3}C\ddot{C}+\left[2\frac{\ddot{b}}{b}-\left(\frac{\dot{b}}{b}\right)^{2}\right]\left[\left(\frac{\pi^{2}}{36}+\frac{1}{3}\right)A^{2}+\frac{2\pi^{2}}{45}(2B^{2}+C^{2})\right]$
$\displaystyle+4\frac{\dot{b}}{b}\left[\left(\frac{\pi^{2}}{36}+\frac{1}{3}\right)A\dot{A}+\frac{2\pi^{2}}{45}(2B\dot{B}+C\dot{C})\right]-\frac{3}{3A^{2}+8C^{2}}\frac{\ell^{2}}{b^{2}}$
$\displaystyle+\left(4-\frac{1}{3b^{2}}\right)A^{2}+\frac{8}{15}\left(5-\frac{1}{b^{2}}\right)(2B^{2}+C^{2})+4A^{2}(2B+C)+\frac{32}{15}B(2B^{2}+3C^{2})$
$\displaystyle+\frac{1}{2}A^{4}+\frac{32}{35}\left(B^{4}+\frac{3}{8}C^{4}+3B^{2}C^{2}\right)+\frac{8}{5}A^{2}(2B^{2}+C^{2}+2BC)=0.$
(13d)

The system of four equations has a Lagrangian

$\mathcal{L}_{\ell}=K-U_{\ell},$ (14a) with
$U_{\ell}=U+\frac{\ell^{2}}{2b\left(A^{2}+\frac{8}{3}C^{2}\right)}.$ (14b)

Here $K$ is as in (10) and $U$ as in (11). Note that the frequency $\omega$
has disappeared from the system (13)-(14), along with $\dot{\theta}$. The role
of the single parameter has been taken over by the quantity $\ell$. We also
note that equations (13) conserve energy:

$H=K+U_{\ell}.$ (15)

###  2.2 Fixed points

Stationary solutions of the system (13) are given by critical points of the
function $U_{\ell}(A,B,C,b)$. The numerical analysis reveals that one such
point, continuously dependent on $\ell$, exists for all $0\leq\ell<\infty$
(see Appendix A). This nontrivial fixed point exists for the same reason why a
classical particle moving in a centrally-symmetric confining potential has an
equilibrium orbit with any nonzero angular momentum. The components of the
fixed point are presented graphically in Fig 1(a-b). The energy of the
stationary point,

$H(\ell)=U_{\ell}\left[A(\ell),B(\ell),C(\ell),b(\ell)\right],$ (16)

is shown in Fig 1(c).

Figure 1:  The fixed-point solutions of the variational equations (13). (a)
amplitude of the first harmonic and width parameter; (b) amplitudes of the
zeroth and second harmonic; (c) energy of the fixed point.

It is worth mentioning here that our numerical continuation algorithm
determines the components of the fixed point as functions of $\omega$ (rather
than $\ell$). Following the branch of fixed points that extends from
$\omega_{0}=2$ to $\omega_{c}=0.417\,\omega_{0}$ before reversing to
$\omega_{b}=0.527\,\omega_{0}$, we compute $\ell(\omega)=\omega
b(A^{2}+\frac{8}{3}C^{2})$ (see Fig 6 in Appendix A). The functions $A(\ell)$,
$B(\ell)$, $C(\ell)$, $b(\ell)$ and $H(\ell)$ in Fig 1 are then plotted as
parametric curves. The asymptotic regime $\omega\to\omega_{0}$ corresponds to
$\ell\to 0$; as the branch $\omega$ approaches its terminal point
$\omega_{b}$, we have $\ell\to\infty$.

Returning to the partial differential equation (2), the energy of its
solutions is given by

$E[\Phi]=\frac{1}{2}\int\left[\Phi_{t}^{2}+\Phi_{x}^{2}+(\Phi^{2}-1)^{2}\right]\,dx.$
(17)

Substituting the ansatz (7) in the integral (17) gives a time-dependent
quantity that cannot be used as a bifurcation measure of the trial function
(7). However when the collective coordinates $A,B,C$ and $b$ take constant
values, the integral (17) becomes periodic (with period $T$). In that case, we
can define the average energy carried by the configuration (7):

${\overline{E}}=\frac{1}{T}\int_{0}^{T}E[\Phi]\,d\mathcal{T}_{0}.$ (18)

Performing integration over $\mathcal{T}_{0}$ and $x$, equations (17) and (18)
give $\overline{E}=U_{\ell}(A,B,C,b)$, where $U_{\ell}$ is as in (14b). Thus,
the average field energy carried by the critical trial function (7) agrees
with the energy function of the variational equations (13) evaluated at the
corresponding fixed point:

$\overline{E}(\ell)=H(\ell)=U_{\ell}\left[A(\ell),B(\ell),C(\ell),b(\ell)\right].$
(19)

###  2.3 Stability of the fixed point

In order to classify the stability of the fixed points, it is sufficient to
examine perturbations preserving the integral $\ell$. Linearising equations
(13) and assuming the time dependence of the form

$(\delta A,\delta B,\delta C,\delta
b)^{T}=e^{(\lambda/\epsilon)\mathcal{T}_{1}}{\vec{y}},$ (20)

where ${\vec{y}}$ is a constant 4-vector, gives a generalised eigenvalue
problem

$M{\vec{y}}=-\lambda^{2}P{\vec{y}}.$ (21)

Here $M$ and $P$ are $4\times 4$ symmetric real matrices. The $M$ matrix is
given in the Appendix B while

$P=\left(\begin{array}[]{cccc}2b&0&0&A\\\ 0&\frac{8}{3}b&0&\frac{4}{3}B\\\
0&0&\frac{4}{3}b&\frac{2}{3}C\\\
A&\frac{4}{3}B&\frac{2}{3}C&P_{44}\end{array}\right),$ (22)
$P_{44}=\pi^{2}\frac{5A^{2}+16B^{2}+8C^{2}}{90b}+\frac{2}{3}\frac{A^{2}}{b}.$

Note that the ansatz (20) tacitly assumes $\lambda$ being of order $\epsilon$.

The determinant of $P$ is given by

$\det
P=\frac{32b^{2}}{405}\left[5(\pi^{2}+3)A^{2}+2(4\pi^{2}-15)(2B^{2}+C^{2})\right].$
(23)

Since $\det P>0$, Sylvester’s criterion implies that $P$ is positive definite.
Therefore, all generalised eigenvalues $-\lambda^{2}$ are real and the
exponents form pairs of real or pure imaginary opposite values, $\lambda$ and
$-\lambda$.

The generalised eigenvalues are computed numerically; the results are in Fig
2. A pair of pure-imaginary eigenvalues emerges from the origin as $\ell$ is
increased from zero. These stable eigenvalues account for the stability of the
Dashen-Hasslacher-Neveu’s oscillon in the limit $\omega\to\omega_{0}$, where
its slowly-varying amplitude satisfies the nonlinear Schrödinger equation. All
other eigenvalues shown in Fig 2 (including three other pairs occurring for
small $\ell$) are of order 1 — which is inconsistent with the ansatz (20).
These eigenvalues do not carry any information on the stability properties of
the oscillon and should be disregarded.

The absence of $O(\epsilon)$ eigenvalues outside the asymptotic regime
$\omega\to\omega_{0}$ implies that the oscillon remains stable as $\omega$ is
reduced to lower values — for as long as our variational approximation remains
valid. Indeed, had the instability set in at some
$\omega_{\mathrm{cr}}<\omega_{0}$, it would have brought along slowly-varying
amplitude perturbations (of the periodic oscillation with the frequency
$\omega_{\mathrm{cr}}$). The associated $O(\epsilon)$ eigenvalues would have
been captured by the eigenvalue problem (21).

Figure 2:  Roots of the characteristic equation $\det(M+\lambda^{2}P)=0$. The
generalised eigenvalues $\lambda^{2}$ are shown as functions of $\ell$. The
blow-up of a segment of the branch emerging from the origin highlights the
fact that the eigenvalue never returns to zero.

##  3 Numerical standing waves

###  3.1 Energy-frequency diagram

To assess the accuracy of the variational approximation (7), we consider
standing-wave solutions of the equation (2). The standing waves (also known as
nanopterons Boyd2  and quasibreathers Fodor2 ) are temporally periodic
solutions assuming prescribed values at the ends of the finite interval
$-R\leq x\leq R$. Confining the analysis to spatially symmetric (even)
standing waves, we determine these as solutions of a boundary-value problem
posed on a rectangular domain $(0,R)\times(0,T)$. Equation (2) with the
boundary conditions

$\Phi_{x}(0,t)=\Phi(R,t)=0;\quad\Phi(x,0)=\Phi(x,T)$ (24)

was solved by a path-following algorithm with the Newtonian iteration. The
half-length of the interval, $R$, was set to $40$.

A typical solution consists of a localised core and a non-decaying small-
amplitude wing, resulting from the interference of the outgoing radiation and
radiation reflected by the boundary at $x=R$.

Fig 3 shows the energy

$E_{R}[\Phi]=\int_{0}^{R}\left[\Phi_{t}^{2}+\Phi_{x}^{2}+(\Phi^{2}-1)^{2}\right]\,dx$
(25)

of solutions of the boundary-value problem (2), (24). There are two features
that attract attention in the figure.

Figure 3:  The energy (25) of the numerical standing-wave solution (blue). The
vertical dashed lines are drawn at the subharmonic resonance frequencies
$\omega=\Omega^{(n)}/2$, where $\Omega^{(n)}$ are the frequencies (27) of the
linear waves. The red dotted arc underlying the numerical curve is the
envelope of the family of spikes. (For visual clarity, it has been shifted
down by a tiny amount from its actual position.) Clearly visible are
discontinuities of the envelope at $\omega/\omega_{0}=1/2$ and
$\omega/\omega_{0}=1/3$. The pink-tinted bands demarcate the frequency
intervals where all Floquet multipliers lie on the unit circle.

First, the energy-frequency diagram exhibits what appears to be a sequence of
spikes. On a closer examination, each “spike” turns out to consist of a pair
of $E(\omega)$ branches rising steeply but not joining together. As the point
$(\omega,E(\omega))$ climbs up the slope of a spike, the amplitude of the
standing wave’s wing grows — this accounts for the rapid growth of the energy
of the solution.

The spikes are caused by the resonance between the double frequency of the
core of the standing wave and the eigenfrequencies $\Omega^{(n)}$ of the
linear standing waves. The linear standing waves are given by

$\Phi=1+\epsilon\sin(\Omega^{(n)}\mathcal{T}_{0})\cos(\kappa^{(n)}x),$ (26)

where

$\Omega^{(n)}=\sqrt{\omega_{0}^{2}+\left(\kappa^{(n)}\right)^{2}},\quad\kappa^{(n)}=\frac{\pi}{R}\left(n+\frac{1}{2}\right)\quad(n=0,1,2,...).$
(27)

(Similar resonances have been detected in the three-dimensional version of the
$\Phi^{4}$ model Alex_PRD .) The positions of the $\frac{1}{2}$ undertones of
the linear waves are marked by the vertical dashed lines in Fig 3. One can
clearly see a correspondence between the positions of the “spikes” and the
points $\omega=\Omega^{(n)}/2$ through which the vertical lines are drawn.

The positions of the spikes are sensitive to the choice of the interval half-
length, $R$. Let $\omega$ be a fixed frequency with
$\omega_{0}/2<\omega<\omega_{0}$ and $R_{a}$ an arbitrarily chosen half-length
($R_{a}\gg 1$). Denote $k$ the wavenumber of the second-harmonic radiation,
satisfying the dispersion relation

$(n\omega)^{2}=\omega_{0}^{2}+k^{2}$ (28)

with $n=2$. By tuning $R$ to a suitably chosen value $R_{\omega}$ within the
interval $(R_{a},R_{a}+\pi/k)$, the amplitude of the “wing” can be minimized
(yet not reduced to zero). The corresponding value of $E_{R}$ gives the
minimum energy of the family of standing waves with frequency $\omega$:
$E_{\rm min}(\omega)=E_{R_{\omega}}$. The graph of $E_{\rm min}(\omega)$
comprises segments of the $E_{R_{a}}(\omega)$ curve outside the neighbourhoods
of the spikes, while the full, gapless, $E_{\rm min}(\omega)$ arc can be
obtained as the envelope of the family of curves with $R$ in
$(R_{a},R_{a}+\pi/k)$.

Another aspect of Fig 3 that is worth commenting on, concerns the
fragmentation of the $E(\omega)$ curve into three disjoint branches with
$\omega$ in the intervals $(\omega_{*},\frac{1}{3}\omega_{0})$,
$(\frac{1}{3}\omega_{0},\frac{1}{2}\omega_{0})$ and
$(\frac{1}{2}\omega_{0},\omega_{0})$, respectively. Here
$\omega_{*}=0.289\,\omega_{0}$ is the lowest frequency value for which we
obtained a solution of the boundary-value problem (2), (24). The endpoints of
the intervals, $\omega_{0}/2$ and $\omega_{0}/3$, are marked by a rapid growth
of the energy $E_{\rm min}$ as $\omega$ approaches these values from the
right, and by finite energy jumps — as $\omega$ approaches these values from
the left.

To explain the fragmentation theoretically, we turn to the dispersion relation
(28). According to (28), the $n$-th harmonic radiation can only be emitted by
the core oscillating with the frequency $\omega>\omega_{0}/n$. In the event of
the coexistence of the $n$-th and $(n+1)$-th harmonics, the lower (the $n$-th)
overtone will be energetically dominant. Therefore the radiation waves should
be dominated by the second, third and fourth harmonic in the frequency ranges
$(\frac{1}{2}\omega_{0},\omega_{0})$,
$(\frac{1}{3}\omega_{0},\frac{1}{2}\omega_{0})$, and
$(\frac{1}{4}\omega_{0},\frac{1}{3}\omega_{0})$, respectively. The harmonic
waves forming the wings of our numerically obtained nanopterons do comply with
this selection rule.

Raising $\omega$ through $\omega_{0}/2$ opens a channel of the quadratic
radiation — a more powerful channel than the $n=3$ channel available for
$\omega<\omega_{0}/2$. The amplitude of the wing increases and this explains
the energy jump in Fig 3. Similarly, the energy jump observed as $\omega$ is
increased through $\omega_{0}/3$ is due to the turning on of the third-
harmonic channel, not accessible to the standing waves with
$\omega<\omega_{0}/3$. We note that the energy jumps have the same origin as
the staccato flashes of radiation from a slowly fading oscillon Dorey ; Nagy .

Assume now that $\omega$ approaches $\omega_{0}/2$ from the right. The
wavelength $\lambda=2\pi/k$ of the second-harmonic radiation grows while the
width of the wave’s core drops from its large values characteristic for
frequencies close to $\omega_{0}$. (The decrease of $b$ as $\ell$ is raised
from zero is clearly visible in the variational results of Fig 1.) The
convergence of $b$ and $\lambda$ produces a resonant growth of the amplitude
of the second-harmonic radiation which, in turn, gives rise to the energy
hike. A similar argument explains the energy growth occurring as $\omega$ is
reduced towards $\omega_{0}/3$.

###  3.2 Practical stability of standing waves

To classify the stability of the standing-wave solution, we linearise equation
(2) about $\Phi(x,t)$:

$y_{tt}-y_{xx}-2y+6\Phi^{2}y=0.$ (29)

Equation (29) is supplemented with the boundary conditions

$y_{x}(0,t)=y(R,t)=0;$ (30)

that is, we confine our study to perturbations sharing the symmetry of the
standing wave and vanishing at the same point on the $x$-line.

Having expanded $y(x,t)$ in the cosine Fourier series in the interval $(0,R)$
and keeping only the first $N$ harmonics, we have evaluated the monodromy
matrix of the $T$-periodic solution for each $\omega=2\pi/T$ in Fig 3. (We
took $N=512$.) If all eigenvalues $\mu_{n}$ ($n=1,2,...,2N$) of the monodromy
matrix satisfy $|\mu_{n}|=1$ , the periodic solution $\Phi(x,t)$ is stable. If
there are Floquet multipliers with $|\mu_{n}|>1$, the standing wave $\Phi$ is
deemed linearly unstable; however, the growth of the unstable perturbations
should not necessarily produce a noticeable deformation of the wave’s core.

The frequency intervals with no multipliers outside the unit circle are
indicated by pink bands in Fig 3. In particular, the entire region
$0.903\,\omega_{0}<\omega<\omega_{0}$ is found to be stable. As $\omega$ is
continuously turned down from $0.903\,\omega_{0}$, a pair of real eigenvalues
($\mu_{m}$ and $\mu_{m}^{-1}$) repeatedly emerge and return to the unit
circle. (The intervals of $\omega$ characterised by the presence of
$\mu_{m}>1$ are left blank in Fig 3.) After the frequency has reached below
$0.798\,\omega_{0}$, the off-circle pair remains in the Floquet spectrum for
any further decrease of $\omega$.

The instability occurring in parts of the range $\omega\gtrsim
0.75\,\omega_{0}$ is weak. (Here we are assuming that the solution is not on
the slope of a resonant peak). Specifically, the growth rate
$\lambda=\frac{1}{T}\ln\mu_{m}$ associated with the Floquet multiplier
$\mu_{m}>1$ is bounded by $3\times 10^{-3}$. As $\omega$ is decreased below
$0.75\,\omega_{0}$, the unstable real eigenvalue becomes larger. In addition,
complex quadruplets $\\{\mu,\mu^{*},\frac{1}{\mu},\frac{1}{\mu^{*}}\\}$ emerge
from the unit circle.

To understand the effect of instability, we have carried out direct numerical
simulations of equation (2) with initial conditions in the form of an unstable
standing wave. In all cases that we examined, the evolution of the instability
affected the amplitude and phase of the wing of the nanopteron, but it never
led to any significant deformation of its core. Since the resulting changes
would not be noticeable in most physical settings, we are referring to the
standing-wave solutions of equation (2) as practically stable.

##  4 Numerical solution vs variational approximation

Fig 4 compares the energy of the standing wave with the average energy (19)
carried by the trial function (7). In the trial function, $A,B,C$ and $b$ are
taken to be the components of the fixed-point solution of equations (13). For
further comparison, we have included the energy of the asymptotic solution
(3),

$E=\frac{4}{3}\epsilon+\frac{32}{81}\epsilon^{3},$ (31)

where $\epsilon=\sqrt{\omega_{0}^{2}-\omega^{2}}$ and corrections higher than
$\epsilon^{3}$ have been dropped.

Figure 4:  The energy (17) of the standing-wave solution determined
numerically (blue), with the envelope curve plotted in red. (Unlike Fig 3, the
envelope is shown in its true position here.) The black line depicts the
average energy (19) carried by the trial function (7) with parameters given by
the fixed-point of the variational equations. The dashed parabola indicates
the asymptotic result (31).

In the frequency range $(0.6\,\omega_{0},\omega_{0})$, the variational ansatz
(7) provides a fairly accurate approximation of the envelope of the numerical
$E(\omega)$ curve. Remarkably, the variational result is much closer to
$E_{\mathrm{min}}(\omega)$ than the asymptotic expansion (31). The stability
of the standing wave with $\omega>0.9\,\omega_{0}$ and its practical stability
for $\omega<0.9\,\omega_{0}$ are also reproduced by the variational approach.

As $\omega$ is reduced from $0.6\,\omega_{0}$ to $0.5\,\omega_{0}$, the
variational $\overline{E}(\omega)$ dependence deviates from its numerical
counterpart and once $\omega$ has fallen under $\omega_{0}/2$, the numerical-
variational correspondence breaks down entirely. The envelope of the numerical
curve in Figs 3 and 4 is split into three fragments, with $E_{\mathrm{min}}$
growing monotonically as $\omega$ decreases within each fragment. There is a
unique value of $E_{\mathrm{min}}$ for each $\omega$. By contrast, the
variational energy $\overline{E}(\omega)$ grows as $\omega$ is reduced from
$\omega_{0}$ to $\omega_{c}=0.417\,\omega_{0}$, but then the curve turns back
so that there are two coexisting branches for each $\omega$ in the interval
$(\omega_{c},\omega_{b})$, where $\omega_{b}=0.527\omega_{0}$.

The numerical analysis suggests two factors that contribute to the failure of
the variational approximation with low $\omega$. First, the amplitude of the
radiation emitted from the core of the wave (and reflected from the boundary
at $x=R$) grows as its frequency is decreased whereas the ansatz (7) does not
take into account the radiation wing at all. Second, as $\omega$ is lowered
below $\omega_{0}/2$, the contribution of the first harmonic to the Fourier
spectrum of the core decreases while the role of higher harmonics grows. As
$\omega$ falls under $0.45\,\omega_{0}$, the amplitude of the second harmonic
becomes larger than, and the amplitude of the third harmonic comparable to,
the amplitude of the first harmonic. When $\omega$ is under $0.4\,\omega_{0}$,
both second and third harmonics make larger contributions to the dynamics than
the first harmonic. As a result, the variational ansatz (7) — which keeps
$\cos(\omega\mathcal{T}_{0}+\theta)$ but does not include $\cos
3(\omega\mathcal{T}_{0}+\theta)$ — becomes inadequate.

##  5 Concluding remarks

Any collective coordinate approach to a localised structure aims to identify
the nonlinear modes that capture the essentials of its dynamics. The choice of
essential coordinates is validated by the agreement between the finite-
dimensional description of the structure and its true properties as revealed
by the numerical analysis. In this study, we utilised the multiscale
variational method to determine the nonlinear modes responsible for the
formation and stability of the one-dimensional $\phi^{4}$ oscillons.

The method was previously applied BA  to oscillons of the Kosevich-Kovalev
model KK , defined by the Lagrangian

$L=\frac{1}{2}\int\left(\phi_{t}^{2}-\phi_{x}^{2}-4\phi^{2}+\phi^{4}\right)dx.$
(32)

An ansatz comprising three variables,

$\phi=A\cos(\omega\mathcal{T}_{0}+\theta)\,\mathrm{sech}\left(\frac{x}{b}\right),$
(33)

was found to provide an accurate agreement with the numerical simulations of
the corresponding equation of motion. The $\phi^{4}$ Lagrangian of the present
paper,

$L=\frac{1}{2}\int\left(\phi_{t}^{2}-\phi_{x}^{2}-4\phi^{2}-4\phi^{3}-\phi^{4}\right)dx,$
(34)

is different from (32) in the presence of the symmetry-breaking $\phi^{3}$
term. This term would not contribute to the effective Lagrangian generated by
the trial function (33); the single-harmonic ansatz “does not see” the cubic
term. To capture the asymmetry in oscillations, we have had to expand the
single-harmonic ansatz (33) into a sum of three harmonics in (7). Accordingly,
the number of collective variables has increased from three to five.

Our variational study was complemented by the numerical analysis of equation
(2) on a periodic two-dimensional domain $(0,R)\times(0,T)$. Solutions of this
boundary-value problem are standing waves consisting of a localised core and
small-amplitude wings formed by the interference of the radiation emitted by
the core and radiation reflected from the boundaries. We obtained solutions
with all $\omega$ between $0.3\,\omega_{0}$ and $\omega_{0}$, where
$\omega=2\pi/T$ is the fundamental frequency of the wave and $\omega_{0}$ is
the endpoint of the continuous spectrum.

Results are summarised in Fig 3 which shows the energy of the standing wave as
a function of its frequency. The diagram features a sequence of spikes
generated by the resonant growth of the wing amplitude. The envelope of the
family of $E_{R}(\omega)$ curves with varied $R$ gives the energy curve of the
standing wave with the thinnest wings. The envelope is found to be fragmented
according to the dominant radiation harmonic (second, third, or higher).
Another noteworthy conclusion of the numerical analysis is the practical
stability of the entire branch of standing waves.

Fig 4 attests to a good agreement between the numerical energy-frequency
dependence and its variational counterpart $\overline{E}(\omega)$ in the
interval $(0.6\,\omega_{0},\omega_{0})$. There also is consistency between the
stability of the variational fixed point and practical stability of the
standing waves.

As $\omega$ is decreased below $0.6\,\omega_{0}$, the agreement deteriorates
and then breaks down completely. While the energy of the numerical solution
changes monotonically within each of the three fragments of its $E(\omega)$
diagram, the variational curve turns back into a coexisting branch of fixed
points (Fig 4). It remains a challenge to determine whether the standing-wave
counterpart of the coexisting branch actually exists.

###### Acknowledgements.

This research was supported by the NRF of South Africa (grant No
SRUG2204285129).

##  Appendix A Appendix: Fixed points of the variational equations

The fixed points of the dynamical system (13) satisfy four simultaneous
algebraic equations

$\displaystyle\frac{1}{3b^{2}}-1+A^{2}+\frac{4}{5}C^{2}+\frac{4}{5}\left(\frac{5}{2}+2B+C\right)^{2}=\omega^{2},$
(35a)
$\displaystyle\frac{1}{3}\left(5+\frac{1}{b^{2}}\right)B+\frac{5}{4}A^{2}+2B^{2}+C^{2}+\frac{1}{2}A^{2}(2B+C)+\frac{2}{7}B(2B^{2}+3C^{2})=0,$
(35b)
$\displaystyle\left(5+\frac{1}{b^{2}}\right)C+\frac{15}{4}A^{2}+12BC+3A^{2}(B+C)+\frac{9}{7}(4B^{2}+C^{2})C=5\omega^{2}C,$
(35c)
$\displaystyle\frac{5}{8}\left(4-\frac{1}{3b^{2}}\right)A^{2}+\frac{1}{3}\left(5-\frac{1}{b^{2}}\right)(2B^{2}+C^{2})+\frac{5}{2}A^{2}(2B+C)+\frac{4}{3}B(2B^{2}+3C^{2})$
$\displaystyle+\frac{5}{16}A^{4}+\frac{1}{14}(8B^{4}+3C^{4}+24B^{2}C^{2})+A^{2}(2B^{2}+C^{2}+2BC)=\frac{5}{24}\omega^{2}\left(3A^{2}+8C^{2}\right).$
(35d)

Note that we have switched from the parametrisation by $\ell$ back to the
frequency $\omega$, where

$\omega=\frac{\ell}{b\left(A^{2}+\frac{8}{3}C^{2}\right)}.$ (36)

While the invariant $\ell$ characterises the dynamics (in particular,
stability) of solutions to equations (13), $\omega$ appears to be a more
convenient parameter in the search for roots of (35).

Expressing $A^{2}$ from equation (35a) and substituting it in (35b), we obtain

$b^{-2}=\frac{f_{\omega}(B,C)}{C+\frac{5}{2}}$ (37)

with

$\displaystyle f_{\omega}=10B+12B^{2}+6C^{2}+\frac{12}{7}B(2B^{2}+3C^{2})$
$\displaystyle+3\left(\frac{5}{2}+2B+C\right)\left[\omega^{2}+1-\frac{4}{5}C^{2}-\frac{4}{5}\left(\frac{5}{2}+2B+C\right)^{2}\right].$
(38)

A similar substitution of $A^{2}$ in (35c) yields

$b^{-2}=\frac{g_{\omega}(B,C)}{B+\frac{5}{4}}$ (39)

with

$\displaystyle
g_{\omega}=\frac{3}{2}(1+\omega^{2})\left(\frac{5}{2}+2B-\frac{4}{3}C\right)+\left[10+12B+\frac{9}{7}(4B^{2}+C^{2})\right]C$
$\displaystyle-\frac{6}{5}\left(\frac{5}{2}+2B+2C\right)\left[\left(\frac{5}{2}+2B+C\right)^{2}+C^{2}\right],$
(40)

while eliminating $A^{2}$ between (35a) and (35d) produces

$b^{-4}+p_{\omega}b^{-2}+q_{\omega}=0,$ (41)

where

$p_{\omega}(B,C)=2\left[\frac{16}{5}\left(B+\frac{5}{4}\right)\left(C+\frac{5}{2}\right)-\omega^{2}-6\right]$
(42)

and

$\displaystyle
q_{\omega}(B,C)=16\left[2B^{2}+(1-\omega^{2})C^{2}+\frac{4}{5}B(2B^{2}+3C^{2})\right.$
$\displaystyle+\left.\frac{3}{70}(8B^{4}+3C^{4}+24B^{2}C^{2})\right]-\frac{48}{25}\left[\left(\frac{5}{2}+2B+C\right)^{2}+C^{2}-\frac{5}{4}(\omega^{2}+1)\right]^{2}.$
(43)

Before reducing the number of equations further, it is fitting to note that
the system (37), (39) and (41) has a root reproducing the asymptotic expansion
(3):

$B=-\frac{\epsilon^{2}}{4},\quad C=\frac{\epsilon^{2}}{12},\quad
b^{-2}=\epsilon^{2}.$ (44)

Here we have defined a small parameter $\epsilon$ by letting, in equations
(37), (39) and (41), $\omega^{2}=4-\epsilon^{2}$. Using (35a) we recover the
amplitude of the first harmonic in the Dashen-Hasslacher-Neveu’s expansion:

$A^{2}=\frac{\epsilon^{2}}{3}.$ (45)

Returning to the system of three equations and using equation (37) to
eliminate $b^{-2}$ from (39) and (41), we arrive at

$\displaystyle\left(B+\frac{5}{4}\right)f_{\omega}(B,C)-\left(C+\frac{10}{4}\right)g_{\omega}(B,C)=0,$
(46a) $\displaystyle
f_{\omega}^{2}(B,C)+\left(C+\frac{10}{4}\right)p_{\omega}(B,C)f_{\omega}(B,C)+\left(C+\frac{10}{4}\right)^{2}q_{\omega}(B,C)=0.$
(46b)

For each $\omega$, equations (46) with
$f_{\omega},g_{\omega},p_{\omega},q_{\omega}$ as in (38), (40), (42) and (43),
comprise a system of two equations with two unknowns, $B$ and $C$.

For much of the $\omega$, the system (46) has multiple roots with real $B$ and
$C$ (Fig 5). However only roots satisfying $A^{2}>0$ and $b^{-2}>0$ correspond
to fixed points of the variational equations (13). Here $b^{-2}$ is given by
(37) and $A^{2}$ by equation

$A^{2}=\frac{h_{\omega}(B,C)}{2C+5}$ (47a) with
$h_{\omega}=-2B^{2}-C^{2}-B\left[\omega^{2}+\frac{8}{3}+\frac{2}{7}(2B^{2}+3C^{2})-\frac{4}{5}C^{2}-\frac{4}{5}\left(\frac{5}{2}+2B+C\right)^{2}\right],$
(47b)

which ensues from (35a)-(35b).

Figure 5: Graphical solution of the system (46) with $\omega=0.71\,\omega_{0}$
(a) and $\omega=0.45\,\omega_{0}$ (b). The red and blue curves are described
by equations (46a) and (46b), respectively. Tinted in light blue is the
“physical” region where the inequalities $A^{2}>0$ and $b^{-2}>0$ are
satisfied simultaneously, with $A^{2}$ and $b^{-2}$ as in (47) and (37).
Circles mark the roots of the system in the physical region.

There is only one such root for $\omega_{b}<\omega<\omega_{0}$, where
$\omega_{b}=0.527\,\omega_{0}$ (and $\omega_{0}=2$). In the vicinity of
$\omega_{0}$, the corresponding fixed point is given by equations (44)-(45).
The branch of fixed points extends from $\omega_{0}$ to
$\omega_{c}=0.417\,\omega_{0}$ where it folds onto itself. As we path-follow
the turning branch back to $\omega_{b}$, the $A$, $B$ and $C$ components of
the fixed point approach finite values while $b$ grows without bound. As a
result, the adiabatic invariant $\ell=\omega b(A^{2}+\frac{8}{3}C^{2})$ and
the energy (16) tend to infinity as well (Fig 6).

Figure 6: The adiabatic invariant $\ell$ and energy $H$ of the fixed point as
functions of $\omega$.

##  Appendix B Appendix: Linearisation matrix

The matrix elements $M_{ij}=M_{ji}$ in equation (21) are given by

$\displaystyle
M_{11}=24\omega^{2}\frac{A^{2}b}{3A^{2}+8C^{2}}+\frac{2}{3b}+2b\left[3A^{2}+\frac{4}{5}\left(\frac{5}{2}+2B+C\right)^{2}+\frac{4C^{2}}{5}-\omega^{2}-1\right];$
$\displaystyle M_{12}=\frac{32}{5}Ab\left(\frac{5}{2}+2B+C\right);\quad
M_{13}=64\omega^{2}\frac{ACb}{3A^{2}+8C^{2}}+\frac{16}{5}Ab\left(\frac{5}{2}+2B+2C\right);$
$\displaystyle
M_{14}=4\omega^{2}A+\frac{8}{5}A\left(\frac{5}{2}+2B+C\right)^{2}+2A\left(A^{2}+\frac{4C^{2}}{5}-\frac{1}{3b^{2}}-\omega^{2}-1\right);$
$\displaystyle
M_{22}=\frac{32}{5}A^{2}b+\frac{32}{15b}+\frac{32}{105}b\left[36\left(B+\frac{7}{6}\right)^{2}+18C^{2}-14\right];\quad
M_{23}=\frac{16}{35}b\left(7A^{2}+24CB+28C\right);$ $\displaystyle
M_{24}=\frac{16}{5}A^{2}\left(\frac{5}{2}+2B+C\right)+\frac{192}{35}\left(B+\frac{7}{6}\right)C^{2}+\frac{8}{15}B\left[\frac{48}{7}\left(B+\frac{7}{4}\right)^{2}-1\right]-\frac{32}{15}\frac{B}{b^{2}};$
$\displaystyle
M_{33}=\frac{512}{3}\omega^{2}\frac{C^{2}b}{3A^{2}+8C^{2}}+\frac{192}{35}b\left(B+\frac{7}{6}\right)^{2}+\frac{16}{15b}+\frac{16}{105}b\left(21A^{2}+27C^{2}-35\omega^{2}-14\right);$
$\displaystyle
M_{34}=\frac{32}{3}\omega^{2}C+\frac{48}{35}C^{3}+\frac{16}{15}\left[3A^{2}+\frac{36}{7}\left(B+\frac{7}{6}\right)^{2}-5\omega^{2}-2\right]C+\frac{16}{5}A^{2}\left(B+\frac{5}{4}\right)-\frac{16}{15}\frac{C}{b^{2}};$
$\displaystyle
M_{44}=\frac{2}{3}\frac{\omega^{2}}{b}(3A^{2}+8C^{2})+\frac{2}{15b^{3}}\left(5A^{2}+16B^{2}+8C^{2}\right).$
(48)

Here $A,B,C$ and $b$ are components of the real root of the system (35).

Note that the linearisation of equations (13) preserves $\ell$:
$\delta\ell=0$. The frequency $\omega$ does not appear in (13) but once the
linearisation procedure has been completed, we reintroduce $\omega$ according
to

$\omega=\frac{\ell}{b\left(A^{2}+\frac{8}{3}C^{2}\right)},$

in agreement with (36). In the matrix elements (48), the variables $A,B,C$ and
$b$ are single-valued functions of $\ell$ — and $\omega$ is also considered as
a function of $\ell$.

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