# TRAVELLING WAVES IN NONLINEAR MAGNETO-INDUCTIVE LATTICES

M. Agaoglou  Lab of Nonlinear Mathematics and Department of Mechanical
Engineering, Faculty of Engineering,
Aristotle University of Thessaloniki, Thessaloniki 54124, Greece  M. Fečkan
Department of Mathematical Analysis and Numerical Mathematics, Comenius
University in Bratislava,
Mlynská dolina, 842 48 Bratislava, Slovakia  Mathematical Institute of Slovak
Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia  M. Pospíšil
Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73
Bratislava, Slovakia  V.M. Rothos  Lab of Nonlinear Mathematics and Department
of Mechanical Engineering, Faculty of Engineering,
Aristotle University of Thessaloniki, Thessaloniki 54124, Greece  H. Susanto
Department of Mathematical Sciences, University of Essex, Wivenhoe Park,
Colchester CO4 3SQ,
United Kingdom

###### Abstract

We consider a lattice equation modelling one-dimensional metamaterials formed
by a discrete array of nonlinear resonators. We focus on periodic travelling
waves due to the presence of a periodic force. The existence and uniqueness
results of periodic travelling waves of the system are presented. Our
analytical results are found to be in good agreement with direct numerical
computations.

Keywords: travelling wave, lattice wave, forced magneto-inductive lattice

##  I Introduction

In this work, we consider [27]

$\begin{gathered}\frac{d^{2}}{dt^{2}}(u_{n}-\lambda u_{n-1}-\lambda
u_{n+1})+\gamma\frac{d}{dt}u_{n}+u_{n}+\frac{d^{2}}{dt^{2}}(u_{n}^{2}-\lambda
u_{n-1}^{2}-\lambda u_{n+1}^{2})+\gamma\frac{d}{dt}u_{n}^{2}-h(\omega
t+pn)=0\end{gathered}$ (1)

where $\gamma\geq 0$, $\lambda,\omega>0$, $p\neq 0$ are parameters and $h\in
C(\mathbb{R},\mathbb{R})$ is such that

$h(z)=\sum_{k\in\mathbb{Z}}h_{k}{\,\textrm{\rm e}}^{k\imath
z},\qquad\sum_{k\in\mathbb{Z}}|h_{k}|<\infty.$

The equation models the dynamics of electromagnetic waves in the so-called
magneto-inductive metamaterials.

Metamaterials are artificial materials that are engineered to have properties
that may not be found in nature [5]. The (structural rather than chemical)
engineering is achieved by composing periodic inhomogeneities to create
desirable effective behaviour. The invention ignited a new paradigm in
electromagnetism, including cloaking devices [21] (see also [18] for a recent
comprehensive review of electromagnetic manipulation enabled by
metamaterials). Magnetic metamaterials are non-magnetic materials exhibiting
magnetic properties in the Terahertz and optical frequencies. They were
predicted theoretically in [12, 29] and demonstrated experimentally in, e.g.,
[7, 11, 14, 28]. When the materials composed of magnetic metamaterials, e.g.,
split-ring resonators (SRRs), are magnetically weakly coupled through their
mutual inductances, one obtains magneto-inductive metamaterials [9, 22, 23,
24]. Magneto-inductive metamaterials consisting of periodic arrays of SRRs
have been made in one-, two- and three dimensions [20]. Both the dimensions of
SRRs and their inter-space distance are small relative to the free space
wavelength and, thus, the dynamics of electromagnetic fields in such settings
are governed by the quasi magnetostatic approximation [13].

Model equations for the propagation of nonlinear electromagnetic waves in
metamaterials can be grouped into two classes. The first approach assumes that
effective metamaterials are homogeneous media with specific physical
properties resulting in partial differential equations, such as coupled short-
pulse equation [25] and higher-order nonlinear Schrödinger equations [26]. In
the second class, metamaterials are modelled by arrays of coupled oscillators,
i.e. lattice equations, such as a nonlinear Klein-Gordon equation [19] and
coupled Klein-Gordon equations [15, 17]. The governing equation (1) falls into
the second approach with $\gamma$ representing the loss coefficient, $\lambda$
is the coupling parameter, $h$ is an external forcing that is periodic in time
and varying in the spatial direction.

In the lattice equation (1) the nonlinearity appears in the coupling terms and
in the damping. This is due to the assumption of the nonlinearity of the
capacitance of the slit-ring resonators that compose the magneto-inductive
materials [27]. Note that the nonlinearity is different from our previous work
[1, 6], where it is in the onsite potential. The present work is to study the
effect of such nonlinearity. Here, we investigate the existence of travelling
periodic solutions of system (1) due to the periodic forcing, and the
bifurcation of such solutions with small $\gamma$, $\lambda$ and $h$. We also
study the modulational stability of the periodic solutions by computing
Floquet multipliers of the linearised systems.

Note that in our governing equation (1) the nonlinearity in the coupling terms
between the sites is akin to that in the Fermi-Pasta-Ulam lattices [10].
However, there is a significant difference in fact that the coupling in our
governing equation is also the derivative term. The bifurcation structures of
periodic solutions in general FPU lattices forced by periodic drive were
studied in [8]. It is imperative to study the effect of the present nonlinear
couplings to periodic solutions caused by the same periodic drive. Using
Lyapunov-Schmidt reduction, we derive the asymptotic expressions of the
bifurcating solutions.

The present paper is organized as follows. In Section II we are looking for a
periodic travelling wave when its amplitude is limited by a function of the
magnitude of the forcing. In the section, we prove the existence of such waves
rigorously. The resonance condition for the parameter values is also derived.
In Section III, we study the resonance region. Using the Lyapunov-Schmidt
reduction, we show that the periodic solutions persist. Several explicit
examples of the general results obtained in the previous sections are
presented in Section IV. In Section V we solve the governing equations
numerically. Comparisons with the analytical results are presented where we
obtain good agreement.

##  II Existence of small periodic solutions

Putting $u_{n}(t)=U(\omega t+pn)$, $z=\omega t+pn$ in (1), we obtain

$\begin{gathered}\omega^{2}(U(z)-\lambda U(z-p)-\lambda
U(z+p))^{\prime\prime}+\gamma\omega U^{\prime}(z)+U(z)\\\
{}+\omega^{2}(U^{2}(z)-\lambda U^{2}(z-p)-\lambda
U^{2}(z+p))^{\prime\prime}+\gamma\omega(U^{2}(z))^{\prime}-h(z)=0.\end{gathered}$
(2)

We take Banach spaces

$\begin{gathered}X:=\left\\{U\in C(\mathbb{R},\mathbb{R})\mid
U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm e}}^{k\imath
z},\,\sum_{k\in\mathbb{Z}}|c_{k}|<\infty\right\\},\\\ Y:=\left\\{U\in
C^{1}(\mathbb{R},\mathbb{R})\mid U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm
e}}^{k\imath
z},\,\sum_{k\in\mathbb{Z}\backslash\\{0\\}}|k||c_{k}|<\infty\right\\},\\\
Z:=\left\\{U\in C^{2}(\mathbb{R},\mathbb{R})\mid
U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm e}}^{k\imath
z},\,\sum_{k\in\mathbb{Z}\backslash\\{0\\}}k^{2}|c_{k}|<\infty\right\\}\end{gathered}$

with the norms

$\|U\|:=\sum_{k\in\mathbb{Z}}|c_{k}|,\quad\|U\|_{1}:=|c_{0}|+\sum_{k\in\mathbb{Z}\backslash\\{0\\}}|k||c_{k}|,\quad\|U\|_{2}:=|c_{0}|+\sum_{k\in\mathbb{Z}\backslash\\{0\\}}k^{2}|c_{k}|,$

respectively. It is easy to verify that $Z\circlearrowleft Y\circlearrowleft
X$ are compact embeddings and $\|U\|\leq\|U\|_{1}\,\forall U\in Y$,
$\|U\|_{1}\leq\|U\|_{2}\,\forall U\in Z$.

The following lemma is clear.

######  Lemma 1.

If $U_{1},U_{2}\in X$ then $U_{1}U_{2}\in X$ and
$\|U_{1}U_{2}\|\leq\|U_{1}\|\|U_{2}\|$.

By setting

$\begin{gathered}{\mathcal{K}}U:=\omega^{2}\left(U^{\prime\prime}(z)-\lambda
U^{\prime\prime}(z-p)-\lambda U^{\prime\prime}(z+p)\right)+\gamma\omega
U^{\prime}(z)+U(z),\\\
{\mathcal{F}}(U,h):=-\omega^{2}\left((U^{2}(z))^{\prime\prime}-\lambda(U^{2}(z-p))^{\prime\prime}-\lambda(U^{2}(z+p))^{\prime\prime}\right)-\gamma\omega(U^{2}(z))^{\prime}+h(z),\end{gathered}$

equation (2) has the form

${\mathcal{K}}U={\mathcal{F}}(U,h)\,.$

We have the next result.

######  Lemma 2.

Function ${\mathcal{F}}:Z\times X\to X$ fulfils

$\|{\mathcal{F}}(U,h)\|\leq
2\omega(2\omega+4\lambda\omega+\gamma)\|U\|_{2}^{2}+\|h\|,$ (3)
$\|{\mathcal{F}}(U_{1},h)-{\mathcal{F}}(U_{2},h)\|\leq
2\omega(2\omega+4\lambda\omega+\gamma)\|U_{1}-U_{2}\|_{2}(\|U_{1}\|_{2}+\|U_{2}\|_{2}),$
(4) $\|{\mathcal{F}}(U,h_{1})-{\mathcal{F}}(U,h_{2})\|\leq\|h_{1}-h_{2}\|$ (5)

for any $U,U_{1},U_{2}\in Z$ and $h,h_{1},h_{2}\in X$.

###### Proof.

Property (5) is obvious. Next, we use $\|U^{\prime}\|=\|U\|_{1}\leq\|U\|_{2}$,
$\|U^{\prime\prime}\|=\|U\|_{2}$ with Lemma 1 to derive

$\|(U^{2}(z+c))^{\prime\prime}\|=2\|(U^{\prime}(z+c))^{2}+U(z+c)U^{\prime\prime}(z+c)\|\leq
4\|U(z+c)\|_{2}^{2}$

for $c=-p,0,p$. Moreover,

$\|U(z\pm p)\|=\sum_{k\in\mathbb{Z}}|c_{k}||{\,\textrm{\rm e}}^{k\imath
p}|=\sum_{k\in\mathbb{Z}}|c_{k}|=\|U\|$

for $U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm e}}^{k\imath z}$. Similarly
we derive

$\|(U^{2}(z))^{\prime}\|=2\|U(z)U^{\prime}(z)\|\leq 2\|U\|_{2}^{2}.$

Hence, property (3) is obtained. Analogically, we get

$\begin{gathered}\|(U_{1}^{2}(z+c)-U_{2}^{2}(z+c))^{\prime\prime}\|\leq\|(U_{1}(z+c)-U_{2}(z+c))^{\prime\prime}(U_{1}(z+c)+U_{2}(z+c))\|\\\
{}+2\|(U_{1}(z+c)-U_{2}(z+c))^{\prime}(U_{1}(z+c)+U_{2}(z+c))^{\prime}\|+\|(U_{1}(z+c)-U_{2}(z+c))(U_{1}(z+c)+U_{2}(z+c))^{\prime\prime}\|\\\
\leq 4\|U_{1}-U_{2}\|_{2}\|U_{1}+U_{2}\|_{2}\end{gathered}$

for $c=-p,0,p$, and

$\begin{gathered}\|(U_{1}^{2}(z)-U_{2}^{2}(z))^{\prime}\|\leq\|(U_{1}(z)-U_{2}(z))^{\prime}(U_{1}(z)+U_{2}(z))\|\\\
{}+\|(U_{1}(z)-U_{2}(z))(U_{1}(z)+U_{2}(z))^{\prime}\|\leq
2\|U_{1}-U_{2}\|_{2}\|U_{1}+U_{2}\|_{2}\end{gathered}$

since $a^{2}-b^{2}=(a-b)(a+b)$ for any $a,b\in\mathbb{C}$. Now, (4) follows
easily. ∎

Next, if $U\in Z$ with $U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm
e}}^{k\imath z}$ then

${\mathcal{K}}U(z)=\sum_{k\in\mathbb{Z}}\left(1-\omega^{2}k^{2}+2\lambda\omega^{2}k^{2}\cos
kp+\imath\gamma\omega k\right)c_{k}{\,\textrm{\rm e}}^{k\imath z}$ (6)

and so ${\mathcal{K}}\in L(Z,X)$ with

$\|{\mathcal{K}}\|_{L(Z,X)}\leq 1+\omega^{2}(1+2\lambda)+\gamma\omega.$

If

$\Theta:=\inf_{k\in\mathbb{Z}\backslash\\{0\\}}\left\\{1,\sqrt{\left(\frac{1}{k^{2}}+\omega^{2}(2\lambda\cos
kp-1)\right)^{2}+\frac{\gamma^{2}\omega^{2}}{k^{2}}}\right\\}>0$ (7)

is a constant depending on $\gamma$, $\lambda$, $\omega$ and $p$, then we also
have ${\mathcal{K}}^{-1}\in L(X,Z)\subset L(X,Y)\subset L(X)$. So
${\mathcal{K}}^{-1}:X\to Z$ is continuous such that

$\|{\mathcal{K}}^{-1}\|_{L(X,Z)}\leq\frac{1}{\Theta}.$ (8)

Now we can prove the following existence result on (2) when all parameters
except $h$ are fixed.

######  Theorem 1.

Assume (7) along with

$0<\|h\|<\frac{\Theta^{2}}{8\omega(2\omega+4\lambda\omega+\gamma)}.$ (9)

Then equation (2) has a unique solution $U(h)\in{B(\rho_{h})}$ in a closed
ball

$B(\rho_{h}):=\\{U\in Z\mid\|U\|_{2}\leq\rho_{h}\\},$

where

$\rho_{h}=\frac{\Theta-\sqrt{\Theta^{2}-8\omega(2\omega+4\lambda\omega+\gamma)\|h\|}}{4\omega(2\omega+4\lambda\omega+\gamma)}.$
(10)

Moreover, $U(h)$ can be approximated by an iteration process. Finally, it
holds

$\|U(h_{1})-U(h_{2})\|_{2}\leq\frac{\|h_{1}-h_{2}\|}{\sqrt{\Theta^{2}-8\omega(2\omega+4\lambda\omega+\gamma)\max\\{\|h_{1}\|,\|h_{2}\|\\}}}$
(11)

for any $h_{1},h_{2}\in X$ satisfying (9).

###### Proof.

We rewrite (2) as a parameterized fixed point problem

$U={\mathcal{R}}(U,h):={\mathcal{K}}^{-1}{\mathcal{F}}(U,h)$

in $Z$. We already know that ${\mathcal{R}}:Z\times X\to Z$ is continuous and
by (3), (8) such that

$\|{\mathcal{R}}(U,h)\|_{2}\leq\frac{1}{\Theta}\left(2\omega(2\omega+4\lambda\omega+\gamma)\|U\|_{2}^{2}+\|h\|\right).$

Next, if there is $\rho_{h}>0$ such that

$A(\rho_{h}):=\Theta\rho_{h}-2\omega(2\omega+4\lambda\omega+\gamma)\rho_{h}^{2}=\|h\|,$
(12)

then ${\mathcal{R}}(\cdot,h)$ maps $B(\rho_{h})$ into itself. So it remains to
study (12). In order to find $H>0$ – the largest right-hand side of (12) when
this equation has a solution $\rho_{H}>0$, we need to solve $A(r)=H$ together
with ${\mathrm{D}}A(r)=0$. This implies

$\rho_{H}=\frac{\Theta}{4\omega(2\omega+4\lambda\omega+\gamma)}$

and (9). So assuming (7), (9), we know that (12) has a positive solution
$\rho_{h}<\rho_{H}$. We take the smallest one which clearly has the form (10).
So ${\mathcal{R}}(\cdot,h)$ maps $B(\rho_{h})$ into itself and, moreover, by
(4), (8)

$\|{\mathcal{R}}(U_{1},h)-{\mathcal{R}}(U_{2},h)\|_{2}\leq\frac{4\omega(2\omega+4\lambda\omega+\gamma)\|U_{1}-U_{2}\|_{2}\rho_{h}}{\Theta}$

for any $U_{1},U_{2}\in B(\rho_{h})$. Hence, ${\mathcal{R}}(\cdot,h)$ is a
contraction on $B(\rho_{h})$ with a contraction constant

$\frac{4\omega(2\omega+4\lambda\omega+\gamma)\rho_{h}}{\Theta}<\frac{4\omega(2\omega+4\lambda\omega+\gamma)\rho_{H}}{\Theta}=1.$

The proof of the existence and uniqueness is finished by the Banach fixed
point theorem [2]. Next, let $h_{1},h_{2}\in X$ satisfy (9). Then $U(h_{i})\in
B(\rho_{h_{i}})\subset B(\rho_{\widetilde{H}})$ for $i=1,2$ and
$\widetilde{H}:=\max\left\\{\|h_{1}\|,\|h_{2}\|\right\\}$. Note
$\widetilde{H}$ satisfies (9). By (4), (5) and (8), we derive

$\begin{gathered}\|U(h_{1})-U(h_{2})\|_{2}=\|{\mathcal{R}}(U(h_{1}),h_{1})-{\mathcal{R}}(U(h_{2}),h_{2})\|_{2}\\\
\leq\|{\mathcal{R}}(U(h_{1}),h_{1})-{\mathcal{R}}(U(h_{2}),h_{1})\|_{2}+\|{\mathcal{R}}(U(h_{2}),h_{1})-{\mathcal{R}}(U(h_{2}),h_{2})\|_{2}\\\
\leq\frac{4\omega(2\omega+4\lambda\omega+\gamma)\|U(h_{1})-U(h_{2})\|_{2}\rho_{\widetilde{H}}+\|h_{1}-h_{2}\|}{\Theta}\end{gathered}$

which implies (11). ∎

######  Remark 1.

In the following special cases, (7) holds and we can replace $\Theta$ with the
corresponding $\Theta_{i}$ in the above considerations:

1\. If $p\in 2\pi\mathbb{Z}$, $\lambda>\frac{1}{2}$, then

$\Theta\geq\Theta_{1}:=\min\\{1,\omega^{2}(2\lambda-1)\\}>0.$

2\. Let $\gamma=0$. If $p\in\pi\mathbb{Q}$, then we can write
$p=\frac{p_{1}}{p_{2}}\pi$ for some $p_{1}\in\mathbb{Z}$,
$p_{2}\in\mathbb{N}$, where $p_{1}$, $p_{2}$ are relatively prime (their only
common divisor is $1$). Moreover,

$M_{1}:=\\{\cos kp\mid
k\in\mathbb{Z}\backslash\\{0\\}\\}\subset\left\\{\cos\frac{k\pi}{p_{2}}\mid
k\in\\{0,1,\dots,2p_{2}-1\\}\right\\}=:M_{2}.$ (13)

Indeed, for each $k\in\mathbb{Z}$ there exist $i,j\in\mathbb{Z}$ such that
$0\leq i\leq 2p_{2}-1$ and $kp_{1}=2jp_{2}+i$. Hence, for $\cos kp\in M_{1}$
we have

$\cos kp=\cos\frac{kp_{1}\pi}{p_{2}}=\cos\frac{i\pi}{p_{2}}\in M_{2}.$

To find a better relationship between $M_{1}$ and $M_{2}$, we need to solve
$\cos\frac{\widetilde{k}_{1}p_{1}}{p_{2}}\pi=\cos\frac{k_{2}\pi}{p_{2}}$ for
$\widetilde{k}_{1}\in\mathbb{Z}\backslash\\{0\\}$ and
$k_{2}\in\\{0,1,\dots,2p_{2}-1\\}$. This is equivalent to
$\frac{k_{1}p_{1}}{p_{2}}\pi=\frac{k_{2}\pi}{p_{2}}+2\pi l$ for
$l\in\mathbb{Z}$ and $k_{1}=\pm\widetilde{k}_{1}$, i.e.,
$k_{1}p_{1}=k_{2}+2lp_{2}$. If $p_{1}$ is odd, then $p_{1}$ and $2p_{2}$ are
relatively prime. Thus, we know [3] that there exist
$\hat{k},\hat{l}\in\mathbb{Z}$ such that $\hat{k}p_{1}=1+2\hat{l}p_{2}$.
Consequently, $\hat{k}k_{2}p_{1}=k_{2}+2\hat{l}k_{2}p_{2}$ and $M_{1}=M_{2}$.
If $p_{1}$ is even, then $p_{1}=2\bar{p}_{1}$ and
$k_{1}2\bar{p}_{1}=k_{2}+2lp_{2}$, so $k_{2}=2\bar{k}_{2}$ and we have
$k_{1}\bar{p}_{1}=\bar{k}_{2}+lp_{2}$. Clearly $\bar{p}_{1}$ and $p_{2}$ are
relatively prime. Thus, there exist $\bar{k},\bar{l}\in\mathbb{Z}$ such that
$\bar{k}\bar{p}_{1}=1+\bar{l}p_{2}$. Then
$\bar{k}\bar{k}_{2}\bar{p}_{1}=\bar{k}_{2}+\bar{l}\bar{k}_{2}p_{2}$. Hence
$M_{1}=\left\\{\cos\frac{2k\pi}{p_{2}}\mid
k\in\\{0,1,\dots,p_{2}-1\\}\right\\}\subsetneqq M_{2}$.

Nevertheless, assuming

$\omega^{2}\notin\left\\{\frac{1}{k^{2}(1-2\lambda\cos
kp)}\right\\}_{k\in\mathbb{Z}\backslash\\{0\\}}$ (14)

and

$0\notin 2\lambda M_{2}-1$ (15)

for $M_{2}$ defined in (13), we get the existence of $\delta>0$ such that

$\sqrt{\left(\frac{1}{k^{2}}+\omega^{2}(2\lambda\cos
kp-1)\right)^{2}+\frac{\gamma^{2}\omega^{2}}{k^{2}}}\geq\omega^{2}\left|\frac{1}{\omega^{2}k^{2}}+2\lambda\cos
kp-1\right|\geq\delta>0$

for each $k\in\mathbb{Z}\backslash\\{0\\}$. Thus, if $p\in\pi\mathbb{Q}$ and
(14), (15) are valid, then

$\Theta\geq\min\left\\{1,\inf_{k\in\mathbb{Z}\backslash\\{0\\}}\omega^{2}\left|\frac{1}{\omega^{2}k^{2}}+2\lambda\cos
kp-1\right|\right\\}=:\Theta_{2}>0.$

Note that $\frac{1}{k^{2}(1-2\lambda\cos kp)}\to 0$ if $|k|\to\infty$. So if
$p\in\pi\mathbb{Q}$ and (15) holds, there is at most a finite number of
resonant modes $k_{0}\in\mathbb{Z}\backslash\\{0\\}$ determined by equation

$\omega^{2}=\frac{1}{k_{0}^{2}(1-2\lambda\cos k_{0}p)}.$ (16)

More precisely, for given $\lambda$, $\omega$ there is not more than
$2(p_{2}+1)$ resonant modes $k_{0}$. Indeed, let $\\{k_{0j}\\}_{j=1}^{J}$ be
an increasing sequence of positive resonant modes corresponding to the fixed
$\lambda$, $\omega$. Then $\\{1-2\lambda\cos k_{0j}p\\}_{j=1}^{J}$ has to be
decreasing, since

$k_{0j}^{2}(1-2\lambda\cos
k_{0j}p)=\frac{1}{\omega^{2}}=\text{const.}\quad\forall j=1,\dots,J,$

i.e., $\\{\cos k_{0j}p\\}_{j=1}^{J}$ is increasing. From properties of $\cos
x$ and (13), it follows that the longest sequence of the form $\cos k_{0j}p$
has the values $\\{\cos\frac{k\pi}{p_{2}}\mid k=p_{2},\dots,2p_{2}\\}$ which
has $p_{2}+1$ elements. Finally, to each $k_{0j}$ corresponds resonant mode
$-k_{0j}$.

3\. If $\gamma>0$, condition (7) is satisfied even without the non-resonance
condition (14), since in each resonant mode $k_{0}$ we have

$\sqrt{\left(\frac{1}{k_{0}^{2}}+\omega^{2}(2\lambda\cos
k_{0}p-1)\right)^{2}+\frac{\gamma^{2}\omega^{2}}{k_{0}^{2}}}=\frac{\gamma\omega}{|k_{0}|}>0$

and there is only the finite number of resonant modes. For each non-resonant
mode $k$ it holds

$\sqrt{\left(\frac{1}{k^{2}}+\omega^{2}(2\lambda\cos
kp-1)\right)^{2}+\frac{\gamma^{2}\omega^{2}}{k^{2}}}\geq\left|\frac{1}{k^{2}}+\omega^{2}(2\lambda\cos
kp-1)\right|>0.$

Summarizing, if $\gamma>0$, $p\in\pi\mathbb{Q}$ and (15) holds, condition (7)
is fulfilled and

$\Theta\geq\min\left\\{1,\min_{k\in
M}\frac{\gamma\omega}{|k|},\inf_{k\in\mathbb{Z}\backslash(\\{0\\}\cup
M)}\omega^{2}\left|\frac{1}{\omega^{2}k^{2}}+2\lambda\cos
kp-1\right|\right\\}=:\Theta_{3}>0$

where $M$ is the set of resonant modes $k_{0}$.

##  III Simple resonances

In this section, we investigate the bifurcation of a small solution of (2)
under the assumption of a simple resonance described in Remark 1.2. So, we
suppose

  * (R)

$p\in\pi\mathbb{Q}$, condition (15) is valid, for some
$k_{0}\in\mathbb{Z}\backslash\\{0\\}$ equation (16) holds and has the only
solutions $\pm k_{0}$,

and we consider the equation

$A(U,\varepsilon):={\mathcal{K}}U-{\mathcal{F}}(U)-\varepsilon{\mathcal{G}}(U)=0$
(17)

with

$\begin{gathered}{\mathcal{K}}U:=\omega^{2}(U^{\prime\prime}(z)-\lambda
U^{\prime\prime}(z-p)-\lambda U^{\prime\prime}(z+p))+U(z)\\\
{\mathcal{F}}(U):=-\omega^{2}((U^{2}(z))^{\prime\prime}-\lambda(U^{2}(z-p))^{\prime\prime}-\lambda(U^{2}(z+p))^{\prime\prime})\\\
{\mathcal{G}}(U)(z):=-\gamma\omega(U^{\prime}(z)+(U^{2}(z))^{\prime})+h(z)\end{gathered}$
(18)

for $U$ and $\varepsilon$ small.

We shall solve equation (17) via Lyapunov-Schmidt reduction method [4].
Clearly, $A(0,0)=0$ and ${\mathrm{D}}_{U}A(0,0)={\mathcal{K}}$. For simplicity
we denote $\kappa:=|k_{0}|$ and $R_{1}^{\kappa}:={\mathcal{N}}{\mathcal{K}}$,
$R_{2}^{\kappa}:={\mathcal{R}}{\mathcal{K}}$ the null space and the range of
the operator ${\mathcal{K}}\in L(Z,X)$, respectively. Equation (6) with
$\gamma=0$ together with $\mathrm{(R)}$ yields that if $U\in Z$ with
$U(z)=\sum_{k\in\mathbb{Z}}c_{k}{\,\textrm{\rm e}}^{k\imath z}$, then

${\mathcal{K}}U(z)=\sum_{k\in\mathbb{Z}\backslash\\{\pm
k_{0}\\}}(1-\omega^{2}k^{2}+2\lambda\omega^{2}k^{2}\cos kp)c_{k}{\,\textrm{\rm
e}}^{k\imath z}.$

Consequently,

$\displaystyle R_{1}^{\kappa}=\left\\{c{\,\textrm{\rm e}}^{\kappa\imath
z}+\bar{c}{\,\textrm{\rm e}}^{-\kappa\imath z}\mid
c\in\mathbb{C}\right\\},\qquad R_{2}^{\kappa}=\left\\{U\in X\mid
U(z)=\sum_{k\in\mathbb{Z}\backslash\\{\pm k_{0}\\}}c_{k}{\,\textrm{\rm
e}}^{k\imath z}\right\\}.$

Let $Q:X\to R_{2}^{\kappa}$ be the projection onto $R_{2}^{\kappa}\subset X$
given by

$QU(z)=\sum_{k\in\mathbb{Z}\backslash\\{\pm k_{0}\\}}c_{k}{\,\textrm{\rm
e}}^{k\imath z}.$

Now, we take the decomposition $U=U_{1}+U_{2}$, $U_{1}\in R_{1}^{\kappa}$,
$U_{2}\in R_{2}^{\kappa}\cap Z$ for $U\in Z$ and decouple (17) to equations

$\displaystyle QA(U_{1}+U_{2},\varepsilon)=0,$ (19)
$\displaystyle(\mathbb{I}-Q)A(U_{1}+U_{2},\varepsilon)=0.$ (20)

Applying the implicit function theorem to the first equation implies the
existence of neighbourhoods $V_{1}\times V_{0}\subset
R_{1}^{\kappa}\times\mathbb{R}$ of the point $(0,0)$ in
$R_{1}^{\kappa}\times\mathbb{R}$, $V_{2}\subset R_{2}^{\kappa}\cap Z$ of $0$
in $R_{2}^{\kappa}\cap Z$ and a unique $C^{\infty}$–function
$U_{2}:V_{1}\times V_{0}\to V_{2}$ such that
$QA(U_{1}+\widetilde{U}_{2},\varepsilon)=0$ for $(U_{1},\varepsilon)\in
V_{1}\times V_{0}$ and $\widetilde{U}_{2}\in V_{2}$ if and only if
$\widetilde{U}_{2}=U_{2}(U_{1},\varepsilon)$. Moreover, $U_{2}(0,0)=0$.

Thus, equation (20) has the form

$(\mathbb{I}-Q)A(U_{1}+U_{2}(U_{1},\varepsilon),\varepsilon)=0$ (21)

for $U_{1}\in V_{1}$ and $\varepsilon\in V_{0}$. Differentiating equation (19)
we obtain

${\mathrm{D}}_{U_{1}}U_{2}(0,0)=0,\qquad{\mathrm{D}}_{\varepsilon}U_{2}(0,0)=-{\mathcal{K}}^{-1}Q{\mathrm{D}}_{\varepsilon}A(0,0)={\mathcal{K}}^{-1}Q{\mathcal{G}}(0).$
(22)

Hence

$U_{2}(U_{1},\varepsilon)={\mathcal{K}}^{-1}Q{\mathcal{G}}(0)\varepsilon+O(\|(U_{1},\varepsilon)\|^{2}).$
(23)

Expanding ${\mathcal{F}}$ and ${\mathcal{G}}$ of (18) into Taylor series using
the last expansion of $U_{2}(U_{1},\varepsilon)$ yields

$\displaystyle{\mathcal{F}}(U_{1}+U_{2}(U_{1},\varepsilon))=\frac{1}{2}{\mathrm{D}}^{2}{\mathcal{F}}(0)(U_{1}+{\mathrm{D}}_{\varepsilon}U_{2}(0,0)\varepsilon)^{2}+O(\|(U_{1},\varepsilon)\|^{3}),$
$\displaystyle{\mathcal{G}}(U_{1}+U_{2}(U_{1},\varepsilon))={\mathcal{G}}(0)+{\mathrm{D}}{\mathcal{G}}(0)(U_{1}+{\mathrm{D}}_{\varepsilon}U_{2}(0,0)\varepsilon)+O(\|(U_{1},\varepsilon)\|^{2}).$

Note that ${\mathrm{D}}^{2}{\mathcal{F}}(0)V^{2}=2{\mathcal{F}}(V)$ and
$\varepsilon O(\|(U_{1},\varepsilon)\|^{2})=O(\|(U_{1},\varepsilon)\|^{3})$.
Thus equation (21) is equivalent to

$\begin{gathered}(\mathbb{I}-Q)({\mathcal{F}}(U_{1}+{\mathrm{D}}_{\varepsilon}U_{2}(0,0)\varepsilon)+\varepsilon{\mathcal{G}}(0)+\varepsilon{\mathrm{D}}{\mathcal{G}}(0)(U_{1}+{\mathrm{D}}_{\varepsilon}U_{2}(0,0)\varepsilon)+O(\|(U_{1},\varepsilon)\|^{3}))=0\end{gathered}$

or, using ${\mathrm{D}}_{\varepsilon}U_{2}(0,0)$ and ${\mathcal{F}}$ of (18),

$\begin{gathered}(\mathbb{I}-Q)(\omega^{2}(((U_{1}(z)+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)\varepsilon)^{2})^{\prime\prime}-\lambda((U_{1}(z-p)+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z-p)\varepsilon)^{2})^{\prime\prime}\\\
{}-\lambda((U_{1}(z+p)+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z+p)\varepsilon)^{2})^{\prime\prime})-\varepsilon{\mathcal{G}}(0)(z)-\varepsilon{\mathrm{D}}{\mathcal{G}}(0)(z)(U_{1}+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)\varepsilon)+O(\|(U_{1},\varepsilon)\|^{3}))=0.\end{gathered}$
(24)

Note that

$\begin{gathered}((U_{1}(z)+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)\varepsilon)^{2})^{\prime\prime}=2(U_{1}^{\prime}(z)+({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime}\varepsilon)^{2}\\\
+2(U_{1}(z)+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)\varepsilon)(U_{1}^{\prime\prime}(z)+({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime\prime}\varepsilon)\end{gathered}$

for any $z\in\mathbb{R}$. Let $U_{1}(z)=c{\,\textrm{\rm e}}^{k_{0}\imath
z}+\bar{c}{\,\textrm{\rm e}}^{-k_{0}\imath z}\in R_{1}^{\kappa}$ and
$h(z)=\sum_{k\in\mathbb{Z}}h_{k}{\,\textrm{\rm e}}^{k\imath z}\in X$. Then

$\begin{gathered}U_{1}^{\prime}(z)=\tilde{c}{\,\textrm{\rm e}}^{k_{0}\imath
z}+\overline{\tilde{c}}{\,\textrm{\rm e}}^{-k_{0}\imath z}\in
R_{1}^{\kappa},\quad{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)=\sum_{k\in\mathbb{Z}\backslash\\{\pm
k_{0}\\}}d_{k}{\,\textrm{\rm e}}^{k\imath z}\in
R_{2}^{\kappa},\quad({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime}=\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0}\\}}\tilde{d}_{k}{\,\textrm{\rm e}}^{k\imath z}\in
R_{2}^{\kappa}\end{gathered}$ (25)

where

$\tilde{c}=\imath k_{0}c,\qquad
d_{k}=\frac{h_{k}}{1-\omega^{2}k^{2}+2\lambda\omega^{2}k^{2}\cos
kp},\qquad\tilde{d}_{k}=\imath kd_{k}.$

Clearly, $d_{-k}=\overline{d_{k}}$, $\tilde{d}_{-k}=\overline{\tilde{d}_{k}}$
for each $k\in\mathbb{Z}\backslash\\{0,\pm k_{0}\\}$. It can be easily shown
that

$\begin{gathered}(\mathbb{I}-Q)(U_{1}^{\prime}(z))^{2}=0=(\mathbb{I}-Q)(U_{1}(z)U_{1}^{\prime\prime}(z)),\\\
(\mathbb{I}-Q)(U_{1}^{\prime}(z)({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime})=\tilde{c}\overline{\tilde{d}_{2k_{0}}}{\,\textrm{\rm
e}}^{-k_{0}\imath z}+\overline{\tilde{c}}\tilde{d}_{2k_{0}}{\,\textrm{\rm
e}}^{k_{0}\imath z}.\end{gathered}$ (26)

Next, setting $\tilde{d}_{\pm k_{0}}=\tilde{d}_{0}=0$,

$\sum_{k\in\mathbb{Z}\backslash\\{0,\pm k_{0}\\}}\tilde{d}_{k}{\,\textrm{\rm
e}}^{k\imath z}\sum_{j\in\mathbb{Z}\backslash\\{0,\pm
k_{0}\\}}\tilde{d}_{j}{\,\textrm{\rm e}}^{j\imath
z}=\sum_{l\in\mathbb{Z}}\sum_{k+j=l}\tilde{d}_{k}\tilde{d}_{j}{\,\textrm{\rm
e}}^{l\imath z}.$

Hence the coefficient by ${\,\textrm{\rm e}}^{k_{0}\imath z}$ in
$(({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime})^{2}$ is

$b:=\sum_{k+j=k_{0}}\tilde{d}_{k}\tilde{d}_{j}=\sum_{k\in\mathbb{Z}}\tilde{d}_{k}\tilde{d}_{k_{0}-k}=\sum_{k\in\mathbb{Z}}\tilde{d}_{k}\overline{\tilde{d}_{k-k_{0}}}=\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\tilde{d}_{k}\overline{\tilde{d}_{k-k_{0}}}.$

Therefore

$(\mathbb{I}-Q)((({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime})^{2})=b{\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{b}{\,\textrm{\rm e}}^{-k_{0}\imath z}.$ (27)

Analogically, denoting

$\tilde{\tilde{d}}_{k}:=-k^{2}d_{k},\;k\in\mathbb{Z}\backslash\\{0,\pm
k_{0}\\},\qquad\tilde{\tilde{c}}:=-k_{0}^{2}c,\qquad\tilde{b}:=\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}d_{k}\overline{\tilde{\tilde{d}}_{k-k_{0}}}$

we have

$\displaystyle U_{1}^{\prime\prime}(z)=\tilde{\tilde{c}}{\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{\tilde{\tilde{c}}}{\,\textrm{\rm
e}}^{-k_{0}\imath z}\in
R_{1}^{\kappa},\qquad({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime\prime}=\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0}\\}}\tilde{\tilde{d}}_{k}{\,\textrm{\rm e}}^{k\imath z}\in
R_{2}^{\kappa},$

and the next identities can be proved

$\begin{gathered}(\mathbb{I}-Q)(U_{1}(z)({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime\prime})=\overline{c}\tilde{\tilde{d}}_{2k_{0}}{\,\textrm{\rm
e}}^{k_{0}\imath z}+c\overline{\tilde{\tilde{d}}_{2k_{0}}}{\,\textrm{\rm
e}}^{-k_{0}\imath z},\\\
(\mathbb{I}-Q)(U_{1}^{\prime\prime}(z){\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))=\left(\overline{\tilde{\tilde{c}}}d_{2k_{0}}+\tilde{\tilde{c}}d_{0}\right){\,\textrm{\rm
e}}^{k_{0}\imath
z}+\left(\tilde{\tilde{c}}\overline{d_{2k_{0}}}+\overline{\tilde{\tilde{c}}}d_{0}\right){\,\textrm{\rm
e}}^{-k_{0}\imath z},\\\
(\mathbb{I}-Q)({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)({\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z))^{\prime\prime})=\tilde{b}{\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{\tilde{b}}{\,\textrm{\rm e}}^{-k_{0}\imath
z}.\end{gathered}$ (28)

Moreover, by definition of ${\mathcal{G}}$ in (18),

$\begin{gathered}(\mathbb{I}-Q){\mathcal{G}}(0)(z)=h_{k_{0}}{\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{h_{k_{0}}}{\,\textrm{\rm e}}^{-k_{0}\imath
z},\\\
(\mathbb{I}-Q){\mathrm{D}}{\mathcal{G}}(0)(U_{1}+{\mathcal{K}}^{-1}Q{\mathcal{G}}(0)(z)\varepsilon)=-\gamma\omega(\tilde{c}{\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{\tilde{c}}{\,\textrm{\rm e}}^{-k_{0}\imath
z}).\end{gathered}$ (29)

Now, we put (26), (27), (28), (29) in (24) and collect coefficients by
${\,\textrm{\rm e}}^{k_{0}\imath z}$ and ${\,\textrm{\rm e}}^{-k_{0}\imath z}$
to obtain

$H{\,\textrm{\rm e}}^{k_{0}\imath z}+\overline{H}{\,\textrm{\rm
e}}^{-k_{0}\imath z}+O(\|(U_{1},\varepsilon)\|^{3})=0$ (30)

where

$\displaystyle
H=2\omega^{2}\left(2\overline{\tilde{c}}\tilde{d}_{2k_{0}}+\overline{c}\tilde{\tilde{d}}_{2k_{0}}+\overline{\tilde{\tilde{c}}}d_{2k_{0}}+\tilde{\tilde{c}}d_{0}\right)(1-\lambda{\,\textrm{\rm
e}}^{-k_{0}\imath p}-\lambda{\,\textrm{\rm e}}^{k_{0}\imath
p})\varepsilon-(h_{k_{0}}-\gamma\omega\tilde{c})\varepsilon+2\omega^{2}(b+\tilde{b})(1-\lambda{\,\textrm{\rm
e}}^{-k_{0}\imath p}-\lambda{\,\textrm{\rm e}}^{k_{0}\imath
p})\varepsilon^{2}.$

Equation (30) is equivalent to

$H+O(\|(c,\overline{c},\varepsilon)\|^{3})=0.$ (31)

Using resonance condition (16), we get

$(1-\lambda{\,\textrm{\rm e}}^{-k_{0}\imath p}-\lambda{\,\textrm{\rm
e}}^{k_{0}\imath p})=1-2\lambda\cos k_{0}p=\frac{1}{\omega^{2}k_{0}^{2}}.$

Dropping tildes yields

$\displaystyle
2\overline{\tilde{c}}\tilde{d}_{2k_{0}}+\overline{c}\tilde{\tilde{d}}_{2k_{0}}+\overline{\tilde{\tilde{c}}}d_{2k_{0}}+\tilde{\tilde{c}}d_{0}=-k_{0}^{2}\overline{c}d_{2k_{0}}-k_{0}^{2}cd_{0},$
$\displaystyle b+\tilde{b}=\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}k_{0}(k-k_{0})d_{k}\overline{d_{k-k_{0}}}.$

Thus we have to solve

$\displaystyle-\left(2(\overline{c}d_{2k_{0}}+cd_{0})+h_{k_{0}}-\gamma\omega\tilde{c}\right)\varepsilon+\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}k_{0}(k-k_{0})d_{k}\overline{d_{k-k_{0}}}+O(\|(c,\overline{c},\varepsilon)\|^{3})=0.$

Since $c,h_{k}\in\mathbb{C}$, we have

$c=x+\imath y,\qquad h_{k}=\mu_{k}+\imath\nu_{k},\qquad
d_{k}=\frac{\mu_{k}+\imath\nu_{k}}{\delta_{k}}$

where $x,y,\mu_{k},\nu_{k}\in\mathbb{R}$ and
$\delta_{k}=1-\omega^{2}k^{2}+2\lambda\omega^{2}k^{2}\cos kp\in\mathbb{R}$.
Obviously, $\delta_{0}=1$, $h_{0}=d_{0}=\mu_{0}\in\mathbb{R}$ and
$\mu_{k}=\mu_{-k}$, $\nu_{k}=-\nu_{-k}$ $\forall k\in\mathbb{N}$. So, we get

$\displaystyle-\left(\frac{2(x-\imath
y)(\mu_{2k_{0}}+\imath\nu_{2k_{0}})}{\delta_{2k_{0}}}+2(x+\imath
y)\mu_{0}+\mu_{k_{0}}+\imath\nu_{k_{0}}-\gamma\omega\imath k_{0}(x+\imath
y)\right)\varepsilon$
$\displaystyle+\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k}+\imath\nu_{k})(\mu_{k-k_{0}}-\imath\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+O(\|(x,y,\varepsilon)\|^{3})=0$

or, separating the real and imaginary parts,

$\displaystyle-\left(\frac{2(x\mu_{2k_{0}}+y\nu_{2k_{0}})}{\delta_{2k_{0}}}+2x\mu_{0}+\mu_{k_{0}}+\gamma\omega
k_{0}y\right)\varepsilon+\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k}\mu_{k-k_{0}}+\nu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+O(\|(x,y,\varepsilon)\|^{3})$
$\displaystyle=0,$
$\displaystyle-\left(\frac{2(x\nu_{2k_{0}}-y\mu_{2k_{0}})}{\delta_{2k_{0}}}+2y\mu_{0}+\nu_{k_{0}}-\gamma\omega
k_{0}x\right)\varepsilon+\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k-k_{0}}\nu_{k}-\mu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+O(\|(x,y,\varepsilon)\|^{3})$
$\displaystyle=0.$

Multiplying both equations $(-1)$ and collecting terms by $x$ and $y$ we
obtain

$\displaystyle
2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)\varepsilon
x+\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right)\varepsilon
y+\mu_{k_{0}}\varepsilon-\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k}\mu_{k-k_{0}}+\nu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+O(\|(x,y,\varepsilon)\|^{3})=0,$
(32) $\displaystyle\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right)\varepsilon
x+2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)\varepsilon
y+\nu_{k_{0}}\varepsilon-\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k-k_{0}}\nu_{k}-\mu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+O(\|(x,y,\varepsilon)\|^{3})=0.$

Now we intend to derive the cubic $O(\|(U_{1},0)\|^{3})$ of (30) as follows.
We denote by $B:=-\frac{1}{2}{\mathrm{D}}^{2}{\mathcal{F}}(0)$ and
$u_{2}(U_{1}):=U_{2}(U_{1},0)$. Then (19) and (20) are equivalent to

$\displaystyle Q({\mathcal{K}}u_{2}(U_{1})+B(U_{1}+u_{2}(U_{1}))^{2})=0,$ (33)
$\displaystyle(\mathbb{I}-Q)B(U_{1}+u_{2}(U_{1}))^{2}=0.$ (34)

First note that $u_{2}(U_{1})=O(\|U_{1}\|^{2})$ by (23),
$(\mathbb{I}-Q)BU_{1}^{2}=0$ and then (34) has the form

$\displaystyle
0=(\mathbb{I}-Q)B(U_{1}+u_{2}(U_{1}))^{2}=(\mathbb{I}-Q)BU_{1}^{2}+2(I-Q)B(U_{1},u_{2}(U_{1}))+(I-Q)B(u_{2}(U_{1}))^{2}$
$\displaystyle=(I-Q)B(U_{1},{\mathrm{D}}_{U_{1}}^{2}u_{2}(0)U_{1}^{2})+O(\|U_{1}\|^{4}),$

hence
$O(\|(U_{1},0)\|^{3})=(I-Q)B(U_{1},{\mathrm{D}}_{U_{1}}^{2}u_{2}(0)U_{1}^{2})$.
To derive ${\mathrm{D}}_{U_{1}}^{2}u_{2}(0)$, we twice differentiate (33) to
obtain

${\mathcal{K}}{\mathrm{D}}_{U_{1}}u_{2}(U_{1})V_{1}+2QB(U_{1}+u_{2}(U_{1}),V_{1}+{\mathrm{D}}_{U_{1}}u_{2}(U_{1})V_{1})=0$

and then

${\mathcal{K}}{\mathrm{D}}_{U_{1}}^{2}u_{2}(0)(V_{1},V_{2})+2QB(V_{1},V_{2})=0,$

hence

${\mathrm{D}}_{U_{1}}^{2}u_{2}(0)U_{1}^{2}=-2{\mathcal{K}}^{-1}QBU_{1}^{2}=2{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1}),$

consequently, we arrive at

$\begin{gathered}O(\|(U_{1},0)\|^{3})=(I-Q)B(U_{1},{\mathrm{D}}_{U_{1}}^{2}u_{2}(0)U_{1}^{2})=-(I-Q){\mathrm{D}}^{2}{\mathcal{F}}(0)(U_{1},{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})).\end{gathered}$
(35)

Let $U_{1}(z)=c{\,\textrm{\rm e}}^{k_{0}\imath z}+\bar{c}{\,\textrm{\rm
e}}^{-k_{0}\imath z}\in R_{1}^{\kappa}$. Then

$\displaystyle{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})={\mathcal{K}}^{-1}Q\left(4\omega^{2}k_{0}^{2}(1-2\lambda\cos
2k_{0}p)\left(c^{2}{\,\textrm{\rm e}}^{2k_{0}\imath
z}+\bar{c}^{2}{\,\textrm{\rm e}}^{-2k_{0}\imath
z}\right)\right)=\frac{1-\delta_{2k_{0}}}{\delta_{2k_{0}}}c^{2}{\,\textrm{\rm
e}}^{2k_{0}\imath
z}+\frac{1-\delta_{2k_{0}}}{\delta_{2k_{0}}}\bar{c}^{2}{\,\textrm{\rm
e}}^{-2k_{0}\imath z}.$

Next we derive

$\displaystyle{\mathrm{D}}^{2}{\mathcal{F}}(0)(V,W)=-2\omega^{2}((V(z)W(z))^{\prime\prime}-\lambda(V(z-p)W(z-p))^{\prime\prime}-\lambda(V(z+p)W(z+p))^{\prime\prime})$
$\displaystyle=-2\omega^{2}\Big{(}V(z)W^{\prime\prime}(z)+V^{\prime\prime}(z)W(z)+2V^{\prime}(z)W^{\prime}(z)$
$\displaystyle{}-\lambda\big{(}V(z-p)W^{\prime\prime}(z-p)+V^{\prime\prime}(z-p)W(z-p)+2V^{\prime}(z)W^{\prime}(z)\big{)}$
$\displaystyle{}-\lambda\big{(}V(z+p)W^{\prime\prime}(z+p)+V^{\prime\prime}(z+p)W(z+p)+2V^{\prime}(z)W^{\prime}(z)\big{)}\Big{)}.$

Denoting $\mu=\frac{1-\delta_{2k_{0}}}{\delta_{2k_{0}}}$ we get

$\displaystyle
U_{1}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]^{\prime\prime}=-4k_{0}^{2}U_{1}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]=-4k_{0}^{2}\Big{(}\mu
c^{3}{\,\textrm{\rm e}}^{3k_{0}\imath z}+\mu\bar{c}^{3}{\,\textrm{\rm
e}}^{-3k_{0}\imath z}+\mu c^{2}\bar{c}{\,\textrm{\rm e}}^{k_{0}\imath z}+\mu
c\bar{c}^{2}{\,\textrm{\rm e}}^{-k_{0}\imath z}\Big{)},$ $\displaystyle
U_{1}^{\prime\prime}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]=-k_{0}^{2}U_{1}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]=-k_{0}^{2}\Big{(}\mu
c^{3}{\,\textrm{\rm e}}^{3k_{0}\imath z}+\mu\bar{c}^{3}{\,\textrm{\rm
e}}^{-3k_{0}\imath z}+\mu c^{2}\bar{c}{\,\textrm{\rm e}}^{k_{0}\imath z}+\mu
c\bar{c}^{2}{\,\textrm{\rm e}}^{-k_{0}\imath z}\Big{)},$ $\displaystyle
U_{1}^{\prime}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]^{\prime}=\Big{(}k_{0}\imath
c{\,\textrm{\rm e}}^{k_{0}\imath z}-k_{0}\imath\bar{c}{\,\textrm{\rm
e}}^{-k_{0}\imath z}\Big{)}\Big{(}2k_{0}\imath\mu c^{2}{\,\textrm{\rm
e}}^{2k_{0}\imath z}-2k_{0}\imath\mu\bar{c}^{2}{\,\textrm{\rm
e}}^{-2k_{0}\imath z}\Big{)}$ $\displaystyle=-2k_{0}^{2}\mu
c^{3}{\,\textrm{\rm e}}^{3k_{0}\imath
z}-2k_{0}^{2}\mu\bar{c}^{3}{\,\textrm{\rm e}}^{-3k_{0}\imath z}+2k_{0}^{2}\mu
c^{2}\bar{c}{\,\textrm{\rm e}}^{k_{0}\imath z}+2k_{0}^{2}\mu
c\bar{c}^{2}{\,\textrm{\rm e}}^{-k_{0}\imath z}.$

So

$(I-Q)\Big{(}U_{1}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]^{\prime\prime}+2U_{1}^{\prime}(z)[{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1})(z)]^{\prime}\Big{)}=0.$

Hence

$\displaystyle-(I-Q){\mathrm{D}}^{2}{\mathcal{F}}(0)(U_{1},{\mathcal{K}}^{-1}Q{\mathcal{F}}(U_{1}))=2\omega^{2}\Big{(}\mu
c^{2}\bar{c}{\,\textrm{\rm e}}^{k_{0}\imath z}(-k_{0}^{2})+\mu
c\bar{c}^{2}{\,\textrm{\rm e}}^{-k_{0}\imath z}(-k_{0}^{2})$
$\displaystyle{}-\lambda\mu c^{2}\bar{c}{\,\textrm{\rm
e}}^{k_{0}\imath(z-p)}(-k_{0}^{2})-\lambda\mu c\bar{c}^{2}{\,\textrm{\rm
e}}^{-k_{0}\imath(z-p)}(-k_{0}^{2})-\lambda\mu c^{2}\bar{c}{\,\textrm{\rm
e}}^{k_{0}\imath(z+p)}(-k_{0}^{2})-\lambda\mu c\bar{c}^{2}{\,\textrm{\rm
e}}^{-k_{0}\imath(z+p)}(-k_{0}^{2})\Big{)}$
$\displaystyle=-2k_{0}^{2}\omega^{2}\mu|c|^{2}\Big{(}U_{1}(z)-\lambda
U_{1}(z-p)-\lambda
U_{1}(z+p)\Big{)}=-2k_{0}^{2}\omega^{2}\mu|c|^{2}(1-\lambda{\,\textrm{\rm
e}}^{k_{0}\imath p}-\lambda{\,\textrm{\rm e}}^{-k_{0}\imath p})U_{1}(z)$
$\displaystyle=-2\mu|c|^{2}U_{1}(z)=-\frac{2(1-\delta_{2k_{0}})}{\delta_{2k_{0}}}|c|^{2}\left(c{\,\textrm{\rm
e}}^{k_{0}\imath z}+\bar{c}{\,\textrm{\rm e}}^{-k_{0}\imath z}\right).$

Thus, by (35) and the last identity, in (31) we have

$O(\|(c,\overline{c},0)\|^{3})=-\frac{2c|c|^{2}(1-\delta_{2k_{0}})}{\delta_{2k_{0}}}.$

Summarizing, the bifurcation equation (32) has a form

$\begin{gathered}2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)\varepsilon
x+\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right)\varepsilon
y+\mu_{k_{0}}\varepsilon+\frac{2(1-\delta_{2k_{0}})x(x^{2}+y^{2})}{\delta_{2k_{0}}}\\\
-\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k}\mu_{k-k_{0}}+\nu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+\varepsilon
O(\|(x,y,\varepsilon)\|^{2})=0,\\\
\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right)\varepsilon
x+2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)\varepsilon
y+\nu_{k_{0}}\varepsilon+\frac{2(1-\delta_{2k_{0}})y(x^{2}+y^{2})}{\delta_{2k_{0}}}\\\
-\frac{2\varepsilon^{2}}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k-k_{0}}\nu_{k}-\mu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}}+\varepsilon
O(\|(x,y,\varepsilon)\|^{2})=0.\end{gathered}$ (36)

Now, we scale (36)

$x\longleftrightarrow\varepsilon x,\qquad y\longleftrightarrow\varepsilon
y,\qquad\varepsilon\longleftrightarrow\varepsilon^{3},\qquad\gamma\longleftrightarrow\gamma$
(37)

and divide both equations by $\varepsilon^{3}$. This leads to

$\displaystyle\mu_{k_{0}}+\frac{2(1-\delta_{2k_{0}})x(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)$
$\displaystyle=0$ (38)
$\displaystyle\nu_{k_{0}}+\frac{2(1-\delta_{2k_{0}})y(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)$
$\displaystyle=0$

and the following result holds.

######  Theorem 2.

Assume $\mathrm{(R)}$ and $(\mu_{k_{0}},\nu_{k_{0}})\neq(0,0)$. Then equation
(17) has a solution $U\in Z$ close to $0$ for any $\varepsilon\neq 0$
sufficiently small.

###### Proof.

Let us denote

$\widetilde{H}_{1}(x,y,\varepsilon):=\left(\mu_{k_{0}}+\frac{2(1-\delta_{2k_{0}})x(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon),\nu_{k_{0}}+\frac{2(1-\delta_{2k_{0}})y(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)\right).$

Implicit function theorem applied to equation (38), i.e.
$\widetilde{H}_{1}(x,y,\varepsilon)=0$ gives the existence of neighbourhoods
$W_{0}\subset V_{0}\subset\mathbb{R}$ of the point $0$ in $\mathbb{R}$,
$W\subset\mathbb{R}^{2}$ of
$-\sqrt[3]{\frac{\delta_{2k_{0}}}{2(1-\delta_{2k_{0}})(\mu_{k_{0}}^{2}+\nu_{k_{0}}^{2})}}(\mu_{k_{0}},\nu_{k_{0}})$
in $\mathbb{R}^{2}$ and a unique pair of continuous functions
$x(\varepsilon)$, $y(\varepsilon)$ defined in $W_{0}$ such that
$\widetilde{H}_{1}(\tilde{x},\tilde{y},\varepsilon)=0$ for $\varepsilon\in
W_{0}$, $(\tilde{x},\tilde{y})\in W$ if and only if $\tilde{x}=x(\varepsilon)$
and $\tilde{y}=y(\varepsilon)$. Moreover,
$(x(0),y(0))=-\sqrt[3]{\frac{\delta_{2k_{0}}}{2(1-\delta_{2k_{0}})(\mu_{k_{0}}^{2}+\nu_{k_{0}}^{2})}}(\mu_{k_{0}},\nu_{k_{0}})$.
After scaling backwards in (37), we have the desired solution of equation
(17). ∎

Now, assume $\varepsilon h$ instead of $h$ in (17). Then we scale (36)

$x\longleftrightarrow\varepsilon x,\qquad y\longleftrightarrow\varepsilon
y,\qquad h_{k}\longleftrightarrow\varepsilon
h_{k},\qquad\gamma\longleftrightarrow\gamma$ (39)

and divide both equations by $\varepsilon^{2}$. This leads to

$\displaystyle\gamma\omega k_{0}y+\mu_{k_{0}}+O(\varepsilon)$
$\displaystyle=0$ (40) $\displaystyle-\gamma\omega
k_{0}x+\nu_{k_{0}}+O(\varepsilon)$ $\displaystyle=0$

and the following result.

######  Theorem 3.

Assume $\mathrm{(R)}$, $\gamma\neq 0$ and $\varepsilon h$ instead of $h$ in
(17) with $(\mu_{k_{0}},\nu_{k_{0}})\neq(0,0)$. Then equation (17) has a
solution $U\in Z$ close to $0$ for any $\varepsilon\neq 0$ sufficiently small.

###### Proof.

Let us denote

$\displaystyle\widetilde{H}_{2}(x,y,\varepsilon):=\left(\gamma\omega
k_{0}y+\mu_{k_{0}}+O(\varepsilon),-\gamma\omega
k_{0}x+\nu_{k_{0}}+O(\varepsilon)\right).$

Implicit function theorem applied to equation (40), i.e.
$\widetilde{H}_{2}(x,y,\varepsilon)=0$ gives the existence of neighbourhoods
$W_{0}\subset V_{0}\subset\mathbb{R}$ of the point $0$ in $\mathbb{R}$,
$W\subset\mathbb{R}^{2}$ of $(\frac{\nu_{k_{0}}}{\gamma\omega
k_{0}},-\frac{\mu_{k_{0}}}{\gamma\omega k_{0}})$ in $\mathbb{R}^{2}$ and a
unique pair of continuous functions $x(\varepsilon)$, $y(\varepsilon)$ defined
in $W_{0}$ such that $\widetilde{H}_{2}(\tilde{x},\tilde{y},\varepsilon)=0$
for $\varepsilon\in W_{0}$, $(\tilde{x},\tilde{y})\in W$ if and only if
$\tilde{x}=x(\varepsilon)$ and $\tilde{y}=y(\varepsilon)$. Moreover,
$x(0)=\frac{\nu_{k_{0}}}{\gamma\omega k_{0}}$,
$y(0)=-\frac{\mu_{k_{0}}}{\gamma\omega k_{0}}$. After scaling backwards in
(39), we have the desired solution of equation (17). ∎

Finally, we deal with a gap between Theorems 2 and 3, namely that $h$ is not
scaled and $\mu_{k_{0}}=\nu_{k_{0}}=0$. Now, we scale (36)

$x\longleftrightarrow\varepsilon x,\qquad y\longleftrightarrow\varepsilon
y,\qquad\varepsilon\longleftrightarrow\pm\varepsilon^{2},\qquad\gamma\longleftrightarrow\gamma$
(41)

and divide both equations by $\varepsilon^{3}$. This leads to

$\displaystyle\pm
2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)x\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right)y+\frac{2(1-\delta_{2k_{0}})x(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)$
$\displaystyle=0,$ (42)
$\displaystyle\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right)x\pm
2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)y+\frac{2(1-\delta_{2k_{0}})y(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)$
$\displaystyle=0$

and the following result.

######  Theorem 4.

Assume $\mathrm{(R)}$ and $(\mu_{k_{0}},\nu_{k_{0}})=(0,0)$. Then equation
(17) has a solution $U\in Z$ close to $0$ for any $\varepsilon\neq 0$
sufficiently small provided

$\delta_{2k_{0}}^{2}(4\mu_{0}^{2}+\gamma^{2}\omega^{2}k_{0}^{2})-4(\mu_{2k_{0}}^{2}+\nu_{2k_{0}}^{2})\neq
0.$ (43)

The order of this solution is $O(\varepsilon)$. If, in addition, it holds

$\delta_{2k_{0}}^{2}(4\mu_{0}^{2}+\gamma^{2}\omega^{2}k_{0}^{2})-4(\mu_{2k_{0}}^{2}+\nu_{2k_{0}}^{2})<0,$
(44)

then (17) has another solution. This one is of order
$O(\sqrt{|\varepsilon|})$.

###### Proof.

Let us denote

$\displaystyle\widetilde{H}_{3}(x,y,\varepsilon)$ $\displaystyle:=\Bigg{(}\pm
2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)x\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right)y+\frac{2(1-\delta_{2k_{0}})x(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon),$
$\displaystyle\qquad\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right)x\pm
2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)y+\frac{2(1-\delta_{2k_{0}})y(x^{2}+y^{2})}{\delta_{2k_{0}}}+O(\varepsilon)\Bigg{)}.$

First we note that
$\widetilde{H}_{3}(x,y,\varepsilon)=A(x,y)+B(x,y)+O(\varepsilon)$ where $A$ is
linear and $B$ is cubic. Since (43) is equivalent to $\det A\neq 0$, implicit
function theorem applied to equation (42), i.e.
$\widetilde{H}_{3}(x,y,\varepsilon)=0$ gives the existence of neighbourhoods
$W_{0}\subset V_{0}\subset\mathbb{R}$ of the point $0$ in $\mathbb{R}$,
$W\subset\mathbb{R}^{2}$ of $(0,0)$ in $\mathbb{R}^{2}$ and a unique pair of
continuous functions $x(\varepsilon)$, $y(\varepsilon)$ defined in $W_{0}$
such that $\widetilde{H}_{3}(\tilde{x},\tilde{y},\varepsilon)=0$ for
$\varepsilon\in W_{0}$, $(\tilde{x},\tilde{y})\in W$ if and only if
$\tilde{x}=x(\varepsilon)$ and $\tilde{y}=y(\varepsilon)$. Moreover,
$(x(0),y(0))=(0,0)$, so $x(\varepsilon)=O(\varepsilon)$,
$y(\varepsilon)=O(\varepsilon)$, which by scaling (41) gives $O(\varepsilon)$
solution for (17). On the other hand, if (44) holds then the Brouwer degree
$\deg(\widetilde{H}_{3},B_{1},(0,0))=\operatorname{sgn}\det A=-1$ while
$\deg(\widetilde{H}_{3},B_{2},(0,0))=1$ for small and large balls $B_{1}$,
$B_{2}$ in $\mathbb{R}^{2}$ centered at $(0,0)$. This gives another solution
of (42) of order $O(1)$ placed in the set $B_{2}\setminus B_{1}$. After
scaling backwards in (41), we have the desired $O(\sqrt{|\varepsilon|})$
solution of (17). ∎

######  Remark 2.

The above $O(\varepsilon)$ solution for (17) can be derived by putting Taylor
expansions $x(\varepsilon)=x^{\prime}(0)\varepsilon+O(\varepsilon^{2})$,
$y(\varepsilon)=y^{\prime}(0)\varepsilon+O(\varepsilon^{2})$ into (36) and
comparing $\varepsilon^{2}$-terms. So we derive

$\displaystyle
2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)x^{\prime}(0)+\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right)y^{\prime}(0)$
$\displaystyle=\frac{2}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k}\mu_{k-k_{0}}+\nu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}},$
(45) $\displaystyle\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right)x^{\prime}(0)+2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right)y^{\prime}(0)$
$\displaystyle=\frac{2}{k_{0}^{2}}\sum_{k\in\mathbb{Z}\backslash\\{0,\pm
k_{0},2k_{0}\\}}\frac{k_{0}(k-k_{0})(\mu_{k-k_{0}}\nu_{k}-\mu_{k}\nu_{k-k_{0}})}{\delta_{k}\delta_{k-k_{0}}},$

which determines the first order approximation of this solution. Hence by (23)
and (25), we get

$U(z)=U_{1}(z)+U_{2}(U_{1},\varepsilon)=\varepsilon\left(c^{\prime}(0){\,\textrm{\rm
e}}^{k_{0}\imath z}+\overline{c^{\prime}(0)}{\,\textrm{\rm e}}^{-k_{0}\imath
z}+\sum_{k\in\mathbb{Z}\backslash\\{\pm k_{0}\\}}d_{k}{\,\textrm{\rm
e}}^{k\imath z}\right)+O(\varepsilon^{2})$ (46)

for $c^{\prime}(0)=x^{\prime}(0)+y^{\prime}(0)\imath$.

######  Remark 3.

1\. Now we specify the $O(\sqrt{|\varepsilon|})$ solution (solutions) of
Theorem 4. Nontrivial roots of equation

$\widetilde{H}_{3}(x,y,0)=0$ (47)

can be explicitly calculated as follows. Denoting

$\begin{gathered}a:=\pm
2\left(\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right),\qquad
b:=\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}+\gamma\omega
k_{0}\right),\\\
c:=\pm\left(\frac{2\nu_{2k_{0}}}{\delta_{2k_{0}}}-\gamma\omega
k_{0}\right),\qquad d:=\pm
2\left(-\frac{\mu_{2k_{0}}}{\delta_{2k_{0}}}+\mu_{0}\right),\qquad\upsilon:=\frac{2(1-\delta_{2k_{0}})}{\delta_{2k_{0}}}\end{gathered}$
(48)

with the same fixed sign in all $a$, $b$, $c$, $d$ depending on the sign of
unscaled $\varepsilon$, this equation is equivalent to the system

$\displaystyle\widetilde{a}x+\widetilde{b}y$ $\displaystyle+x(x2+y^{2})=0$
(49) $\displaystyle\widetilde{c}x+\widetilde{d}y$
$\displaystyle+y(x2+y^{2})=0$

for $\widetilde{a}=a/\upsilon$, $\widetilde{b}=b/\upsilon$,
$\widetilde{c}=c/\upsilon$, $\widetilde{d}=d/\upsilon$. Note $\upsilon\neq 0$,
i.e., $\delta_{2k_{0}}\neq 1$. It is easy to see that $y=0$ whenever $x=0$ and
vice versa. Hence we assume $x\neq 0\neq y$. Dividing the first equation by
$x$, the second equation by $y$, and comparing the resulting equations, one
derives

$\widetilde{a}+\widetilde{b}\frac{y}{x}=\widetilde{c}\frac{x}{y}+\widetilde{d}.$

Therefrom $x=q_{1,2}y$ for

$q_{1,2}=\frac{(\widetilde{a}-\widetilde{d})\pm\sqrt{(\widetilde{a}-\widetilde{d})^{2}+4\widetilde{b}\widetilde{c}}}{2\widetilde{c}}$

(here $q_{1}$ corresponds to “$+$”-sign and $q_{2}$ to “$-$”-sign). As a
consequence, putting $x$ in the second equation of (49) one obtains

$y_{1,2}=\pm\sqrt{-\frac{\widetilde{c}q_{1,2}+\widetilde{d}}{1+q_{1,2}^{2}}}$

(again $y_{1}$ corresponds to “$+$”, $y_{2}$ to “$-$”), i.e., one has four
points $(q_{i}y_{j}(q_{i}),y_{j}(q_{i}))$, $i,j=1,2$.

Condition (44) means $ad-bc<0$, i.e.,
$\widetilde{a}\widetilde{d}-\widetilde{b}\widetilde{c}<0$. So if (44) holds,
then

$\sqrt{(\widetilde{a}-\widetilde{d})^{2}+4\widetilde{b}\widetilde{c}}=\sqrt{(\widetilde{a}+\widetilde{d})^{2}-4(\widetilde{a}\widetilde{d}-\widetilde{b}\widetilde{c})}>|\widetilde{a}+\widetilde{d}|.$

Accordingly,

$\widetilde{c}q_{1,2}+\widetilde{d}=\frac{1}{2}\left(\widetilde{a}+\widetilde{d}\pm\sqrt{(\widetilde{a}-\widetilde{d})^{2}+4\widetilde{b}\widetilde{c}}\right)$
(50)

which is negative only for the minus sign (without any respect to the sign of
unscaled $\varepsilon$). Thus the nontrivial solutions of (47) are

$(q_{2}y_{1,2}(q_{2}),y_{1,2}(q_{2})).$ (51)

On the other hand, assuming

$0<\delta_{2k_{0}}^{2}(4\mu_{0}^{2}+\gamma^{2}\omega^{2}k_{0}^{2})-4(\mu_{2k_{0}}^{2}+\nu_{2k_{0}}^{2})<4\mu_{0}^{2},$
(52)

equation (47) still has nontrivial solutions. Indeed, in this case

$0<\sqrt{(\widetilde{a}-\widetilde{d})^{2}+4\widetilde{b}\widetilde{c}}=\sqrt{(\widetilde{a}+\widetilde{d})^{2}-4(\widetilde{a}\widetilde{d}-\widetilde{b}\widetilde{c})}<|\widetilde{a}+\widetilde{d}|,$

so $q_{1,2}$ are both real, again. However, (50) is negative for both signs on
suppose that $\widetilde{a}+\widetilde{d}<0$, i.e.,
$\pm\frac{\mu_{0}}{\upsilon}<0$ where the sign depends on the sign of unscaled
$\varepsilon$, e.g. if originally $\varepsilon>0$ and
$\frac{\mu_{0}}{\upsilon}<0$, equation (47) has four distinct nontrivial
solutions

$(q_{i}y_{j}(q_{i}),y_{j}(q_{i})),\qquad i,j=1,2,$ (53)

while for $\frac{\mu_{0}}{\upsilon}>0$, (47) has no nontrivial solutions.

In conclusion, if condition (44) holds, there might exist two solutions of
(17) of order $O(\sqrt{|\varepsilon|})$ for $\varepsilon\neq 0$ small, and if
(52) is valid, there might exist four solutions of (17) of order
$O(\sqrt{|\varepsilon|})$ for
$\varepsilon\operatorname{sgn}\frac{\mu_{0}\delta_{2k_{0}}}{1-\delta_{2k_{0}}}<0$
and no solutions of order $O(\sqrt{|\varepsilon|})$ for
$\varepsilon\operatorname{sgn}\frac{\mu_{0}\delta_{2k_{0}}}{1-\delta_{2k_{0}}}>0$.
We recall by Theorem 4, that there exists a small solution in all cases of
lower order than $O(\sqrt{|\varepsilon|})$, namely of order $O(\varepsilon)$.
So we might have 3, 5 and 1 small solutions, respectively. To apply the
implicit function theorem and so to confirm the existence of these solutions,
one has to show that the derivative of $\widetilde{H}_{3}$ at points (51) or
(53) is surjective, which is rather awkward for general $a$, $b$, $c$, $d$,
and so we do not go into details in this paper. But surely these solutions
generically exist. Indeed, the determinant of the Jacobian of the left hand
side of (49) at any $(qy,y)$ is

$-\widetilde{b}\widetilde{c}+\widetilde{a}\widetilde{d}+\left(3\widetilde{a}+\widetilde{d}+q(-2(\widetilde{b}+\widetilde{c})+(\widetilde{a}+3\widetilde{d})q)\right)y^{2}+3\left(1+q^{2}\right)^{2}y^{4}.$
(54)

Now we fix all parameters only $\mu_{0}$ is variable. Then $q=q_{1,2}$,
$\widetilde{b}$, $\widetilde{c}$ are constant (independent of $\mu_{0}$) and
$y_{1,2}^{2}$, $\widetilde{a}$, $\widetilde{d}$ linearly depend on $\mu_{0}$:
$\widetilde{a}=\pm\frac{2}{\upsilon}(r_{1}+\mu_{0})$,
$\widetilde{d}=\pm\frac{2}{\upsilon}(-r_{1}+\mu_{0})$ and
$y^{2}=y_{1,2}^{2}=\frac{-\widetilde{c}\upsilon q\pm
2(r_{1}-\mu_{0})}{\upsilon(1+q^{2})}$ for constants $r_{1}$, $r_{2}$
independent of $\mu_{0}$. Inserting these expressions into (54), we have

$\frac{1}{\upsilon^{2}(1+q^{2})}\left(a_{\pm}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)+b_{\pm}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)\mu_{0}\right)$
(55)

for polynomials

$\displaystyle a_{+}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)$
$\displaystyle=\widetilde{c}\left(-\widetilde{b}+(\widetilde{b}+5\widetilde{c})q^{2}+3\widetilde{c}q^{4}\right)\upsilon^{2}-4q\left(\widetilde{b}+\widetilde{c}\left(5+2q^{2}\right)\right)\upsilon
r_{1}+16r_{1}^{2},$ $\displaystyle
b_{+}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)$
$\displaystyle=4\left(q\left(\widetilde{b}+\widetilde{c}\left(2+q^{2}\right)\right)\upsilon-4r_{1}\right),$
$\displaystyle a_{-}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)$
$\displaystyle=c\left(-\widetilde{b}+(\widetilde{b}+5\widetilde{c})q^{2}+3\widetilde{c}q^{4}\right)\upsilon^{2}+4r_{1}\left(q\left(\widetilde{b}+c\left(5+2q^{2}\right)\right)\upsilon+4r_{1}\right),$
$\displaystyle b_{-}(r_{1},q,\widetilde{b},\widetilde{c},\upsilon)$
$\displaystyle=-4\left(q\left(\widetilde{b}+\widetilde{c}\left(2+q^{2}\right)\right)\upsilon+4r_{1}\right).$

Now it is clear that $a_{\pm}$, $b_{\pm}$ are generically nonzero and then
(55) is nonzero for generic $\mu_{0}$. This shows that the above solutions
exist in generic cases.

2\. Let $p=\frac{\pi}{4}$, $\lambda=\frac{12}{37\sqrt{2}}$. Then by (13) we
have

$M_{2}=\left\\{\cos\frac{k\pi}{4}\mid
k=\\{0,1,\dots,7\\}\right\\}=\left\\{0,\pm\frac{\sqrt{2}}{2},\pm{1}\right\\}$
(56)

and condition (15) is fulfilled. If $\omega^{2}=\frac{37}{25}$, then
$k_{0}=\pm 1$ satisfies the resonance condition (16). Note that

$\frac{1}{k^{2}(1-2\lambda\cos
kp)}\leq\frac{1}{k^{2}(1-2\lambda)}=\frac{37}{k^{2}(37-12\sqrt{2})}<\frac{37}{13k^{2}}<\frac{37}{25}$

whenever $|k|\geq 2$. Thus $k_{0}=\pm 1$ is the only resonant couple
corresponding to $\omega^{2}=\frac{37}{25}$. This is the case of a simple
resonance discussed in this section.

On the other side, if $\omega^{2}=\frac{37}{1225}$, then there are at least
two couples $k_{0}=\pm 5,\pm 7$ such that (16) is fulfilled. Since

$\frac{37}{k^{2}(37-12\sqrt{2})}<\frac{37}{20k^{2}}<\frac{37}{1225}$

whenever $|k|\geq 8$, it is sufficient to evaluate
$\frac{1}{k^{2}(1-2\lambda\cos kp)}$ at $k=\\{1,\dots,7\\}$ to determine the
exact number of resonant modes. From Fig. 1, one can see that for
$\omega^{2}=\frac{37}{1225}$ there are two resonant couples $k_{0}=\pm 5,\pm
7$. This is a double resonance which can be investigated analogically to the
present section.

Figure 1: The graph of functions $\frac{37}{1225}$ and
$\frac{1}{x^{2}(1-2\lambda\cos xp)}$ with values at nonzero integers depicted
with asterisks.

##  IV Examples

In this section we present several examples to illustrate the above results.

######  Example 1.

First we study the non-resonant case from Section II. We consider the equation

$\begin{gathered}\omega^{2}\left(U(z)-\lambda
U\left(z-\frac{\pi}{4}\right)-\lambda
U\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\gamma\omega
U^{\prime}(z)+U(z)\\\ {}+\omega^{2}\left(U^{2}(z)-\lambda
U^{2}\left(z-\frac{\pi}{4}\right)-\lambda
U^{2}\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\gamma\omega(U^{2}(z))^{\prime}-\varepsilon\cos
z=0\end{gathered}$ (57)

for $z\in\mathbb{R}$ and parameters $\gamma,\lambda,\omega>0$,
$\frac{1}{2}\neq\lambda\neq\frac{\sqrt{2}}{2}$, and apply Theorem 1. Note that
equation (57) is in the form of (2) with $p=\frac{\pi}{4}$,
$h(z)=\varepsilon\cos z$, and $M_{2}$ of (13) is equivalent to (56), i.e.,
condition (15) is satisfied. Hence using Remark 1.3 and Theorem 1, we obtain
the existence of a unique solution $U(\varepsilon,z)=O(\varepsilon)$ of
equation (57) for any $\varepsilon$ sufficiently close to $0$. Since
$U(\varepsilon,z)$ is $C^{\infty}$–smooth, we can expand $U(\cdot,z)$ into
Taylor series
$U(\varepsilon,z)=\varepsilon\widetilde{U}_{1}(z)+O(\varepsilon^{2})$, and
look for the function $\widetilde{U}_{1}$. It is easy to see that
$\widetilde{U}_{1}$ is uniquely determined by the equation

$\omega^{2}\left(\widetilde{U}_{1}(z)-\lambda\widetilde{U}_{1}\left(z-\frac{\pi}{4}\right)-\lambda\widetilde{U}_{1}\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\gamma\omega\widetilde{U}_{1}^{\prime}(z)+\widetilde{U}_{1}(z)-\cos
z=0.$

We look for its solution in the form of $\widetilde{U}_{1}(z)=A\cos z+B\sin
z$, and, by comparing coefficients at $\cos z$ and $\sin z$, we obtain

$-\gamma\omega
A+(1+\omega^{2}(\sqrt{2}\lambda-1))B=0,\quad(1+\omega^{2}(\sqrt{2}\lambda-1))A+\gamma\omega
B=1.$

Therefrom,

$A=\frac{1+\omega^{2}(\sqrt{2}\lambda-1)}{\gamma^{2}\omega^{2}+(1+\omega^{2}(\sqrt{2}\lambda-1))^{2}},\quad
B=\frac{\gamma\omega}{\gamma^{2}\omega^{2}+(1+\omega^{2}(\sqrt{2}\lambda-1))^{2}}.$

In particular, if

$\lambda=\frac{12}{37\sqrt{2}},\quad\omega=\frac{\sqrt{37}}{5},$ (58)

then $A=0$, $B=\frac{5\sqrt{37}}{37\gamma}$, i.e.,

$U(\varepsilon,z)=\frac{5\sqrt{37}}{37\gamma}\varepsilon\sin
z+O(\varepsilon^{2}).$ (59)

######  Example 2.

Now, moving onto resonant cases represented by equation (17), we consider the
equation

$\begin{gathered}\omega^{2}\left(U(z)-\lambda
U\left(z-\frac{\pi}{4}\right)-\lambda
U\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega
U^{\prime}(z)+U(z)\\\ {}+\omega^{2}\left(U^{2}(z)-\lambda
U^{2}\left(z-\frac{\pi}{4}\right)-\lambda
U^{2}\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega(U^{2}(z))^{\prime}-\varepsilon\cos
z=0,\end{gathered}$ (60)

i.e. when $\gamma$ in (57) is replaced by $\varepsilon\gamma$. Proceeding with
point 2 of Remark 3 for parameters (58), then we obtain that $k_{0}=\pm 1$,
$\delta_{2k_{0}}=-\frac{123}{25}$ and
$\upsilon=\frac{2(1-\delta_{2k_{0}})}{\delta_{2k_{0}}}=-\frac{296}{123}$.

Since $h(z)=\cos z=\frac{1}{2}{\,\textrm{\rm e}}^{\imath
z}+\frac{1}{2}{\,\textrm{\rm e}}^{-\imath z}$ in (18), then
$\mu_{k_{0}}=\frac{1}{2}$ and $\nu_{k_{0}}=0$. So Theorem 2 can be applied to
yield

$(x(0),y(0))=-\sqrt[3]{\frac{\delta_{2k_{0}}}{2(1-\delta_{2k_{0}})(\mu_{k_{0}}^{2}+\nu_{k_{0}}^{2})}}(\mu_{k_{0}},\nu_{k_{0}})=\left(\sqrt[3]{\frac{123}{74}}\frac{1}{2},0\right)\doteq(0.592281,0).$

Using (22), (25) and (37), we derive that the solution from Theorem 2 of (60)
with parameters (58) is given by

$\begin{gathered}\sqrt[3]{\varepsilon}c(\sqrt[3]{\varepsilon}){\,\textrm{\rm
e}}^{\imath
z}+\sqrt[3]{\varepsilon}\bar{c}(\sqrt[3]{\varepsilon}){\,\textrm{\rm
e}}^{-\imath
z}+U_{2}\left(\sqrt[3]{\varepsilon}c(\sqrt[3]{\varepsilon}){\,\textrm{\rm
e}}^{\imath
z}+\sqrt[3]{\varepsilon}\bar{c}(\sqrt[3]{\varepsilon}){\,\textrm{\rm
e}}^{-\imath z},\varepsilon\right)\\\
=\sqrt[3]{\varepsilon}\left(c(0){\,\textrm{\rm e}}^{\imath
z}+\bar{c}(0){\,\textrm{\rm e}}^{-\imath
z}+O\left(\sqrt[3]{\varepsilon}\right)\right)=\sqrt[3]{\varepsilon}\left(2x(0)\cos
z+O\left(\sqrt[3]{\varepsilon}\right)\right)\\\
=\sqrt[3]{\frac{123\varepsilon}{74}}\cos
z+O\left(\sqrt[3]{\varepsilon^{2}}\right)\doteq\sqrt[3]{\varepsilon}1.18456\cos
z+O\left(\sqrt[3]{\varepsilon^{2}}\right)\end{gathered}$ (61)

for a smooth function $c(\cdot)$ with $c(0)=x(0)+y(0)\imath$ and
$\varepsilon\neq 0$ small.

######  Example 3.

Similarly, Theorem 3 can be applied in the equation

$\begin{gathered}\omega^{2}\left(U(z)-\lambda
U\left(z-\frac{\pi}{4}\right)-\lambda
U\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega
U^{\prime}(z)+U(z)\\\ {}+\omega^{2}\left(U^{2}(z)-\lambda
U^{2}\left(z-\frac{\pi}{4}\right)-\lambda
U^{2}\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega(U^{2}(z))^{\prime}-\varepsilon^{2}\cos
z=0,\end{gathered}$ (62)

with

$\lambda=\frac{12}{37\sqrt{2}},\quad\omega=\frac{\sqrt{37}}{5},\quad\gamma=0.1.$
(63)

Note that $\cos z$-term in (60) has been replaced by $\varepsilon\cos z$ in
our case here, i.e., $h(z)$ in (18) is replaced by $\varepsilon h(z)$. Then
using the proof of Theorem 3, we obtain that
$(x(0),y(0))=\left(0,-\frac{25}{\sqrt{37}}k_{0}\right)$, since $k_{0}=\pm 1$.
Using (22), (25) and (39), we derive that the solution from Theorem 3 of (62)
with parameters (63) is given by

$\begin{gathered}\varepsilon c(\varepsilon){\,\textrm{\rm e}}^{\imath
k_{0}z}+\varepsilon\bar{c}(\varepsilon){\,\textrm{\rm e}}^{-\imath
k_{0}z}+U_{2}(\varepsilon c(\varepsilon){\,\textrm{\rm e}}^{\imath
k_{0}z}+\varepsilon\bar{c}(\varepsilon){\,\textrm{\rm e}}^{-\imath
k_{0}z},\varepsilon)\\\ =\varepsilon(c(0){\,\textrm{\rm e}}^{\imath
k_{0}z}+\bar{c}(0){\,\textrm{\rm e}}^{-\imath
k_{0}z}+O(\varepsilon))=\varepsilon(-2y(0)\sin k_{0}z+O(\varepsilon))\\\
=\frac{50\varepsilon}{\sqrt{37}}k_{0}\sin k_{0}z+O(\varepsilon^{2})\doteq
8.21995\varepsilon\sin z+O(\varepsilon^{2}).\end{gathered}$ (64)

######  Example 4.

Next, we consider

$\begin{gathered}\omega^{2}\left(U(z)-\lambda
U\left(z-\frac{\pi}{4}\right)-\lambda
U\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega
U^{\prime}(z)+U(z)\\\ {}+\omega^{2}\left(U^{2}(z)-\lambda
U^{2}\left(z-\frac{\pi}{4}\right)-\lambda
U^{2}\left(z+\frac{\pi}{4}\right)\right)^{\prime\prime}+\varepsilon\gamma\omega(U^{2}(z))^{\prime}-\varepsilon(\mu_{0}+2(\cos
2z-\sin 2z))=0\end{gathered}$ (65)

with parameters (63), i.e., equation (17) with the particular driving function
$h(z)=\mu_{0}+2(\cos 2z-\sin 2z)$. So, $\mu_{k_{0}}=\nu_{k_{0}}=0$ and
$\mu_{2k_{0}}=\nu_{2}=-\nu_{-2}=1$. Hence, Theorems 2 and 3 cannot be used.
However, using Theorem 4 one can obtain that (43) is of the form

$-8+\frac{15129}{625}\left(\frac{37}{2500}+4\mu_{0}^{2}\right)\neq 0,$

i.e. $|\mu_{0}|\neq\frac{\sqrt{11940227}}{12300}\doteq 0.280932$. Moreover,
(44) is equivalent to

$|\mu_{0}|<\frac{\sqrt{11940227}}{12300}.$ (66)

First, let $k_{0}=1$ and consider $\varepsilon>0$, i.e. use “$+$”-sign in
(48). Then we can derive that $q_{2}\doteq-0.294153$ and
$y_{1,2}=\pm\sqrt{0.214884+0.764897\mu_{0}}$. Note that
$0<0.214884+0.764897\mu_{0}\leq 0.429768$. Using formula (54), we can estimate
the Jacobian of $\widetilde{H}_{3}(q_{2}y_{1,2}(q_{2}),y_{1,2}(q_{2}),0)$ as

$0<(0.218047+0.776154\mu_{0})v^{2}\leq 2.52553,$ (67)

since the formula (54) is equal to
$\frac{1}{v^{2}}\det{\mathrm{D}}\widetilde{H}_{3}(qy,y,0)$. Therefore, we have
3 different small solutions for $k_{0}=1$, $\varepsilon>0$ small. Using
$k_{0}=-1$, we obtain $q_{2}\doteq 0.294153$,
$y_{1,2}=\pm\sqrt{0.214884+0.764897\mu_{0}}$ and by (54), estimate (67)
follows. Hence for $k_{0}=\pm 1$, $\varepsilon>0$ small and $\mu_{0}$
satisfying (66), by (51) we obtain nontrivial solutions of (47). Scaling
backwards in (41), we obtain the following small solutions of (65) with
parameters (63)

$\begin{gathered}\sqrt{\varepsilon}c(\sqrt{\varepsilon}){\,\textrm{\rm
e}}^{\imath
k_{0}z}+\sqrt{\varepsilon}\bar{c}(\sqrt{\varepsilon}){\,\textrm{\rm
e}}^{-\imath
k_{0}z}+U_{2}(\sqrt{\varepsilon}c(\sqrt{\varepsilon}){\,\textrm{\rm
e}}^{\imath
k_{0}z}+\sqrt{\varepsilon}\bar{c}(\sqrt{\varepsilon}){\,\textrm{\rm
e}}^{-\imath k_{0}z},\varepsilon)\\\ =\sqrt{\varepsilon}(2x(0)\cos
k_{0}z-2y(0)\sin k_{0}z)+O(\varepsilon)=2\sqrt{\varepsilon}y(0)(q_{2}\cos
k_{0}z-\sin k_{0}z)+O(\varepsilon)\\\ =\pm
2\sqrt{\varepsilon(0.214884+0.764897\mu_{0})}(0.294153\cos z+\sin
z)+O(\varepsilon).\end{gathered}$ (68)

Moreover, there is another (smallest) solution of (65) with parameters (63) of
order $U(z)=O(\varepsilon)$. Using (45), we obtain
$x^{\prime}(0)=y^{\prime}(0)=0$ and by (46), the solution is

$U(\varepsilon)=\varepsilon\left(\mu_{0}-\frac{50}{123}\cos
2z+\frac{50}{123}\sin 2z\right)+O(\varepsilon^{2}).$ (69)

Considering $k_{0}=1$ and $\varepsilon<0$, i.e. using “$-$”-sign in (48), we
derive $q_{2}\doteq 1.83348$ and
$y_{1,2}=\pm\sqrt{0.0535297-0.190543\mu_{0}}$. Note that
$0<0.0535297-0.190543\mu_{0}\leq 0.107059$. Formula (54) yields

$0<0.218047-0.776154\mu_{0}\leq 0.436093,$ (70)

which implies that we also have 3 different small solutions for $k_{0}=1$,
$\varepsilon<0$ small and $\mu_{0}$ satisfying (66). Note that if $k_{0}=-1$,
$\varepsilon<0$, we obtain $q_{2}\doteq-1.83348$,
$y_{1,2}=\pm\sqrt{0.0535297-0.190543\mu_{0}}$ and, by (54), estimate (70). The
3 solutions of (65) with parameters (63) have the forms (69) and

$\begin{gathered}\sqrt{-\varepsilon}c(\sqrt{-\varepsilon}){\,\textrm{\rm
e}}^{\imath
k_{0}z}+\sqrt{-\varepsilon}\bar{c}(\sqrt{-\varepsilon}){\,\textrm{\rm
e}}^{-\imath
k_{0}z}+U_{2}\left(\sqrt{-\varepsilon}c(\sqrt{-\varepsilon}){\,\textrm{\rm
e}}^{\imath
k_{0}z}+\sqrt{-\varepsilon}\bar{c}(\sqrt{-\varepsilon}){\,\textrm{\rm
e}}^{-\imath k_{0}z},\varepsilon\right)\\\ =\sqrt{-\varepsilon}(2x(0)\cos
k_{0}z-2y(0)\sin k_{0}z)+O(\varepsilon)=2\sqrt{-\varepsilon}y(0)(q_{2}\cos
k_{0}z-\sin k_{0}z)+O(\varepsilon)\\\ =\pm
2\sqrt{-\varepsilon(0.0535297-0.190543\mu_{0})}(1.83348\cos z-\sin
z)+O(\varepsilon).\end{gathered}$

Next, (52) implies

$0.280932\doteq\frac{\sqrt{11940227}}{12300}<|\mu_{0}|<\frac{\sqrt{\frac{11940227}{74}}}{1400}\doteq
0.286921.$ (71)

From the above computations, we can verify that formula (54) has one of the
forms $0.218047\pm 0.776154\mu_{0}\neq 0$. Since
$\operatorname{sgn}\upsilon=-1$, we have that: if $\varepsilon$ is small,
$\varepsilon\operatorname{sgn}\mu_{0}>0$ and $\mu_{0}$ fulfills (71), then
there are 5 different small solutions ($4\times O(\sqrt{|\varepsilon|})$ and
$1\times O(\varepsilon)$), if $\varepsilon\operatorname{sgn}\mu_{0}<0$ there
is only 1 small solution of order $O(\sqrt{|\varepsilon|})$, actually (69).
For instance, if $\varepsilon>0$ and $\mu_{0}$ satisfies (71), we obtain

$\begin{array}[]{c|c|c}k_{0}&q_{i}&y_{i}\\\ \hline\cr
1&q_{1}=1.83348&y_{1,2}(q_{1})=\pm\sqrt{-0.0535297+0.190543\mu_{0}}\\\
&q_{2}=-0.294153&y_{1,2}(q_{2})=\pm\sqrt{0.214884+0.764897\mu_{0}}\\\
\hline\cr-1&q_{1}=-1.83348&y_{1,2}(q_{1})=\pm\sqrt{-0.0535297+0.190543\mu_{0}}\\\
&q_{2}=0.294153&y_{1,2}(q_{2})=\pm\sqrt{0.214884+0.764897\mu_{0}}\end{array}$

Hence, equation (65) with parameters (63) has the solutions (69) and

$\displaystyle\pm 2\sqrt{\varepsilon(-0.0535297+0.190543\mu_{0})}(1.83348\cos
z-\sin z)+O(\varepsilon),$ $\displaystyle\pm
2\sqrt{\varepsilon(0.214884+0.764897\mu_{0})}(0.294153\cos z+\sin
z)+O(\varepsilon)$

for $\mu_{0}>0$, and the only solution (69) of order $O(\sqrt{\varepsilon})$
for $\mu_{0}<0$.

##  V Numerical results

To illustrate the theoretical results obtained in the previous sections, we
have solved the governing equations (1) and (2) numerically. The advance-delay
equation (2) is solved using a pseudo-spectral method, i.e. we express the
solution $U$ in the Fourier series

$U(z)=\sum_{j=1}^{J}\left[B_{j}\cos\left((j-1)\tilde{k}z\right)+C_{j}\sin\left(j\tilde{k}z\right)\right],$
(72)

where $\tilde{k}=2\pi/L$ and $-L/2<z<L/2$. Substituting the series in (2) and
projecting it onto the Fourier modes in a reminiscence of the Galerkin
truncation method, we obtain a system of algebraic equations for the Fourier
coefficients $B_{j}$ and $C_{j}$, which are then solved using, e.g., a Newton-
Raphson method. Typically, we use $L=2\pi$ and large $J$ that will be
indicated in each calculation below.

The stability of a solution obtained from (2) is determined numerically
through calculating its Floquet multipliers, which are eigenvalues of the
monodromy matrix. As a first-order system, the linearized equation of (1) is

$\displaystyle\dot{u}_{n}$ $\displaystyle=v_{n},$ (73)
$\displaystyle\Delta[\dot{v}_{n}(1+2U_{n})]$ $\displaystyle=-\gamma
v_{n}(1+2U_{n})-u_{n}(1+2\gamma\dot{U}_{n})-2\Delta[\ddot{U}_{n}u_{n}+2\dot{U}_{n}v_{n}],$
(74)

where
$\Delta[\bullet_{n}]=\bullet_{n}-\lambda\bullet_{n-1}-\lambda\bullet_{n+1}$
and $U_{n}(t)=U(\omega t+np)$ (cf. (72)) is a periodic solution of (1). The
linear system is integrated using a Runge-Kutta method of order four with
periodic boundary conditions over the period $T=2\pi/\omega$ and with the site
index $n=1,2,\dots,N$. The $i$th column of the monodromy matrix $M$ is vector
$[u_{1}(T),\dots,u_{N}(T),v_{1}(T),\dots,v_{N}(T)]^{\mathrm{t}}$, that
corresponds to the initial condition
$[u_{1}(0),\dots,u_{N}(0),v_{1}(0),\dots,v_{N}(0)]^{\mathrm{t}}$ equal to the
$i$th column vector of the identity matrix $I_{2N}$. A periodic solution is
asymptotically stable if all Floquet multipliers except the trivial Floquet
multiplier are strictly smaller than one in modulus. The (in)stability result
obtained from calculating the monodromy matrix is also confirmed by
integrating (1). Note that the physically relevant interval for the coupling
parameter is $|\lambda|<1/2$ [6]. It is because there will be a band of
unstable multipliers when $|\lambda|>1/2$, implying that all solutions of the
system will be modulationally unstable. Following Section IV, we set
$\lambda=12/\left(37\sqrt{2}\right)$ and $p=\pi/4$. Computations for different
parameter values have been performed, where we did not see any significant
difference.

Figure 2: (a) The oscillation amplitude of periodic solutions of (2) as a
function of the driving amplitude $h_{0}$ for $\omega=1.23$ with $\gamma=0$
(black thick lines) and $\gamma=0.001$ (blue thin lines). (b) Solution
profiles at the driving amplitudes indicated in the legend for vanishing
$\gamma$.

First, we consider the case of

$h(z)=h_{0}\cos z.$ (75)

From (16) for $\gamma=0$ the first resonance of the system is $k=\pm 1$, which
corresponds to $\omega=\sqrt{37}/5$. In Fig. 2, we depict the bifurcation
diagrams of periodic solutions for a value of $\omega=1.23>\sqrt{37}/5$. On
the vertical axis is the oscillation amplitude $A$ of the periodic solutions
defined as

$A=\frac{1}{2}\left(\text{max}\,(U(z))-\text{min}\,(U(z))\right)$

as a function of the driving amplitude $h_{0}$.

To check the convergence and dependence of the method on the number of modes,
we computed the bifurcation diagrams for several values of $J$. Interestingly
we observed that there is always a critical value of $h_{0}$ beyond which the
bifurcation diagrams depend significantly on the parameter. The diagrams for
three values of $J$ are shown in Fig. 2(a). Using the figure, we conclude that
to determine the validity region of our numerical method, we need to compute
bifurcation diagrams using at least two different values of $J$. The ’breaking
point’ of the method is when the curves start branching out. Studying the
solution profiles near the breaking point as shown in Fig. 2(b), the breaking
point seemingly corresponds to an extreme point of $U(z)$ having discontinuous
first derivative. We therefore infer that our equation (2) may have peaked-
periodic solutions.

As shown as black thick lines in Fig. 2(a), when $\gamma=0$ there is one
saddle-node bifurcation that occurs as we vary $h_{0}$. Note that the
bifurcation diagram has a mirror symmetric (with respect to the vertical axis
$h_{0}=0$) counterpart, i.e. the system is symmetric with the transformation
$h_{0}\to-h_{0},\,U\to-U$. When $\gamma$ is turned on to a non-zero value, the
bifurcation curve interacts with its symmetry and at the intersection point
$h_{0}=0$ for non-vanishing $A$ an additional turning-point is formed, i.e.
there is a curve-splitting-and-merging. In that case, as shown as blue thin
lines in Fig. 2(b), there are now two saddle-node bifurcations.

Figure 3: The same as Fig. 2(a) for $\omega=\sqrt{37}/5$ with (a) $\gamma=0$
and (b) $\gamma=0.1$. In both panels, the solid black and the dashed blue
lines are the numerics and the analytical results (61) (a) and (59), (64) (b),
respectively.

When $\omega$ is decreased toward the resonance frequency, the critical drive
of $h_{0}$ for the occurrence of the first saddle-node bifurcation, i.e. the
bifurcation that also exists when $\gamma=0$, decreases towards zero. In the
limit $\omega=\sqrt{37}/5$, we present in Fig. 3(a) the bifurcation curve of
periodic solutions at the resonance. The numerics (shown as solid black
curves) is in good agreement with the analytical result (dashed line) in
Example 2, i.e. (61), that when $\gamma=0$ the slope of the existence curve is
singular at $h_{0}=0$.

In panel (b) of the same figure, we depict the bifurcation curve of the
periodic solutions for the same parameter values, but with $\gamma\neq 0$. In
agreement with the analytical result in Examples 1 and 3, i.e. (59) and (64),
adding dissipation will regularize the slope at the origin.

Figure 4: The same as Fig. 2(a), but for the periodic drive (76). The dashed
lines are the analytical amplitudes from (68) and (69). Here, $\gamma=0$.

(a) (b) (c)

Figure 5: Floquet multipliers of periodic solutions for some parameter values
(see the text). The dashed curve is the unit circle, showing as guide to the
eye.

We have also considered the multiple resonance case discussed in Section IV.
For the numerics, we take the drive

$h(z)=h_{0}\left(\mu_{0}+2(\cos(2z)-\sin(2z))\right),$ (76)

with $\mu_{0}=1/4$, which satisfies the condition (66). From Example 4, we
know that in this resonance case, there are three small solutions given by
(68) and (69). Shown in Fig. 4 are numerically computed bifurcation curves of
two of the three periodic solutions.

In panel (a), we show only one bifurcation diagram of the solutions (68). It
is because the amplitudes of the two solutions are almost the same. In panel
(b), we depict the smaller solution with amplitudes that scale as $O(h_{0})$.
One can note that in both plots our analytical results from Lyapunov-Schmidt
reduction agree with the numerics.

We have also determined the stability of the solutions obtained numerically
and found that in the non-resonant condition, the periodic solutions are
generally stable for small enough driving amplitude $h_{0}$.

Shown in Fig. 5 are the Floquet multipliers of some periodic solutions,
calculated using $N=200$. Panels (a-b) depict the multipliers of the periodic
solutions corresponding to the bifurcation curve in Fig. 3(b) for $h_{0}=0.01$
and $0.04$, respectively. As all the multipliers in the first panel are inside
the unit circle, we can deduce that the solution is stable. However, as
$h_{0}$ increases further, some multipliers leave the circle as shown in panel
(b) and induce modulational instability. Panel (c) of the same figure
corresponds to a solution along the existence curve in Fig. 3(a), i.e. with
$h_{0}=0.01$ and $\gamma=0$. It is clear that the solution is unstable. For
this resonant case, we found that all solutions for any driving amplitude are
modulationally unstable.

We have also simulated the dynamics of the unstable solutions by solving the
governing equation (1). We observed that the typical dynamics of the
instability is in the form of solution blow-up, similarly to that in [1].

As a conclusion, we found that the qualitative behaviour of the system (1) is
similar to that of forced FPU lattices discussed in [8], despite the
significant difference in the coupling terms as explained in Section I.

## Acknowledgments

The work of MA has been co-financed from resources of the operational program
”Education and Lifelong Learning” of the European Social Fund and the National
Strategic Reference Framework (NSRF) 2007-2013 within the framework of the
Action State Scholarships Foundation’s (IKY) mobility grants programme for the
short term training in recognized scientific/research centres abroad for
candidate doctoral or postdoctoral researchers in Greek universities or
research centres. MF is partially supported by Grants VEGA-MS 1/0071/14, VEGA-
SAV 2/0029/13 and by the Slovak Research and Development Agency under the
contract No. APVV-14-0378. MP is supported by Grant VEGA-SAV 2/0029/13. The
work of VR has been co-financed by the European Union (European Social Fund
ESF) and Greek national funds through the Operational Program Education and
Lifelong Learning of the National Strategic Reference Framework (NSRF)
Research Funding Program: THALES Investing in knowledge society through the
European Social Fund. VR and HS acknowledge partial support from the London
Mathematical Society through a visitor grant.

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