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\begin{document}
\vglue-1cm \hskip1cm
\title[Multiple periodic waves of the Schrödinger system with cubic nonlinearity]{Spectral Stability of multiple periodic waves for the Schrödinger system with cubic nonlinearity}

\begin{center}

\subjclass[2000]{76B25, 35Q51, 35Q70.}

\keywords{Spectral stability, Periodic waves, Schrödinger system.}

\maketitle

{\bf Fábio Natali}

{Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
	Avenida Colombo, 5790, CEP 87020-900, Maring\'a, PR, Brazil.}\\
{ fmanatali@uem.br}

{\bf Gabriel E. Bittencourt Moraes}

{Departamento de Matem\'atica - Universidade Estadual de Maring\'a\\
	Avenida Colombo, 5790, CEP 87020-900, Maring\'a, PR, Brazil.}\\
{ pg54546@uem.br}

\vspace{3mm}

\end{center}

\begin{abstract}
Results concerning the existence and spectral stability and instability of multiple periodic wave solutions for the nonlinear Schrödinger system with \textit{dnoidal} and \textit{cnoidal} profile will be determined in this manuscript. The spectral analysis for the corresponding linearized operator is established by using the comparison theorem and tools of Floquet theory. The main results are determined by applying the spectral stability theory in \cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII} via Krein signature.
\end{abstract}

\section{Introduction}

In this paper, we show the spectral stability of periodic standing waves for the cubic nonlinear Schrödinger system (NLS-system)
\begin{equation}\label{NLS-system}
	\left\{ \begin{array}{l}
		i u_t + u_{xx} + \kappa_1 |u|^2 u + \gamma v^2 \overline{u} = 0 \\
		i v_t + v_{xx} + \kappa_2 |v|^2 v + \gamma u^2 \overline{v} = 0.
	\end{array}
\right.
\end{equation}
Here $u,v: \mathbb{R} \times (0,+\infty) \rightarrow \mathbb{C}$ are complex-valued functions which are $L$-periodic in the spatial variable, $\kappa_1$ and $\kappa_2$ are positive constants and $\gamma \geq 0$.

The NLS equation ($v=0$ in $(\ref{NLS-system})$) appears in many applications in physics and engineering as in nonlinear optics, quantum mechanics and nonlinear waves (see \cite{Boyd} and \cite{Fibich}). The cubic nonlinearity has been used as a model equation in field theory (see \cite{DashenHasslacherNeveu}). The NLS system also appears in these areas such as nonlinear optics and Bose-Einstein condesates (see \cite{Agrawal, IedaMiyakawaWadati, KannaSakkaravarthi, WangCui}).

For the NLS equation with cubic nonlinearity in the periodic setting, Angulo in \cite{angulo2007} established the orbital stability in the energy space $H_{per}^1$ of periodic standing waves solutions of dnoidal type with respect to perturbations of period $L$. Also, the existence of cnoidal waves were also obtained in \cite{angulo2007}. However, the author was not capable to prove the orbital stability/instability in the energy space $H^1_{per}$ for this kind of waves. Recently, Natali \textit{et al.} determined in \cite{NataliMoraesLorenoPastor} the orbital stability of cnoidal waves restricted to the subspace $H_{per,m}^1$ of zero mean periodic functions contained in $H_{per}^1$. Gallay and H\u ar\u agu\c s in \cite{GallayHaragusJDE} and  \cite{GallayHaragusJDDE} showed that the cnoidal waves are orbitally stable with respect to perturbations $w(x+L/2) = -w(x)$ for all $x \in \mathbb{R}$. In addition, also in \cite{GallayHaragusJDE} the authors established a result about the spectral stability with respect to bounded or localized perturbations. Gustafson, Le Coz and Tsai in \cite{GustafsonLeCozTsai} have established spectral stability results for the cnoidal waves with respect to perturbations with the same period $L$ and orbital stability results in the space constituted by anti-periodic functions with period $L/2$. The spectral stability follows by relating the coercivity of the linearized action with the number of eigenvalues with negative Krein signature of an operator $J \mathcal{L}$.

In the context of NLS-system \eqref{NLS-system}, Kawahara and Ohta in \cite{KawaharaOhta} showed the orbital stability and instability of standing solitary wave solutions for the system \eqref{NLS-system}. In that approach, the authors studied the orbital stability properties of semi-trivial standing wave solutions of the form
\begin{equation}\label{semitrivial-ko}(u(x,t),v(x,t)) = (e^{i\omega t} \varphi_\omega(x),0),
\end{equation} where $\varphi_\omega(x) = \sqrt{2 \omega} {\rm \,sech}(\sqrt{\omega} x)$ is a positive and even solution of the equation
\begin{equation}\label{ODE1}
	-\varphi_{\omega}'' + \omega \varphi_{\omega} - \varphi_{\omega}^3 = 0.
\end{equation}
They proved that the semi-trivial standing wave solution \eqref{semitrivial-ko} is orbitally stable if $\gamma < \kappa_1$ and orbitally unstable if $\gamma > \kappa_1$. In addition, if $\gamma = \kappa_1$, the authors concluded the orbital stability when $\kappa_2 < \kappa_1$ and orbital instability when $\kappa_2 > \kappa_1$. Unfortunately, for the case $\gamma = \kappa_1 = \kappa_2$, they were not capable to decide about orbital stability (see \cite[Remark 2]{KawaharaOhta}).

In the periodic setting, Hakkaev \cite{Hakkaev} studied  the spectral stability of the NLS-system by considering semi-trivial standing waves as in \eqref{semitrivial-ko} where $\varphi_\omega$ is an $L$-periodic function with dnoidal profile which solves equation $(\ref{ODE1})$ and it is given by
\begin{equation}\label{varphiw-hakkaev}
	\varphi_\omega(x) = \frac{2 \sqrt{2} {\rm K}(k)}{L} {\rm dn} \left( \frac{2 {\rm K}(k)}{L} x, k\right),
\end{equation}
where $k \in (0,1)$ is the modulus of the elliptic function and ${\rm K}(k)$ is the complete elliptic integral of the first kind. More precisely, the author showed that the semi-trivial periodic waves are orbitally stable for $\gamma < \kappa_1$. In addition, he also obtained results of spectral stability and instability for the semi-trivial waves. In fact, for $\kappa_1 < \gamma \leq 3\kappa_1$, he concluded that the semi-trivial periodic waves are spectrally unstable and for $\gamma = \kappa_1$ the semi-trivial periodic wave solutions of \eqref{ODE} are spectrally stable.

Now, we give the main topics of our paper. Motivated by \cite{Hakkaev} and \cite{KawaharaOhta}, we consider the multiple periodic wave solutions of \eqref{NLS-system} given by
\begin{equation}\label{multiple}
(u(x,t),v(x,t)) = (e^{i\omega t} \varphi_\omega(x), e^{i\omega t} B \varphi_\omega(x))
\end{equation}
where $\varphi_\omega: \mathbb{R} \rightarrow \mathbb{R}$ is an $L$-periodic function, $\omega \in \mathbb{R}$ is the frequency wave and $B \in \mathbb{R}$ is a real constant.

In our paper, we study two kind of profiles for the wave $\varphi=\varphi_\omega$ in the periodic setting. First, we complete the study realized in \cite{Hakkaev} by considering $\varphi$ with (positive) dnoidal profile. On the other hand, equation $(\ref{ODE})$ has a sign-changed periodic solution of cnoidal type. In this specific setting, we study the cases $B=0$ (semitrivial solutions) and $B \neq 0$ (multiple solutions).

Let's start by constructing our two kind of periodic solutions. First, by substituting the form \eqref{multiple} into \eqref{NLS-system}, and considering $B \neq 0$, we get
\begin{equation}\label{ODE-system}
	\left\{ \begin{array}{l}
		- \omega \varphi + \varphi'' + (\kappa_1 + \gamma B^2) \varphi^3 = 0 \\
		- \omega \varphi + \varphi'' + (\kappa_2 B^2 + \gamma) \varphi^3 = 0.
		\end{array}
	\right.
\end{equation}

To produce multiple equations, we assume that $\kappa_1 + \gamma B^2 = \kappa_2 B^2 + \gamma$. Thus, for $\kappa_2\neq\gamma$, we see that $B$ can be expressed by
\begin{equation}\label{B}
	B = \pm\sqrt{ \frac{\kappa_1 - \gamma}{\kappa_2 - \gamma}}.
\end{equation}
Let us consider $B>0$ (the case $B<0$ is similar). In our study, we first consider the three basic cases:
\begin{equation}\label{cases1}
\gamma \in (0,\min\{\kappa_1,\kappa_2\}), \; \; \gamma \in (\max\{\kappa_1,\kappa_2\},+\infty) \; \; \text{ and } \; \; \gamma = 0.
\end{equation}
\indent In \eqref{ODE-system} we can also consider the case $\gamma = \kappa_1$ or $\gamma=\kappa_2$. We obtain in both cases $\gamma = \kappa_1 = \kappa_2$, so that $B \in \mathbb{R}$ is a free parameter which does not depend on $\kappa_1$, $\kappa_2$, $\gamma$ and $\omega$.

In all cases above, the periodic wave $\varphi$ is a solution of the ODE
\begin{equation}\label{ODE}
	- \varphi'' + \omega \varphi + (\kappa_1 + \gamma B^2) \varphi^3 = 0.
\end{equation}
In the case of solutions with dnoidal profile, we can determine the explicit solution as
\begin{equation}\label{dn}
	\varphi(x) = \frac{2\sqrt{2} {\rm K}(k)}{L} \frac{1}{(\kappa_1 + \gamma B^2)^{1/2}} {\rm dn}\left( \frac{2 {\rm K}(k)}{L} x, k\right).
\end{equation}
The frequency of the wave $\omega \in \mathbb{R}$ can be expressed as
\begin{equation}\label{omega-dnoidal}
	\omega = \frac{4 (2-k^2) {\rm K}(k)^2}{L^2}.
\end{equation}
By \eqref{omega-dnoidal} we can see from the dependence of $\omega$ in terms of the parameter $k \in (0,1)$ that $\omega \in \left(\frac{2 \pi^2}{L^2}, +\infty\right)$.\\
\indent  On the other hand, to obtain solutions with cnoidal type, we can proceed similarly as in \cite{angulo2007} to obtain the periodic solution $\varphi$ given by
\begin{equation}\label{cn}
	\varphi(x) = \frac{ \sqrt{2 \omega} k}{\sqrt{2k^2 - 1}} \frac{1}{(\kappa_1 + \gamma B^2)^{1/2}} {\rm cn}\left( \frac{ 4 {\rm K}(k)}{L} x, k \right).
\end{equation}
The frequency wave $\omega > 0$ is
\begin{equation}\label{omega-cnoidal}
	\omega = \frac{ 16 {\rm K}(k)^2 (2k^2 - 1)}{L^2}.
\end{equation}

In both cases, we are enabled to construct a smooth curve of $L$-periodic waves $\omega \in \mathcal{I} \longmapsto \varphi\in H^2_{per}$ which solves \eqref{ODE} (see Theorems $\ref{dnoidalcurve}$ and $\ref{cnoidalcurve}$).

System \eqref{NLS-system} admits the conserved quantity $E: H^1_{per} \times H^1_{per} \rightarrow \mathbb{R}$ defined by
\begin{equation}\label{E}
	E(u,v) = \frac{1}{2} \int_{0}^{L} \left( |u_x|^2 + |v_x|^2 - \frac{\kappa_1}{2} |u|^4 - \frac{\kappa_2}{2} |v|^4 \right) dx - \frac{\gamma}{2} {\rm Re} \int_{0}^L u^2 \overline{v}^2 dx.
\end{equation}
Moreover, \eqref{NLS-system} has another conserved quantity $F: H^1_{per}\times H^1_{per} \rightarrow \mathbb{R}$ as
\begin{equation}\label{F}
	F(u,v) = \frac{1}{2} \int_{0}^L \left( |u|^2 + |v|^2 \right) dx.
\end{equation}
Then, following similar arguments as in \cite{Cazenave} and using standard fixed point arguments, we can conclude by the conservation laws in $(\ref{E})$ and $(\ref{F})$ that the NLS-system \eqref{NLS-system} is globally well posed in $H^1_{per}\times H_{per}^1$ (see, for instance, \cite{Colin, Hakkaev, Whitham}).

Now, we present how to obtain the spectral stability of multiple periodic waves with respect to perturbation with the same period. Consider $U(x,t) = (u(x,t),v(x,t)) = ({\rm Re}\,u, {\rm Re}\, v, {\rm Im}\, u, {\rm Im}\, v)$, $\Phi = (\varphi, B\varphi,0,0)$ and the perturbation
\begin{equation}\label{U-1}
	U(x,t) = e^{i \beta t} (\Phi(x) + W(x,t))
\end{equation} where $W(x,t) = (w_1(x,t), w_2(x,t)) \in \mathbb{R}^4$. Substituting \eqref{U-1} into \eqref{NLS-system} and neglecting all the nonlinear terms, we get the following linearized equation:
\begin{equation}\label{spectral}
	\frac{d}{dt} W(x,t) = J \mathcal{L} W(x,t),
\end{equation}
where
\begin{equation}\label{J}
	J = \left( \begin{array}{cccc}
		0 & 0 & 1 & 0 \\
		0 & 0 & 0 & 1 \\
		-1 & 0 & 0 & 0 \\
		0 & -1 & 0 & 0
		\end{array}
	\right)
\end{equation} and $\mathcal{L}$ is the operator given by
\begin{equation}\label{L-1}
	\mathcal{L}= (-\partial_x^2 + \omega){\rm Id} - \varphi^2 S,
\end{equation}
where ${\rm Id} \in \mathbb{M}_{4 \times 4}(\mathbb{R})$ and $S$ is given by
\begin{equation}
	S = \left(
	\begin{array}{cccc}
		(3\kappa_1 + \gamma B^2) & 2\gamma B & 0 & 0 \\
		2\gamma B & (3\kappa_2 B^2 + \gamma) & 0 & 0 \\
		0 & 0 & (\kappa_1 - \gamma B^2) & 2\gamma B \\
		0 & 0 & 2\gamma B & (\kappa_2 B^2 - \gamma)
	\end{array}
\right).
\end{equation}

As far as we know, it is more convenient to work with the operator $\mathcal{L}$ in a diagonal form. To do so, we need to obtain the existence of an  orthogonal matrix operator $U$ (see the definition of $U$ in Section 4) and a matrix $M$ such that
\begin{equation}\label{S}
	S:= U M U^{-1}.
\end{equation}
Here, $M \in \mathbb{M}_{4 \times 4}(\mathbb{R})$ is given by
\begin{equation}\label{M}
	M = \left( \begin{array}{cccc}
		\beta_1 & 0 & 0 & 0 \\
		0 & \beta_3 & 0 & 0 \\
		0 & 0 & \beta_2 & 0 \\
		0 & 0 & 0 & \beta_4
		\end{array}
		\right),
\end{equation}
and the constants $\beta_i \in \mathbb{R}$ are given in terms of $\gamma$, $\kappa_1$ and $\kappa_2$ (see $(\ref{betai})$). Substituting \eqref{S} into \eqref{L-1}, we have
\begin{equation}\label{ULU}
	\mathcal{L} = U \left( \begin{array}{cccc}
		\mathcal{L}_1 & 0 & 0 & 0 \\
		0 & \mathcal{L}_3 & 0 & 0 \\
		0 & 0 & \mathcal{L}_2 & 0 \\
		0 & 0 & 0 & \mathcal{L}_4
	\end{array}
\right) U^{-1}=U\tilde{\mathcal{L}}U^{-1},
\end{equation}
where, for $i \in \{1,\ldots, 4\}$
\begin{equation}\label{Li}
	\mathcal{L}_i = -\partial_x^2 + \omega - \beta_i \varphi^2,
\end{equation}
are Hill operators. To evaluate all the pictures for the spectral stability and instability, we also need to study the non-positive spectrum of the operator $\mathcal{L}$ and so, it is convenient to obtain all these properties for the operators $\mathcal{L}_i$ for $i = \{1,\ldots,4\}$.

Denoting ${\rm n}(\mathcal{A})$ as the number of negative eigenvalues of a certain linear operator $\mathcal{A}$, we obtain for the case of dnoidal waves that ${\rm n}(\mathcal{L}_1) = 1$ and ${\rm n}(\mathcal{L}_2) = 0$ (see \cite{angulo2007} and \cite{Hakkaev}). In addition, we have that ${\rm Ker}(\mathcal{L}_1) = [\varphi']$ and ${\rm Ker}(\mathcal{L}_2) = [\varphi]$. Then, using the comparison theorem, we can study the behaviour of the non-positive spectrum concerning the operators $\mathcal{L}_3$ and $\mathcal{L}_4$ (see details in Section 4). Next, by considering $\varphi \in H^1_{per}$ with cnoidal profile, we have ${\rm n}(\mathcal{L}_1) = 2$ and ${\rm n}(\mathcal{L}_2) = 1$ (see \cite{angulo2007}). However, since we have a considerable quantity of negative eigenvalues in this case, we can not use the comparison theorem to decide the behaviour of $\mathcal{L}_3$ and $\mathcal{L}_4$ as we have determined in the case of dnoidal solutions. To overcome this difficulty, we determine the quantity of non-zero eigenvalues of $\mathcal{L}_3$ and $\mathcal{L}_4$ by restricting the analysis to the subspace of odd periodic functions in $L_{per}^2$. In this setting, we are enabled to prove that ${\rm n}(\mathcal{L}_{1}) = 1$, ${\rm n}(\mathcal{L}_{2}) = 0$, ${\rm Ker}(\mathcal{L}_{1}) = \{0\}$, and ${\rm Ker}(\mathcal{L}_{2}) = [\varphi]$. These facts allow us to use the comparison theorem to obtain the behaviour of the non-positive spectrum for the operators $\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ restricted to the odd sector of $L_{per}^2$.

Before stating our spectral stability results, we need to present some basic facts. In fact, let us consider $W(x,t)=e^{\lambda t}u(x)$ in the linear problem $(\ref{spectral})$. We obtain the following spectral problem
\begin{equation}\label{spectralproblem}
	J \mathcal{L} u = \lambda u
\end{equation}
\begin{definition}\label{def-spectralstability}
	The stationary wave $\Phi$ is spectrally stable if $\sigma(J \mathcal{L}) \subset i \mathbb{R}$. Otherwise, $\Phi$ is spectrally unstable if there exists at least one eigenvalue $\lambda$ associated to the operator $J \mathcal{L}$ with a positive real part.
\end{definition}

To obtain our results, we use the approach developed by Kapitula, Kevrekidis and Sandstede in \cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII}. First, we denote by $z(\mathcal{A})$ the dimension of the kernel of a certain linear operator $\mathcal{A}$. Consider $\Theta_l \in {\rm Ker}(\mathcal{L})$ and let $V$ be the $n\times n$ matrix whose entries are given by
\begin{equation}\label{V}
V_{jl} = ( \mathcal{L}^{-1} J \Theta_j, J \Theta_l )_{\mathbb{L}^2_{per}},
\end{equation}
where $1\leq j,l\leq n$ and $z(\mathcal{L})=n$.
The formula
\begin{equation}\label{krein}
	k_r + k_c + k_{-} = {\rm n}(\mathcal{L}) - {\rm n}(V),
\end{equation}
is given in \cite{KapitulaKevrekidisSandstedeII} and the left-hand side of $(\ref{krein})$ is exactly the hamiltonian Krein index, an important tool to decide about the spectral stability and instability of waves. In equality $(\ref{krein})$, $k_r$ the number of real-valued and positive eigenvalues (counting multiplicities), $k_c$ denotes the number of complex-valued eigenvalues with a positive real part, whereas $k_-$ is the number of pairs of purely imaginary eigenvalues with negative Krein signature. Since $k_c$ and $k_-$ are always even numbers, we obtain that if the right-hand side in \eqref{krein} is an odd number, then $k_r \geq 1$ and we have automatically the spectral instability. Moreover, if the difference ${\rm n}(\mathcal{L}) - {\rm n}(V)$ is zero, then $k_c = k_- = k_r = 0$ which implies the spectral stability.

Summarizing the comments above, our main and novel results concerning the spectral stability of multiple periodic waves of the form $(\ref{multiple})$ with $B\neq0$ are then established:

\begin{theorem}[\textit{Spectral stability/instability for the multiple wave solution with dnoidal profile}]\label{teo-1}
	Let $L>0$ be fixed. Consider the periodic wave solution $\varphi \in H^1_{per}$ of \eqref{ODE} with dnoidal profile given by \eqref{dn}. For $B$ given in \eqref{B} and for all $\omega \in (\frac{2\pi^2}{L^2},+\infty)$, the wave $\Phi = (\varphi, B \varphi,0,0)$ is spectrally unstable if $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$  and spectrally stable if $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty) \cup \{0\}$. In addition, if one considers $\gamma = \kappa_1 = \kappa_2$ with $B\in\mathbb{R}$ being a free parameter, we obtain that $\Phi$ is spectrally stable for all $\omega \in \left(\frac{2\pi^2}{L^2},+\infty\right).$
\end{theorem}

\begin{theorem}[\textit{Spectral instability for the multiple wave solution with cnoidal profile}]\label{teo-2}
	Let $L>0$ be fixed and consider $\omega>0$. Let $\varphi \in H^1_{per}$ be the  periodic solution of \eqref{ODE} with cnoidal profile given by \eqref{cn}. For $\gamma = \kappa_1 = \kappa_2$ with $B\in\mathbb{R}$ being a free parameter, the multiple wave $\Phi = (\varphi, B \varphi,0,0)$ is spectrally unstable.
\end{theorem}

\begin{theorem}[\textit{Spectral stability/instability for the multiple wave solution with cnoidal profile and restricted to the subspace of odd functions}]\label{teo-odd}
	Let $L>0$ be fixed and consider $\omega>0$. Let $\psi=\varphi(\cdot-L/4) \in H^1_{per}$ be the periodic solution of \eqref{ODE} with cnoidal profile given by \eqref{cnoidal-odd} and restricted to the odd sector. For $B$ given in \eqref{B}, the multiple wave $\Psi = (\psi, B \psi,0,0)$ is spectrally unstable if $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$ and spectrally stable if $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty) \cup \{0\}$. In addition, if one considers $\gamma = \kappa_1 = \kappa_2$ with $B\in\mathbb{R}$ being a free parameter, we have that $\Psi$ is spectrally stable.
\end{theorem}

To finish, we give an extension of the results obtained in \cite{Hakkaev} by showing the spectral stability concerning the semi-trivial periodic wave solution
	$(u(x,t),v(x,t)) = (e^{i \omega t} \varphi(x),0),
$
where $\varphi \in H^1_{per}$ has a cnoidal profile. Here, we also analyze the spectral properties concerning the operator $\mathcal{L}$ to obtain the spectral stability results in the same setting of parameters determined in \cite{Hakkaev}. As we have already established for the case of multiple solutions $(\ref{multiple})$ with $B\neq0$, some difficulties appear in the spectral analysis of $\mathcal{L}$. To overcome all these difficulties, we also study the case where $\varphi(\cdot-L/4) $ is restricted to the odd sector of $\mathbb{L}_{per}^2$.

\begin{theorem}[Spectral instability for the semi-trivial wave solution with cnoidal profile] Let $L>0$ be fixed and consider $\omega > 0$. The semi-trivial wave solution $\Psi = (\psi,0,0,0)$ where $\psi=\varphi(\cdot-L/4)$ is obtained by the translation of the cnoidal profile $\varphi$ is spectrally unstable when $\gamma = \kappa_1$.
\label{teosemi}	\end{theorem}

\begin{theorem}[Spectral stability/instability for the semi-trivial wave solution with cnoidal profile and restricted to the subspace of odd functions] Let $L>0$ be fixed and consider $\omega>0$. The semi-trivial wave solution $\Psi = (\psi,0,0,0)$ where $\psi=\varphi(\cdot-L/4)$ is obtained by the translation of the cnoidal profile $\varphi$ is spectrally stable in the odd sector of $\mathbb{L}_{per}^2$ provided that $\gamma \in (0,\kappa_1]$. In addition, $\Psi = (\psi,0,0,0)$ is spectrally unstable in the odd sector of $\mathbb{L}_{per}^2$ if
$\gamma \in (\kappa_1,3\kappa_1]$.
\label{teooddsemi}\end{theorem}

Our paper is organized as follows:
In Section \ref{section3}, we show the existence of a smooth curve of periodic standing wave solutions of \textit{dnoidal} and \textit{cnoidal} type for the equation \eqref{ODE}. The spectral analysis for the operators $\mathcal{L}$ is determined in Section \ref{section4}. The spectral stability/instability for the multiple wave solutions with dnoidal and cnoidal profile are then established in Section \ref{section5}. Finally, in Section \ref{section6}, we prove the spectral stability/instability result concerning the semi-trivial wave with cnoidal profile.\\

\textbf{Notation:} For $s\geq0$ and $L>0$, the Sobolev space
$H^s_{per}:=H^s_{per}([0,L])$
consists of all periodic functions $f$ such that
$$
\|f\|^2_{H^s_{per}}:= L \sum_{k=-\infty}^{\infty}(1+k^2)^s|\hat{f}(k)|^2 <\infty
$$
where $\hat{f}$ is the periodic Fourier transform of $f$. The space $H^s_{per}$ is a  Hilbert space with the inner product denoted by $(\cdot, \cdot)_{H^s}$. When $s=0$, the space $H^s_{per}$ is isometrically isomorphic to the space  $L^2([0,L])$ and will be denoted by $L^2_{per}:=H^0_{per}$ (see, e.g., \cite{Iorio}). The norm and inner product in $L^2_{per}$ will be denoted by $\|\cdot \|_{L^2_{per}}$ and $(\cdot, \cdot)_{L^2_{per}}$. We need to emphasize here, for $r,s \geq 0$ and $(f,g) \in H^r_{per} \times H^s_{per}$ (complex) we can write $(f,g)=({\rm Re}\,f, {\rm Re}\,g,{\rm Im}\,f, {\rm Im}\,g).$

For $s\geq0$, we denote
$$
H^s_{per,odd}:=\{ f \in H^s_{per} \; ; \; f \:\; \text{is an odd function}\}.
$$
endowed with the norm and inner product in $H^s_{per}$. In addition,
$$
\mathbb{H}^s_{per}:= H^s_{per} \times H^s_{per} \times H^s_{per} \times H^s_{per} \; \; \text{ and } \; \; \mathbb{H}^s_{per,odd}:=H^s_{per,odd} \times H^s_{per,odd} \times H^s_{per,odd} \times H^s_{per,odd}
$$
equipped with their usual norms and scalar products.

The symbols ${\rm \,sn}(\cdot, k), {\rm \,dn}(\cdot, k)$ and ${\rm \,cn}(\cdot, k)$ represent the Jacobi elliptic functions of \textit{snoidal}, \textit{dnoidal}, and \textit{cnoidal} type, respectively. For $k \in (0, 1)$, ${\rm F}(\phi, k)$ and ${\rm \,E}(\phi, k)$  denote the complete elliptic integrals of the first and second kind, respectively, and  we denote by ${\rm \,K}(k)={\rm F}\left(\frac{\pi}{2},k\right)$ and ${\rm \,E}(k)={\rm \,E}\left(\frac{\pi}{2},k\right)$ (for additional details, see \cite{byrd}).

We indicate by $\text{p}(\mathcal{A})$  the number of positive eigenvalues of a certain linear operator $\mathcal{A}$, respectively.

\section{Existence of a Smooth Curve of Periodic Waves}\label{section3}
Our purpose in this section is to present the existence of $L$-periodic solutions $\varphi: \mathbb{R} \longrightarrow \mathbb{R}$ for the following ODE
\begin{equation}\label{ode1}
- \varphi''+\omega\varphi- (\kappa_1 + \gamma B^2)\varphi^{3}=0,
\end{equation}
where $\omega > 0$.

\subsection{$L$-periodic wave solutions with dnoidal profile}

Consider the ODE
$$
	-\phi'' + \omega \phi - \phi^3 = 0.
$$
By \cite{angulo2007}, we obtain periodic solutions with dnoidal profile as
\begin{equation}\label{psi}
\phi(x)=\frac{2 \sqrt{2} {\rm K}(k)}{L} {\rm dn}\left( \frac{2 {\rm K}(k)}{L} x, k \right),
\end{equation}
where $k\in(0,1)$. The frequency $\omega$ depends smoothly on $k \in (0,1)$ and $L>0$ is defined by
\begin{equation}\label{omega-1}
	\omega = \frac{ 4(2-k^2) {\rm K}(k)^2}{L^2}.
\end{equation}
Then, considering the transformation
\begin{equation}\label{dnoidalsol}\varphi(x) = \frac{1}{(\kappa_1 + \gamma B^2)^{1/2}} \phi(x),
\end{equation} we obtain that $\varphi$ is an $L$-periodic solution for the equation \eqref{ode1} with $\omega$ given by the relation \eqref{omega-1} and defined in a subset of $(0,+\infty)$. All these consideration establish the following result:

\begin{theorem}[Smooth Curve of Dnoidal Waves]\label{dnoidalcurve}
Let $L>0$ be fixed and consider  $\omega \in (\frac{2\pi^2}{L^2},+\infty)$. If $\varphi=\varphi_{\omega}$ is the solution of \eqref{ode1} with the dnoidal profile in $(\ref{dnoidalsol})$, the family
	$$
	\omega \in \left(\frac{2\pi^2}{L^2},+\infty\right) \longmapsto \varphi \in H^2_{per}([0,L])
	$$
of periodic solutions of \eqref{ode1} depends smoothly on $\omega$.
\end{theorem}

 \subsection{$L$-periodic wave solutions with cnoidal profile}

 For $\omega> 0$, Angulo in \cite{angulo2007} obtained that the ODE
 \begin{equation}\label{ode2}
 	-\phi'' + \omega \phi - \phi^3 = 0
 \end{equation}
 also admits periodic solutions with cnoidal profile as
\begin{equation}\label{phi}
 \phi(x)=\frac{ \sqrt{2 \omega} k}{\sqrt{2k^2 -1}} {\rm cn}\left( \frac{4 {\rm K}(k)}{L} x, k \right)
 \end{equation}
 where $k\in(0,1)$. The frequency $\omega \in \mathbb{R}$ depends smoothly on $k \in (0,1)$ and $L>0$ is defined by
 \begin{equation}\label{omega-2}
 	\omega = \frac{ 16 {\rm K}(k)^2 (2k^2 - 1)}{L^2}.
 \end{equation}
 \indent Motivated by this, we consider the same scaling transformation as in $(\ref{dnoidalsol})$

 \begin{equation}\label{cnoidalsol}\varphi(x) = \frac{1}{(\kappa_1 + \gamma B^2)^{1/2}} \phi(x).
 \end{equation} We obtain that $\varphi$ is an $L$-periodic solution of \eqref{ode1} for $\omega >0$ given by \eqref{omega-2} and the similar result as in Theorem $\ref{dnoidalcurve}$ reads as follows:

 \begin{theorem}[Smooth Curve of Cnoidal Waves]\label{cnoidalcurve}
 	Let $L>0$ be fixed and consider  $\omega >0$. If $\varphi=\varphi_{\omega}$ is the solution of \eqref{ode1} with cnoidal profile given by $(\ref{cnoidalsol})$, then the family
 	$$
 	\omega \in (0,+\infty) \longmapsto \varphi=\varphi_{\omega} \in H^2_{per}([0,L])
 	$$
 	of periodic solutions of \eqref{ode1} depends smoothly on $\omega \in (0,+\infty)$.
 \end{theorem}

\begin{remark}\label{cnoidal-odd}
	Notice that we can consider the periodic cnoidal solution obtained in Theorem $\ref{cnoidalcurve}$ in the odd sector of $H_{per}^2$. In fact, consider
	\begin{equation}\label{cnoidal-2}
		\psi(x)=\varphi\left(x-\frac{L}{4}\right)=\frac{\sqrt{2 \omega} k}{\sqrt{2k^2 - 1}} \frac{1}{(\kappa_1 + \gamma B^2)^{1/2}} {\rm cn}\left( \frac{4 {\rm K}(k)}{L} \left(x - \frac{L}{4} \right), k \right).
	\end{equation}
Thus, the family $$
\omega \in (0,+\infty) \longmapsto \psi=\psi_{\omega} \in H^2_{per,odd}([0,L])
$$
of odd periodic solutions of \eqref{ode1} depends smoothly on $\omega \in (0,+\infty)$.
\end{remark}

\section{Spectral analysis}\label{section4}

In this section, we calculate the non-positive spectrum of the linear operator $\mathcal{L}$ by using the information of the non-positive spectrum of the Hill operators $\mathcal{L}_i$, $i=1,2,3,4$, given by \eqref{Li}. For the case of dnoidal solutions, we borrow the results determined by Hakkaev \cite{Hakkaev} and Angulo \cite{angulo2007}. On the other hand, for the case of cnoidal solutions, we use some results in Angulo in \cite{angulo2007} and Natali, \textit{et. al.} \cite{NataliMoraesLorenoPastor}.

\subsection{Spectral analysis with dnoidal profile}

Considering $B>0$ given by \eqref{B} (the case $B<0$ is similar). The entries of the matrix $M$ in $(\ref{M})$ are given by
\begin{equation}\label{betai}
	\begin{array}{ll}
		\beta_1 = \dfrac{3(\gamma^2 - \kappa_1\kappa_2)}{\gamma - \kappa_2}, & \beta_3 = \dfrac{-\gamma^2 + 2\gamma(\kappa_1 + \kappa_2) - 3\kappa_1\kappa_2}{\gamma - \kappa_2}, \\
		\beta_2 = \dfrac{(\gamma^2 - \kappa_1\kappa_2)}{\gamma-\kappa_2}, & \beta_4 = \dfrac{-3\gamma^2 + 2\gamma(\kappa_1 + \kappa_2) - \kappa_1\kappa_2}{\gamma - \kappa_2}.
	\end{array}
\end{equation}
Also, the matrix $U$ is given by
\begin{equation}\label{U-Bfixed}
	U = \left( \begin{array}{cccc}
		-\kappa_2 + \gamma & \frac{1}{2\gamma -\kappa_1 - \kappa_2} & 0 & 0 \\
		\sqrt{(\kappa_1-\gamma)(\kappa_2-\gamma)} & - \frac{1}{2 \gamma - \kappa_1 - \kappa_2}  \frac{1}{B} & 0 & 0 \\
		0 & 0 & - \frac{1}{2\gamma - \kappa_1 - \kappa_2} & \kappa_1 - \gamma \\
		0 & 0 & - \frac{1}{2\gamma - \kappa_1 - \kappa_2} \frac{1}{B} & \sqrt{(\kappa_1-\gamma)(\kappa_2-\gamma)}
	\end{array}
		\right).
\end{equation}
For $B$ defined in \eqref{B}, operators $\mathcal{L}_i = -\partial_x^2 + \omega - \beta_i \varphi^2$ can be expressed in terms of the parameters $\beta_i$ in  \eqref{betai}. We have
\begin{equation}\label{L1-cn}
	\mathcal{L}_1 = -\partial_x^2 + \omega - \beta_1 \varphi^2 = -\partial_x^2 + \omega - 3(\kappa_1 + B^2 \gamma) \varphi^2 = -\partial_x^2 + \omega - 3 \phi^2
\end{equation}
and
\begin{equation}\label{L2-cn}
	\mathcal{L}_2 = -\partial_x^2 + \omega - \beta_2 \varphi^2 = -\partial_x^2 + \omega - (\kappa_1 + B^2 \gamma) \varphi^2 = -\partial_x^2 + \omega -  \phi^2,
\end{equation}
where $\phi$ is the solution with dnoidal profile given by \eqref{psi}. So, we can use the spectral properties for the operators $\mathcal{L}_1$ and $\mathcal{L}_2$ as obtained in \cite{angulo2007} and \cite{Hakkaev} to obtain
	${\rm n}(\mathcal{L}_1) = 1,\ {\rm n}(\mathcal{L}_2) = 0,$
	${\rm Ker}(\mathcal{L}_1) = [\varphi']$ and ${\rm Ker}(\mathcal{L}_2) = [\varphi].$
 We need to study the spectral analysis of the operator $\mathcal{L}$ in some different cases:

\textbf{Case I:} $\gamma \in (0, \min\{\kappa_1,\kappa_2\})$. In this case, after some calculations with the parameters $\beta_i$, $i=1,2,3,4$, we conclude that for all $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$, we have
	\begin{equation}\label{ineq-1}
		\beta_4 < \beta_2 < \beta_3 < \beta_1.
	\end{equation}

We obtain by the comparison theorem the following order of operators in the sense of inner product in $L_{per}^2$
\begin{equation}\label{comparison}
	\mathcal{L}_1 < \mathcal{L}_3 < \mathcal{L}_2 < \mathcal{L}_4.
\end{equation} By the inequalities in $(\ref{comparison})$, we have that
	${\rm n}(\mathcal{L}_3) = 1,\ {\rm n}(\mathcal{L}_4) = 0$
and ${\rm z}(\mathcal{L}_3) = {\rm z}(\mathcal{L}_4) = \{0\}$.

Therefore, it follows by $(\ref{ULU})$ that ${\rm n}(\mathcal{L}) = 2$ and ${\rm z}(\mathcal{L})=2$ with
$$
	{\rm Ker}(\mathcal{L}) = \left[(\varphi', B\varphi',0,0), (0,0,\varphi, B\varphi)\right].
$$

\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\}, +\infty)$. By considering $\gamma \in (\max\{\kappa_1,\kappa_2\}, +\infty)$, one has
\begin{equation}\label{ineq-2}
	\beta_4 < \beta_3 < \beta_2 < \beta_1.
\end{equation}

So, we obtain by $(\ref{ineq-2})$
\begin{equation}\label{comparison2}
	\mathcal{L}_1 < \mathcal{L}_2 < \mathcal{L}_3 < \mathcal{L}_4.
\end{equation}
Thus, using the comparison theorem and $(\ref{comparison2})$, it follows that
${\rm n}(\mathcal{L}_3)={\rm n}(\mathcal{L}_4)=0$ and ${\rm z}(\mathcal{L}_3)={\rm z}(\mathcal{L}_4)=0$.

Again, by $(\ref{ULU})$ we get ${\rm n}(\mathcal{L}) = 1$ and ${\rm z}(\mathcal{L})=2$ where
$$
	{\rm Ker}(\mathcal{L}) = \left[(\varphi', B\varphi',0,0), (0,0,\varphi, B\varphi)\right].
$$

\textbf{Case III:} $\gamma = 0$. First, we have to notice that for $B>0$, we obtain $B = \sqrt{\frac{\kappa_1}{\kappa_2}}$ (the case $B<0$ is similar). By the expression of the matrix $S$ given by \eqref{S}, we do not need to use the similar transformation $U$ since in this case, the operator $\mathcal{L}$ is given by
\begin{equation}\label{L-gamma0}
	\mathcal{L} = \left( \begin{array}{cccc}
	\mathcal{L}_1 & 0 & 0 & 0 \\
	0 & \mathcal{L}_1 & 0 & 0 \\
	0 & 0 & \mathcal{L}_2 & 0 \\
	0 & 0 & 0 & \mathcal{L}_2
	\end{array}
\right),
\end{equation}
that is, $\mathcal{L}_3=\mathcal{L}_1$ and $\mathcal{L}_4=\mathcal{L}_2$.\\
\indent The spectral analysis can be directly determined by the behaviour of the linearized operators $\mathcal{L}_i$, $i=1,2,3,4$. Since ${\rm n}(\mathcal{L}_1) = 1,\ {\rm n}(\mathcal{L}_2) = 0$ and
	${\rm z}(\mathcal{L}_1)={\rm z}(\mathcal{L}_2) = 1,$  we have that ${\rm n}(\mathcal{L}) = 2$ and ${\rm z}(\mathcal{L}) = 4$ where
$$
	{\rm Ker}(\mathcal{L}) = \left[ (\varphi', 0,0,0), (0,\varphi', 0,0), (0,0,\varphi,0), (0,0,0,\varphi) \right].
$$

\textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. In this case, we have that $B$ is a free real parameter. Moreover, we also use the similar transformation $S = U M U^{-1}$, where $S$ is given by \eqref{S}. The matrix $M$ and $U$ are then given respectively by
\begin{equation}\label{M-Bfree}
	M = \left( \begin{array}{cccc}
		3(B^2+1)\gamma & 0 & 0 & 0 \\
		0 & (B^2 + 1) \gamma & 0 & 0 \\
		0 & 0 & (B^2 + 1) \gamma & 0 \\
		0 & 0 & 0 & -(B^2 + 1)\gamma
	\end{array}
\right)
\end{equation}
and
\begin{equation}\label{U-Bfree}
	U = \left( \begin{array}{cccc}
		-\frac{1}{B^2+1} & -B & 0 & 0 \\
		-\frac{B}{B^2+1} & 1 & 0 & 0 \\
		0 & 0 & 1 & \frac{B}{B^2+1}  \\
		0 & 0 & B & -\frac{1}{B^2+1}
	\end{array}
\right).
\end{equation}
Thus, operator $\mathcal{L}$ becomes in this case
\begin{equation}\label{L-Bfree}
	\mathcal{L} = U \left( \begin{array}{cccc}
	\mathcal{L}_1 & 0 & 0 & 0 \\
	0 & \mathcal{L}_2 & 0 & 0 \\
	0 & 0 & \mathcal{L}_2 & 0 \\
	0 & 0 & 0 & \mathcal{L}_3
	\end{array}
\right) U^{-1},
\end{equation}
where
\begin{equation}\label{L1L2L3-Bfree}
	\mathcal{L}_1 = -\partial_x^2 + \omega - 3(B^2+1)\gamma \varphi^2, \; \mathcal{L}_2 = -\partial_x^2 + \omega - (B^2+1) \gamma \varphi^2 \text{ and } \mathcal{L}_3 = -\partial_x^2 + \omega + (B^2+1) \gamma \varphi^2.
\end{equation}

As far as we can see, we have that $(B^2+1)\gamma = \kappa_1 + B^2 \gamma$. So, we obtain the same spectral properties concerning the operators $\mathcal{L}_1$ and $\mathcal{L}_2$. In addition, being $\mathcal{L}_3$ a positive operator, we have that ${\rm n}(\mathcal{L}) = 1$ and ${\rm z}(\mathcal{L}) = 3$, where
$$
	{\rm Ker}(\mathcal{L}) = \left[ (\varphi', B\varphi',0,0), (-B\varphi, \varphi,0,0), (0, 0, \varphi, B\varphi) \right].
$$
\begin{remark}\label{remB}
It is worth to be mentioning that the case $B<0$ can be treated in the same way as in the case $B>0$. In fact, since the transformation in $(\ref{dnoidalsol})$ contains the square of $B$, the solution remains the same. When we are analysing the linear operators $\mathcal{L}_i$, $i=1,2,3,4$, we have the same scenario, that is, it appears only the the square of $B$ and this fact does not change the quantity and multiplicity of the non-positive eigenvalues. The matrix transformation $U$ has some terms modified but this fact does not intervene with the analysis of $\mathcal{L}$ since what is important is the analysis of the diagonal operators $M$ in $(\ref{M})$ formed by $\beta_i$ and the diagonal matrix formed by $\mathcal{L}_i$, $i=1,2,3,4$.
\end{remark}
\subsection{Spectral analysis with cnoidal profile}\label{cnoidalprof}

Let $\varphi$ be the solution with cnoidal profile given by Theorem \ref{cnoidalcurve}. For $B>0$ given by \eqref{B} (the case $B<0$ is similar), the transformation $U$ such that $S = U M U^{-1}$ is also given by \eqref{U-Bfixed}. As a consequence, the parameters $\beta_i$, $i=1,2,3,4$ are also given by \eqref{betai} and operators $\mathcal{L}_1$ and $\mathcal{L}_2$ can be expressed as
\begin{equation}\label{eq-L1}
	\mathcal{L}_1 = -\partial_x^2 + \omega - \beta_1 \varphi^2 = -\partial_x^2 + \omega - 3(\kappa_1 + B^2 \gamma) \varphi^2 = -\partial_x^2 + \omega - 3 \phi^2
\end{equation}
and
\begin{equation}
	\mathcal{L}_2 = -\partial_x^2 + \omega - \beta_2 \varphi^2 = -\partial_x^2 + \omega - (\kappa_1 + B^2 \gamma) \varphi^2 = -\partial_x^2 + \omega -  \phi^2,
\end{equation}
where $\phi$ is the solution with cnoidal profile in \eqref{phi}. So, we can use the spectral properties for the operators $\mathcal{L}_1$ and $\mathcal{L}_2$ as obtained in \cite{angulo2007}. Indeed, we have
${\rm n}(\mathcal{L}_1) = 2,\ {\rm n}(\mathcal{L}_2) = 1$ and $z(\mathcal{L}_1)=z(\mathcal{L}_2)=1$. As determined in the case dnoidal solutions, we can use the comparison theorem to obtain spectral properties concerning operators $\mathcal{L}_3$ and $\mathcal{L}_4$ for specific cases.

 Here it is necessary to understand that when the number of negative eigenvalues increases compared with the case of dnoidal solutions, we obtain some problems to obtain the spectral properties for the full operator $\mathcal{L}$ in $(\ref{L-1})$. So, we can study some different cases:

\textbf{Case I:} $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$. By \eqref{ineq-1}, we have that
\begin{equation}
	\mathcal{L}_1 < \mathcal{L}_3 < \mathcal{L}_2 < \mathcal{L}_4.
\end{equation}
Then, using the comparison theorem one has ${\rm n}(\mathcal{L}_3) = 2$ and ${\rm z}(\mathcal{L}_3)=0$. In addition, we have that $\beta_4 < 0$ for all $\gamma \in (\frac{1}{3} ( \kappa_1+\kappa_2-\sqrt{\kappa_1^2-\kappa_1\kappa_2+\kappa_2^2}), \min\{\kappa_1,\kappa_2\})$ and this implies that ${\rm n}(\mathcal{L}_4) = {\rm z}(\mathcal{L}_4) = 0$. Therefore, we have in this case
${\rm n}(\mathcal{L}) = 5 \text{ and } {\rm z}(\mathcal{L}) = 2$ with
$${\rm Ker}(\mathcal{L}) = \left[ (\varphi', B\varphi', 0,0), (0,0,\varphi, B\varphi) \right].$$
\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty)$. By \eqref{ineq-2}, we obtain
\begin{equation}
	\mathcal{L}_1 < \mathcal{L}_2 < \mathcal{L}_3 < \mathcal{L}_4.
\end{equation}
But, using the comparison theorem and the previous knowledge of the quantity of negative eigenvalues for $\mathcal{L}_1$ and $\mathcal{L}_2$, we are no able to determine the same quantity for the operator $\mathcal{L}_3$. In fact, we obtain in this case three different cases:
\begin{equation}\label{tricotomy}
	{\rm n}(\mathcal{L}_3) = 1 \text{ and } {\rm z}(\mathcal{L}_3) = 0, \; \; \; \;	{\rm n}(\mathcal{L}_3) = 0 \text{ and } {\rm z}(\mathcal{L}_3) = 1 \; \; \text{ or } \; \; 	{\rm n}(\mathcal{L}_3) = {\rm z}(\mathcal{L}_3) = 0.
\end{equation}
Thus, the spectral analysis becomes inconclusive and we can not obtain the required spectral stability  for the cnoidal waves.

\textbf{Case III:} $\gamma = 0$. As in the dnoidal case, we have $B = \sqrt{\frac{\kappa_1}{\kappa_2}}$ and consequently, the operator $\mathcal{L}$ is a diagonal operator as in \eqref{L-gamma0}. Thus, from the comparison theorem we obtain
	\begin{equation}\label{Ker-caseIII-cnoidal}
		{\rm n}(\mathcal{L}) = 6 \text{ and } {\rm Ker}(\mathcal{L}) = \left[ (\varphi',0,0,0), (0,\varphi',0,0), (0,0,\varphi,0), (0,0,0,\varphi) \right].
	\end{equation}

\textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. Here $B \in \mathbb{R}$ is a free parameter and we obtain as in case of dnoidal solutions the similar transformation $S = UMU^{-1}$, where $M$ and $U$ are given by \eqref{M-Bfree} and \eqref{U-Bfree}, respectively. Thus, the operator $\mathcal{L}$ can be also expressed by \eqref{L-Bfree} where the operators $\mathcal{L}_i$, $i=1,2,3,4$ are given by \eqref{L1L2L3-Bfree}.

We have to notice that $(B^2 + 1)\gamma = \kappa_1 + B^2 \gamma$, ${\rm n}(\mathcal{L}_1) = 2$ and ${\rm n}(\mathcal{L}_2) = 1$ and $\mathcal{L}_3$ being a positive operator. Summarizing all mentioned results, we get
\begin{equation}\label{Ker-caseIV-cnoidal}
	{\rm n}(\mathcal{L}) = 4 \text{ and } {\rm Ker}(\mathcal{L}) = \left[ (\varphi', B\varphi', 0,0), (-B\varphi, \varphi,0,0), (0,0,\varphi, B \varphi)\right].
\end{equation}
\begin{remark}\label{remB1}
As in the case of dnoidal solutions, it is worth to be mentioning that the case $B<0$ can be treated in the same way as in the case $B>0$ (see Remark $\ref{remB}$ for further details).
\end{remark}

\subsection{Spectral analysis in the odd sector of $H_{per}^2$}\label{subsection-odd} To overcome the excessive quantity of negative eigenvalues, we shall make some restriction in the operator $\mathcal{L}$ defined in $(\ref{L-1})$. In fact, since $\psi\in H^1_{per}$ in $(\ref{cnoidal-2})$ is an odd solution of \eqref{ODE}, we can proceed with a similar analysis for the (restricted) linearized operator $\mathcal{L}$ as determined in the last subsection. Indeed, by the spectral analysis for the cnoidal solution in \cite{angulo2007}, we obtain that the first three eigenvalues and the corresponding eigenfunctions of $\mathcal{L}_1$ are
\begin{align}
\lambda_0 = (1 - 6k^2 - 2a(k))\left(\frac{16 {\rm K}(k)^2}{L^2} \right) \; \; \; & \; \; \; \phi_0(x) = k^2 {\rm sn}^2\left( \frac{4 {\rm K}(k)}{L} x,k\right) - \frac{1}{3} (1+k^2+a(k)), \label{eigf-1} \\
\lambda_1 = -3k^2 \left(\frac{16 {\rm K}(k)^2}{L^2} \right) \; \; \; & \; \; \; \phi_1(x) =  {\rm cn}\left( \frac{4 {\rm K}(k)}{L} x,k\right) {\rm dn}\left( \frac{4 {\rm K}(k)}{L} x,k\right), \\
\lambda_2 = 0 \; \; \; & \; \; \; \phi_2(x) = \partial_x {\rm cn}\left( \frac{4 {\rm K}(k)}{L} x,k\right),
\end{align}
where $a(k):=\sqrt{1-k^2+k^4}$. In addition, the first two eigenvalues and the corresponding eigenfunctions for the linearized operator $\mathcal{L}_2$ are
\begin{align}
	\lambda_0 = \frac{16 {\rm K}(k)^2 (k^2-1)}{L^2} \; \; \; & \; \; \; \phi_0(x) = {\rm dn}\left( \frac{4 {\rm K}(k)}{L} x,k\right), \\
	\lambda_1 = 0 \; \; \; & \; \; \; \phi_1(x) = {\rm cn}\left( \frac{4 {\rm K}(k)}{L} x,k\right). \label{eigf-2}
\end{align}

Considering $\psi$ as in \eqref{cnoidal-2}, we have that the operators
 $$\mathcal{L}_{i,odd}:= \mathcal{L}_i: H^2_{per,odd} \cap L^2_{per,odd} \subset L^2_{per,odd} \rightarrow L^2_{per,odd},$$
 are well defined. Thus, by applying the transformation $f=g(\cdot-L/4)$ in all eigenfunctions given in \eqref{eigf-1}-\eqref{eigf-2}, we conclude
\begin{equation}\label{L1L2-odd}
	{\rm n}(\mathcal{L}_{1,odd}) = 1 \; \; \text{ and } {\rm n}(\mathcal{L}_{2,odd}) = 0.
\end{equation}
In addition,
\begin{equation}\label{L1L2-ker-odd}
	{\rm Ker}(\mathcal{L}_{1,odd}) = \{0\} \; \; \text{ and } \; \; {\rm Ker}(\mathcal{L}_{2,odd}) = [\varphi].
\end{equation}

Therefore, since we have almost the same scenario as determined for the case of dnoidal solutions, we can use the comparison theorem without further problems. We can analyse again the cases in this new perspective:

\textbf{Case I:} $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$. Since in this case, we have $\mathcal{L}_{1,odd} < \mathcal{L}_{3,odd} < \mathcal{L}_{2,odd} < \mathcal{L}_{4,odd}$, we obtain by the comparison theorem and \eqref{L1L2-odd} that ${\rm n}(\mathcal{L}_{3,odd}) = 1$, ${\rm n}(\mathcal{L}_{4,odd}) = 0$ and ${\rm Ker}(\mathcal{L}_{3,odd}) = {\rm Ker}(\mathcal{L}_{4,odd}) = \{0\}$. Therefore,
\begin{equation}\label{Ker-caseI-cnoidal-odd}
	{\rm n}(\mathcal{L}_{odd}) = 2 \text{ and } {\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\varphi,B\varphi)\right].
\end{equation}

\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty)$. Here, we have $\mathcal{L}_{1,odd} < \mathcal{L}_{2,odd} < \mathcal{L}_{3,odd} < \mathcal{L}_{4,odd}$. By the comparison theorem and \eqref{L1L2-odd}, we obtain ${\rm n}(\mathcal{L}_{3,odd}) = {\rm n}(\mathcal{L}_{4,odd}) = 0$ and ${\rm Ker}(\mathcal{L}_{3,odd}) = {\rm Ker}(\mathcal{L}_{4,odd}) = \{0\}$. Thus,
\begin{equation}\label{Ker-caseII-cnoidal-odd}
	{\rm n}(\mathcal{L}_{odd}) = 1 \text{ and } {\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\varphi,B\varphi)\right].
\end{equation}

\textbf{Case III}: $\gamma = 0$. In this case, we have that $B = \sqrt{\frac{\kappa_1}{\kappa_2}}$ and
\begin{equation}
	\mathcal{L}_{odd} = \left(
	\begin{array}{cccc}
		\mathcal{L}_{1,odd} & 0 & 0 & 0 \\
		0 & \mathcal{L}_{1,odd} & 0 & 0 \\
		0 & 0 & \mathcal{L}_{2,odd} & 0 \\
		0 & 0 & 0 & \mathcal{L}_{2,odd}
	\end{array}
\right).
\end{equation}
So, we get
\begin{equation}\label{Ker-caseIII-cnoidal-odd}
	{\rm n}(\mathcal{L}_{odd}) = 2 \text{ and } {\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\varphi,0), (0,0,0,\varphi) \right].
\end{equation}

\textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. In this case, $B \in \mathbb{R}$ is a free parameter and we can use the similar transformation $S = UMU^{-1}$ where $M$ and $U$ given in \eqref{M-Bfree} and \eqref{U-Bfree}, respectively (similar to the case of dnoidal waves). Thus,  since $\mathcal{L}_{3,odd}$ is now positive, the operator $\mathcal{L}$ in \eqref{L-Bfree} restricted in the odd sector of $H_{per}^2$ satisfies
\begin{equation}\label{Ker-caseIV-cnoidal-odd}
	{\rm n}(\mathcal{L}_{odd})=1 \text{ and } {\rm Ker}(\mathcal{L}_{odd}) = \left[ (-B\varphi, \varphi,0,0), (0,0,\varphi, B\varphi) \right].
\end{equation}

\section{Spectral stability}\label{section5}
In this section, we obtain spectral stability results for the periodic multiple solution $\Phi = (\varphi, B\varphi,0,0)$ in three different scenarios:  when $\varphi \in H^1_{per}$ has a dnoidal profile, when $\varphi$ has a cnoidal profile in $H^1_{per}$ and when $\psi=\varphi(\cdot-L/4)$ has a cnoidal profile and restricted to the odd sector of $H^1_{per}$. To do so, we need to obtain the entries of the matrix $V$ in \eqref{V}. In fact, we can obtain a simplified way to obtain the entries of the matrix $V$ using the transformation $(\ref{ULU})$ and the fact that $\Theta_l\in {\rm \,Ker}(\mathcal{L})$. Thus, we have
\begin{equation}
V_{jl} = ( \mathcal{L}^{-1} J \Theta_j, J \Theta_l )_{\mathbb{L}^2_{per}} = ( \tilde{\mathcal{L}}^{-1} U^{-1} J \Theta_j, U^{-1} J \Theta_l )_{\mathbb{L}^2_{per}}.
\end{equation}

We analyse the spectral stability by considering all the four cases for dnoidal profile, cnoidal profile and cnoidal profile in the odd sector as determined in the last section. In the present section, we also consider for the spectral analysis, the four cases for each kind of solution and it is necessary to link the result of spectral stability with the characterization of the respective non-positive spectrum. Before obtaining all possible cases of the matrix $V$, we need to present some important results.

\begin{lemma}\label{positivity}
Let  $\mathcal{A}$ be any self-adjoint operator defined in a Hilbert space $H$ with dense domain $D(\mathcal{A})$ such that $\mathcal{A}$ does not have negative eigenvalues. There exists $\delta > 0$ such that
\begin{equation}
	(\mathcal{A} v, v)_{H} \geq \delta \|v\|_{H}^2
\end{equation}
for all $v \in D(\mathcal{A})$ satisfying $v \in {\rm Ker}(\mathcal{A})^\perp$.
\end{lemma}

\begin{proof}
	Since $H$ is a Hilbert space, we have the decomposition $H = {\rm Ker}(\mathcal{A}) \oplus {\rm Ker}(\mathcal{A})^\perp$. From Theorem 6.17 in \cite[page 178]{Kato}, we have
	\begin{equation}
		\sigma( \mathcal{A} ) = \sigma \left( \mathcal{A} \big{|}_{{\rm Ker}(\mathcal{A})} \right) \cup \left( \mathcal{A} \big{|}_{{\rm Ker}(\mathcal{A})^\perp} \right).
	\end{equation}
	On the other hand, we have that
	\begin{equation}
	\left( \mathcal{A} \big{|}_{{\rm Ker}(\mathcal{A})^\perp} \right) = \sigma(\mathcal{A}) \setminus \{0\},
	\end{equation}
	that is, the spectrum is bounded from below. The arguments in \cite[page 279]{Kato} imply that $\mathcal{A}$ is also bounder from below. Therefore, there exists $\delta > 0$ satisfying
	\begin{equation}
		(\mathcal{A} v, v) \geq \delta \|v\|_{L^2_{per}}^2 \text{ for all } v \in H^2_{per} \cap {\rm Ker}(\mathcal{A})^\perp.
	\end{equation}
\end{proof}

\begin{lemma}\label{dif-norma}
	Let $L>0$ be fixed. Consider the periodic wave $\varphi \in H^1_{per}$ with the dnoidal profile given by Theorem \ref{dnoidalcurve}. We have $\frac{d}{d\omega} \|\varphi\|_{L^2_{per}}^2 > 0$.
\end{lemma}
\begin{proof}
	Let $L>0$ be fixed and consider $\varphi$ as the dnoidal profile given by Theorem $\ref{dnoidalcurve}$. Then, by \cite[Formula 314.02]{byrd} we get
	\begin{align*}
		\int_{0}^{L} \varphi(x)^2 dx & = \frac{ 8{\rm K}(k)^2}{L^2(\kappa_1 + \gamma B^2)} \int_{0}^L {\rm dn}^2 \left( \frac{2 {\rm K}(k)}{L} x,k \right) dx \\
		& = \frac{ 8 {\rm K}(k)}{L (\kappa_1 + \gamma B^2)} \int_{0}^{{\rm K}(k)} {\rm dn}^2 (u,k) du \\
		& = \frac{8}{L(\kappa_1 + \gamma B^2)} {\rm E}(k) {\rm K}(k).
	\end{align*}
Thus, we obtain
$$
	\frac{d}{d\omega} \|\varphi\|_{L^2_{per}}^2 = \frac{d}{d\omega} \left( \frac{8}{L(\kappa_1 + \gamma B^2)} {\rm E}(k) {\rm K}(k) \right) = \frac{8}{L(\kappa_1 + \gamma B^2)} \frac{d}{dk} \left( {\rm E}(k) {\rm K}(k) \right) \left( \frac{d\omega}{d k} \right)^{-1} > 0
$$
for all $k \in (0,1)$.
\end{proof}

\begin{lemma}\label{lemmaL1-cn}
	Let $L>0$ be fixed. Consider the periodic wave $\varphi \in H^1_{per}$ with the cnoidal profile given by Theorem \ref{cnoidalcurve}. Thus, we have $\frac{d}{d \omega} \|\varphi\|_{L^2_{per}}^2 > 0$.
\end{lemma}
\begin{proof}
	Let $L>0$ be fixed and consider $\varphi$ as the cnoidal profile given by Theorem
	$\ref{cnoidalcurve}$. By \cite[Formula 312.02]{byrd}, we have
	\begin{align*}
		\int_{0}^{L} \varphi(x)^2 dx & = \frac{ 32 k^2 {\rm K}(k)^2}{L^2(\kappa_1 + \gamma B^2)} \int_{0}^L {\rm cn}^2 \left( \frac{4 {\rm K}(k)}{L} x,k \right) dx \\
		& = \frac{ 32 k^2 {\rm K}(k)}{L (\kappa_1 + \gamma B^2)} \int_{0}^{{\rm K}(k)} {\rm cn}^2 (u,k) du \\
		& = \frac{32}{L(\kappa_1 + \gamma B^2)} \left( {\rm K}(k) \left( {\rm E}(k) - (1-k^2) {\rm K}(k) \right) \right).
	\end{align*}
	Thus, we obtain that
	$$
		\frac{d}{d\omega} \|\varphi\|_{L^2_{per}}^2 = \frac{32}{L (\kappa_1 + \gamma B^2)} \frac{d}{dk} \left[ {\rm K}(k) \left( {\rm E}(k) - (1-k^2) {\rm K}(k) \right) \right] \left( \frac{ d\omega}{dk} \right)^{-1} > 0
	$$
	for all $k \in (0,1)$.
\end{proof}

\begin{remark}
	The result obtained in Lemma \ref{lemmaL1-cn} can be applied in the case of odd periodic waves $\psi$ as in Remark $\ref{cnoidal-odd}$.
\end{remark}

\subsection{Spectral stability for the multiple periodic wave solution with dnoidal profile}
In what follows, we consider the dnoidal wave solution $\varphi$ determined by Theorem \ref{dnoidalcurve}. Since we have separated the spectral analysis into four cases, we need to consider the same four cases in order to establish the spectral stability for the multiple solution $\Phi=(\varphi,B\varphi,0,0)$.

\textbf{Case I:} $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$. Since $z(\mathcal{L})=2$ with $\Theta_1=(\varphi',B\varphi',0,0)$ and $\Theta_2=(0,0,\varphi,B\varphi)$, we obtain that the matrix $V$ is $2\times 2$ and given by
\begin{equation}\label{V-1}
\begin{array}{lllll}
V &=& \left( \begin{array}{cc}
		(\tilde{\mathcal{L}}^{-1} U^{-1} J \Theta_1, U^{-1} J \Theta_1)_{L_{per}^2} & (\tilde{\mathcal{L}}^{-1} U^{-1} J \Theta_1, U^{-1} J \Theta_2)_{L_{per}^2} \\
		(\tilde{\mathcal{L}}^{-1} U^{-1} J \Theta_2, U^{-1} J \Theta_1)_{L_{per}^2} & (\tilde{\mathcal{L}}^{-1} U^{-1} J \Theta_2, U^{-1} J \Theta_2)_{L_{per}^2}
	\end{array}
\right).\\\\
	&=&\left( \begin{array}{cc}
	(2\gamma - \kappa_1 - \kappa_2)^2 ( \mathcal{L}_2^{-1} \varphi', \varphi' )_{L^2_{per}} & 0 \\
	0 & (\gamma - \kappa_2)^{-2} ( \mathcal{L}_1^{-1} \varphi,\varphi )_{L^2_{per}}
\end{array}
	\right),
\end{array}
\end{equation}
where we are using the similar transformation $\mathcal{L}=U\tilde{\mathcal{L}}U^{-1}$ to obtain a more convenient expression for the entries of the matrix $V$.\\
\indent Since $\varphi' \in ({\rm Ker}(\mathcal{L}_2))^{\perp} = {\rm Range}(\mathcal{L}_2)$, there exists $\xi \in D(\mathcal{L}_2)$ such that $\mathcal{L}_2 \xi = \varphi'$. Since $\mathcal{L}_2$ does not have negative eigenvalues, we obtain that $\xi$ satisfy the conditions of Lemma \ref{positivity}, so that $( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} > 0$. On the other hand, by Theorem $\ref{dnoidalcurve}$, we can derive the equation \eqref{ODE} with respect to $\omega$ to obtain that $\mathcal{L}_1 \left( \frac{d}{d\omega} \right) = -\varphi$. Thus, by Lemma \ref{dif-norma} we get
\begin{equation}
	( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}} = - \left( \frac{d}{d\omega} \varphi, \varphi \right)_{L^2_{per}} = - \frac{1}{2} \frac{d}{d\omega} \|\varphi\|_{L^2_{per}}^2 < 0.
\end{equation}
Thus, we have ${\rm n}(V) = 1$. Since ${\rm n}(\mathcal{L}) = 2$, we conclude that the multiple solution $\Phi = (\varphi, B\varphi,0,0)$ is spectrally unstable.

\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty)$. The kernel of $\mathcal{L}$ in this case has the same elements as in the last case, so that the matrix $V$ is the same as in $(\ref{V-1})$. Since we have ${\rm n}(\mathcal{L})={\rm n}(V) = 1$, we conclude that the multiple solution $\Phi = (\varphi, B \varphi,0,0)$ is spectrally stable.

\textbf{Case III:} $\gamma = 0$. Since $z(\mathcal{L})=4$ with $\Theta_1=(\varphi', 0,0,0)$, $\Theta_2=(0,\varphi', 0,0)$, $\Theta_3=(0,0,\varphi,0)$, and $\Theta_4=(0,0,0,\varphi)$, we obtain that the matrix $V$ is $4\times 4$ and given by
\begin{equation}
	V = \left( \begin{array}{cccc}
		( \mathcal{L}_2^{-1} \varphi', \varphi' )_{L^2_{per}} & 0 & 0 & 0 \\
		0 & ( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} & 0 & 0 \\
		0 & 0 & ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}} & 0 \\
		0 & 0 & 0 & ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}}
	\end{array}
\right).
\end{equation}
Doing the same calculations as we have already performed in the first case, we obtain ${\rm n}(V) = 2$. Since ${\rm n}(\mathcal{L}) = 2$, we have that the difference ${\rm n}(\mathcal{L}) - {\rm n}(V)$ is zero and the periodic multiple solution is spectrally stable.\\
\indent  \textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. Now, we have $z(\mathcal{L})=3$ with
$\Theta_1=(\varphi', B\varphi',0,0)$, $\Theta_2=(-B\varphi, \varphi,0,0)$ and
$\Theta_3=(0, 0, \varphi, B\varphi)$. We obtain that the matrix $V$ is $3\times 3$ and given by
\begin{equation}
	V = \left( \begin{array}{ccc}
		( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} & 0 & 0 \\
		0 & (B^2 + 1)^2 ( \mathcal{L}_3^{-1} \varphi,\varphi )_{L^2_{per}} & 0 \\
		0 & 0 & (B^2+1)^2 ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}}
	\end{array}
\right).
\end{equation}
We see in this case that $\mathcal{L}_3$ is positive, so that $( \mathcal{L}_3^{-1} \varphi,\varphi )_{L^2_{per}}>0$. By similar arguments as determined in the first case, we then obtain ${\rm n}(V) = 1$. Since ${\rm n}(\mathcal{L}) = 1$, we get that the multiple solution $\Phi = (\varphi, B \varphi,0,0)$ is spectrally stable.

Summarizing the above, we have proved Theorem \ref{teo-1}.

\subsection{Spectral stability for the multiple periodic wave solution with cnoidal profile}
Here, we consider the periodic multiple wave solution $\Phi = (\varphi, B \varphi,0,0)$ where $\varphi$ has the cnoidal profile and we determine spectral stability results. After that, we consider the translation solution $\psi=\varphi(\cdot-L/4)$ to study the spectral stability in the odd sector of $H_{per}^1$ for the case where the operator $\mathcal{L}$ has too many negative eigenvalues.

We have to notice that in the case of dnoidal profile, we have used Lemma \ref{positivity} to obtain that $( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} > 0$. To obtain this fact, we need to use the crucial fact that ${\rm n}(\mathcal{L}_2) = 0$. However, when we are considering the cnoidal profile, we obtain ${\rm n}(\mathcal{L}_2) = 1$ and this property does not allow us to use directly Lemma \ref{positivity} to evaluate the positiveness of $( \mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}}$. This difficulty can be avoided by the following lemma:

\begin{lemma}\label{LemmaL2-cn}
	Let $L>0$ be fixed and consider $\varphi$ the periodic wave with cnoidal profile given by Theorem $\ref{cnoidalcurve}$. We have that
	\begin{equation}
		( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} > 0.
	\end{equation}
\end{lemma}
\begin{proof}
By \eqref{eq-L1}, we can rewrite $\mathcal{L}_2 = -\partial_x^2 + \omega - \phi^2$ where $\phi$ is the cnoidal wave solution in \eqref{phi}. Since $\varphi$ in $(\ref{cnoidalsol})$ is a multiple of $\phi$, there exists $\chi \in D(\mathcal{L}_2)$ such that $\mathcal{L}_2 \chi = \phi'$ and to calculate the value of $( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}}$, it suffices to evaluate the quantity $(\chi,\phi')_{L^2_{per}}$. To this end, we use a similar approach as in \cite[Section 3]{NataliCardosoAmaral} (see also \cite{BittencourtLoreno} and \cite{NataliMoraesLorenoPastor}).

We can start by noticing that $\lambda = 0$ is a simple eigenvalue with associated eigenfunction $\phi$. Thus, there exists a non-periodic function $y$ satisfying the Hill equation
\begin{equation}\label{y-hillequation}
	- y'' + \omega y + \phi^2 y = 0,
\end{equation}
where $y \in C_b^\infty([0,L])$ and $\{\varphi, y\}$ is the fundamental set of solutions for the equation $(\ref{y-hillequation})$. Since $\phi$ is even, we have that $y$ is odd and it satisfies the following
\begin{equation}\label{y-ivp}
	\left\{ \begin{array}{l}
		-y'' + \omega y - \phi^2 y = 0 \\
		y(0) = 0 \\
		y'(0) = \frac{1}{\phi(0)}.
		\end{array}
	\right.
\end{equation}
\indent Next, we see that $\chi \in D(\mathcal{L}_2)$ satisfies the equation $\mathcal{L}_2 \chi = \phi'$, so that
\begin{equation}\label{chi-hill}
	-\chi'' + \omega \chi - \phi^2 \chi = \phi'.
\end{equation}
Multiplying \eqref{chi-hill} by ${y}$, integrating over $[0,L]$ and using integration by parts, we obtain
\begin{equation}
	\chi'(0) = - \frac{ \int_{0}^L \phi'(x) {y(x)} dx}{{y}(L)},
	\end{equation}
where we are using the fact that $y$ is not periodic, so that $y(L)\neq0$. The fact that $\chi$ is an odd function gives us the following IVP:
\begin{equation}\label{chi-pvi}
	\left\{ \begin{array}{l}
		-\chi'' + \omega \chi - \phi^2 \chi = \phi' \\
		\chi(0) = 0 \\
		\chi'(0) = -\frac{1}{{y}(L)} \int_{0}^L \phi'(x) {y(x)} dx.
	\end{array}
\right.
\end{equation}
Problem \eqref{chi-pvi} is suitable to perform some numeric calculations. In fact, we can deduce that
\begin{equation}
(\chi, \phi')_{L^2_{per}} = \frac{\eta(k)}{L}
\end{equation}
where $\eta$ is a positive constant depending only on $k \in (\frac{1}{\sqrt{2}},1)$. We conclude that $(\chi, \phi')_{L^2_{per}} > 0$ for all $k \in (\frac{1}{\sqrt{2}},1)$. Thus, we have concluded that
\begin{equation}
	(\mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}} = \frac{1}{(\kappa_1 + \gamma B^2)} (\mathcal{L}_2^{-1} \phi',\phi')_{L^2_{per}} = (\chi, \phi')_{L^2_{per}} > 0.
\end{equation}
\end{proof}

\begin{remark} The illustration of the behaviour of $\eta(k)$ for $k \in (\frac{1}{\sqrt{2}},1)$ appears in Figure \ref{Figura}.
	\end{remark}

\begin{figure}[!h]
	\includegraphics[width=5.5cm,height=6cm]{gnk-eps-converted-to.pdf}
	\caption{Behaviour of the quantity $\eta(k)$ with $k \in (\frac{1}{\sqrt{2}}, 1)$.}
	\label{Figura}
\end{figure}

With the results above we are able to evaluate the spectral stability of the multiple periodic wave $\Phi = (\varphi, B\varphi,0,0)$ with cnoidal profile. The analysis is quite similar as determined in the last subsection, and so we only give the main steps.

\textbf{Case I:} $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$. Since $z(\mathcal{L})=2$, we have that $V$ is a $2\times 2$ matrix and given by
\begin{equation}
	V = \left( \begin{array}{cc}
	(2\gamma - \kappa_1 - \kappa_2)^2 ( \mathcal{L}_2^{-1} \varphi', \varphi' )_{L^2_{per}} & 0 \\
		0 & (\gamma - \kappa_2)^{-2} ( \mathcal{L}_1^{-1} \varphi,\varphi )_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
Using the results in Lemmas \ref{lemmaL1-cn} and \ref{LemmaL2-cn}, we obtain respectively that $(\mathcal{L}_1^{-1}\varphi,\varphi) = - \frac{d}{d\omega} \|\varphi\|_{L^2_{per}}^2 < 0$ and $(\mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}} > 0$, so that ${\rm n}(V) = 1$. On the other hand, since we have  ${\rm n}(\mathcal{L}) = 5$, we see that the difference ${\rm n}(\mathcal{L}) - {\rm n}(V) = 4$ is an even number. Therefore, we can not conclude that the multiple solution $\Phi = (\varphi,B\varphi,0,0)$ is spectrally stable or not.

\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty)$. In this specific case, it is possible to see that we can not decide the exact quantity of negative eigenvalues (see $(\ref{tricotomy})$) in order to apply the results in \cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII}. Therefore, we can not conclude a precise result of spectral stability for the multiple solution $\Phi = (\varphi,B\varphi,0,0)$.

\textbf{Case III:} $\gamma = 0$. Since $z(\mathcal{L})=4$, we can proceed as in the third case of the last subsection to obtain
\begin{equation}
	V = \left( \begin{array}{cccc}
		( \mathcal{L}_2^{-1} \varphi', \varphi' )_{L^2_{per}} & 0 & 0 & 0 \\
		0 & ( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} & 0 & 0 \\
		0 & 0 & ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}} & 0 \\
		0 & 0 & 0 & ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
Since by Lemmas \ref{lemmaL1-cn} and \ref{LemmaL2-cn} we have ${\rm n}(V) = 2$, the fact that ${\rm n}(\mathcal{L}) = 6$ gives us that the spectral stability result is also inconclusive.

\textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. Since in this case $z(\mathcal{L})=3$, we can proceed as in the fourth case of the last subsection with cnoidal profile instead of dnoidal profile to obtain $V$ as
\begin{equation}
	V = \left( \begin{array}{ccc}
		( \mathcal{L}_2^{-1} \varphi',\varphi' )_{L^2_{per}} & 0 & 0 \\
		0 & (B^2 + 1)^2 ( \mathcal{L}_3^{-1} \varphi,\varphi )_{L^2_{per}} & 0 \\
		0 & 0 & (B^2+1)^2 ( \mathcal{L}_1^{-1} \varphi, \varphi )_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
Using Lemmas \ref{positivity}, \ref{lemmaL1-cn} and \ref{LemmaL2-cn}, we have ${\rm n}(V) = 1$ and since ${\rm n}(\mathcal{L}) = 4$, we obtain that the difference ${\rm n}(\mathcal{L}) - {\rm n}(V)$ is an odd number. Consequently, the multiple solution $\Phi = (\varphi, B \varphi,0,0)$ is spectrally unstable.\\
\indent Summarizing the results above, we can conclude the statement of Theorem \ref{teo-2}.

\subsubsection{Spectral stability of cnoidal waves in the odd sector} Here we are going to answer some unclear points left behind concerning the spectral stability of periodic multiple solutions $\Phi$ with cnoidal profile. The arguments will be the same, but we need to pay attention with the spectral analysis established in the odd sector of $\mathbb{L}_{per}^2$.

From now on, let us consider the periodic solution $\psi \in H^1_{per,odd}$ given by \eqref{cnoidal-2}.

\textbf{Case I:} $\gamma \in (0,\min\{\kappa_1,\kappa_2\})$. Since in this case $z(\mathcal{L}_{odd})=1$, we have that $V$ is given in a simple way as
\begin{equation}\label{V-2}
	V =  ( \mathcal{L}_1^{-1} \psi,\psi )_{L^2_{per}}.
\end{equation}

By Lemma \ref{lemmaL1-cn} we have ${\rm n}(V) = 1$ and since ${\rm n}(\mathcal{L}_{odd}) = 2$, we obtain that the difference ${\rm n}(\mathcal{L}_{odd}) - {\rm n}(V) = 1$ is an odd number. Therefore, we conclude that the multiple solution $\Psi = (\psi,B\psi,0,0)$ is spectrally unstable.

\textbf{Case II:} $\gamma \in (\max\{\kappa_1,\kappa_2\},+\infty)$. In this case, the matrix $V$ is also one-dimensional and it is given by same expression in $(\ref{V-2})$. Since we also have ${\rm n}(V) = 1$ and in this case ${\rm n}(\mathcal{L}_{odd}) = 1$, we deduce that the multiple solution $\Psi = (\psi,B\psi,0,0)$  is spectrally stable.

\textbf{Case III:} $\gamma = 0$. We have $z(\mathcal{L}_{odd})=2$ and the matrix $V$ is now given by
\begin{equation}
	V = \left( \begin{array}{cc}
	 ( \mathcal{L}_1^{-1} \psi, \psi )_{L^2_{per}} & 0 \\
	 0 & ( \mathcal{L}_1^{-1} \psi, \psi )_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
By Lemma $\ref{lemmaL1-cn}$ and since $( \mathcal{L}_1^{-1} \psi, \psi )_{L^2_{per}}=- \frac{1}{2} \frac{d}{d\omega} \|\psi\|_{L^2_{per}}^2=-\frac{1}{2}\frac{d}{d\omega}||\varphi||_{L_{per}^2}^2 < 0$, we conclude ${\rm n}(V) = 2$. On the other hand, the fact ${\rm n}(\mathcal{L}_{odd}) = 2$ gives us that the difference ${\rm n}(\mathcal{L}_{odd}) -{\rm n}(V) =0$ and the multiple solution $\Psi = (\psi,B\psi,0,0)$  is then spectrally stable.

\textbf{Case IV:} $\gamma = \kappa_1 = \kappa_2$. Again we have $z(\mathcal{L}_{odd})=2$ and $V$ is given by
\begin{equation}
	V = \left( \begin{array}{cc}
		 (B^2 + 1)^2 ( \mathcal{L}_3^{-1} \psi,\psi )_{L^2_{per}} & 0 \\
		 0 & (B^2+1)^2 ( \mathcal{L}_1^{-1} \psi, \psi )_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
By Lemma \ref{positivity} and since $\mathcal{L}_3$ is positive, it follows that $( \mathcal{L}_3^{-1} \psi,\psi )_{L^2_{per}}>0$. On the other hand, by Lemma \ref{lemmaL1-cn} we obtain $( \mathcal{L}_1^{-1} \psi, \psi )_{L^2_{per}}=-\frac{1}{2}\frac{d}{d\omega}||\psi||_{L_{per}^2}^2<0$ and so, ${\rm n}(V) = 1$. Since ${\rm n}(\mathcal{L}_{odd}) = 1$, we deduce that the multiple solution $\Psi = (\psi,B\psi,0,0)$  is also spectrally stable.

Summarizing the above, we conclude the result in Theorem \ref{teo-odd}.

\section{Spectral stability for the semi-trivial periodic solution with cnoidal profile}\label{section6}

An important aspect concerning the solution $(u,v)$ of the NLS system \eqref{NLS-system} is the existence of semi-trivial solutions
\begin{equation}\label{semitrivial-sol}
	(u(x,t),v(x,t)) = (e^{i\omega t} \varphi(x), 0)
\end{equation}
of \eqref{NLS-system}. As we have already mentioned in the introduction, Hakkaev in \cite{Hakkaev} studied the spectral stability for the semi-trivial wave solution \eqref{semitrivial-sol} where $\varphi$ has a dnoidal profile. Our intention is to prove the spectral stability when $\varphi$ as a cnoidal profile.\\
\indent We follow as done in Section 2. First, we substitute \eqref{semitrivial-sol} into \eqref{NLS-system} to obtain the ODE
\begin{equation}\label{ode-semitrivial}
	-\varphi'' + \omega \varphi - \kappa_1 \varphi^3 = 0.
\end{equation}

\indent A similar result as determined in Theorem $\ref{cnoidalcurve}$ is now presented.

\begin{theorem}\label{prop-varphi-semitrivial}
	Let $L > 0$ be fixed. The equation \eqref{ode-semitrivial} has an $L$-periodic solution with cnoidal profile of the form
	\begin{equation}\label{phi-semitrivial}
		\varphi_\omega(x) = \frac{ \sqrt{2 \omega} k}{(2k^2 -1 )} \frac{1}{\sqrt{\kappa_1}} {\rm cn} \left( \frac{4 {\rm K}(k)}{L} x, k \right),
	\end{equation}
where $\omega > 0$ is given by \eqref{omega-cnoidal} depends smoothly on $k \in (\frac{1}{\sqrt{2}}, 1)$ and $L>0$. In addition, the family
\begin{equation}
	\omega \in (0,+\infty) \longmapsto \varphi = \varphi_\omega \in H^2_{per}([0,L])
\end{equation}
of $L$-periodic solutions of \eqref{ode-semitrivial} depends smoothly on $\omega \in (0,+\infty)$.
\end{theorem}

Let us consider the standard spectral problem to be studied as $J \mathcal{L} u = \lambda u$. In  this case, $J$ is given by $(\ref{J})$ and $\mathcal{L}$ is defined as
\begin{equation}\label{L-semitrivial}
	\mathcal{L} = \left( \begin{array}{cccc}
		\mathcal{L}_1 & 0 & 0 & 0 \\
		0 & \mathcal{L}_3 & 0 & 0 \\
		0 & 0 & \mathcal{L}_2 & 0 \\
		0 & 0 & 0 & \mathcal{L}_4
	\end{array}
\right).
\end{equation}
Since $\mathcal{L}$ is a diagonal operator, we only need to analyze the spectral properties of the operators $\mathcal{L}_i$, $i=1,2,3,4$, where
\begin{align}
\mathcal{L}_1 & = -\partial_x^2 + \omega - 3\kappa_1 \varphi^2, \label{L1-semitrivial} \\
\mathcal{L}_2 & = -\partial_x^2 + \omega - \kappa_1 \varphi^2, \\
\mathcal{L}_3 & = -\partial_x^2 + \omega - \gamma \varphi^2, \\
\mathcal{L}_4 & = -\partial_x^2 + \omega + \gamma \varphi^2. \label{L4-semitrivial}
\end{align}

As determined in Subsection $\ref{cnoidalprof}$, we have that
\begin{equation}\label{nL-semitrivial}
	{\rm n}(\mathcal{L}_1) = 2, \; \; {\rm n}(\mathcal{L}_2) = 1 \; \; \text{ and } \; \; {\rm n}(\mathcal{L}_4) = 0.
\end{equation}
In addition,
\begin{equation}\label{kerL-semitrivial}
	{\rm Ker}(\mathcal{L}_1) = [\varphi'], \; \; {\rm Ker}(\mathcal{L}_2) = [\varphi] \; \; \text{ and } \; \; {\rm Ker}(\mathcal{L}_4) = \{0\}.
\end{equation}
Thus, the spectral analysis of the operator $\mathcal{L}$ changes according to the spectral analysis of the operator $\mathcal{L}_3$. Here, we consider the same cases for $\gamma$ as determined in \cite{Hakkaev}.

\textbf{Case I:} $\gamma \in (0,\kappa_1)$. Here, we have that
\begin{equation}
	\mathcal{L}_1 < \mathcal{L}_2 < \mathcal{L}_3 < \mathcal{L}_4.
\end{equation}
Using the comparison theorem, we can not obtain the exact values of ${\rm n}(\mathcal{L}_3)$ and ${\rm z}(\mathcal{L}_3)$ since we obtain three different different scenarios:
\begin{equation}
	{\rm n}(\mathcal{L}_3) = 1 \text{ and } {\rm z}(\mathcal{L}_3) = 0, \; \; \; \;	{\rm n}(\mathcal{L}_3) = 0 \text{ and } {\rm z}(\mathcal{L}_3) = 1 \; \; \text{ or } \; \; 	{\rm n}(\mathcal{L}_3) = {\rm z}(\mathcal{L}_3) = 0.
\end{equation}
Thus, the spectral analysis becomes inconclusive.

\textbf{Case II:} $\gamma = k_1$. In this case, it follows that $\mathcal{L}_2 = \mathcal{L}_3$ and we obtain
\begin{equation}
	{\rm n}(\mathcal{L}) = 4 \text{ and } {\rm Ker}(\mathcal{L}) = \left[ (\varphi',0,0,0), (0,\varphi,0,0), (0,0,\varphi,0) \right].
\end{equation}
To determine the spectral stability result, we need to obtain ${\rm n}(V)$. Since $z(\mathcal{L})=3$, one has
\begin{equation}
	V = \left( \begin{array}{ccc}
		( \mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}} & 0 & 0 \\
		0 & (\mathcal{L}_1^{-1} \varphi,\varphi)_{L^2_{per}} & 0 \\
		0 & 0 & (\mathcal{L}_4^{-1} \varphi, \varphi)_{L^2_{per}}
	\end{array}
\right).
\end{equation}
Since $\mathcal{L}_4$ is a positive operator, we obtain by Lemma $\ref{positivity}$ that $(\mathcal{L}_4^{-1}\varphi,\varphi)_{L^2_{per}} > 0$. On the other hand, using Lemma \ref{LemmaL2-cn}, we also obtain $(\mathcal{L}_2^{-1}\varphi,\varphi)_{L^2_{per}} > 0$ and by Lemma \ref{lemmaL1-cn}, we deduce $(\mathcal{L}_{1}^{-1} \varphi,\varphi) < 0$. Gathering all, we conclude that ${\rm n}(V) = 1$ and the difference ${\rm n}(\mathcal{L}) - {\rm n}(V) = 4 - 1 = 3$ is an odd number. The periodic semi-trivial solution $\Phi=(\varphi,0,0,0)$ is then spectrally unstable concluding the desired result in Theorem \ref{teosemi}.

\textbf{Case III:} $\gamma \in (\kappa_1,3\kappa_1)$. In this case, we obtain the following inequality
\begin{equation}
	\mathcal{L}_1 < \mathcal{L}_3 < \mathcal{L}_2 < \mathcal{L}_4.
\end{equation}
From \eqref{nL-semitrivial}, \eqref{kerL-semitrivial} and using the comparison theorem, we have that ${\rm n}(\mathcal{L}_3) = 2$ and ${\rm Ker}(\mathcal{L}_3) = \{0\}$. Thus,
\begin{equation}
	{\rm n}(\mathcal{L}) = 5 \text{ and } {\rm Ker}(\mathcal{L}) = \left[ (\varphi',0,0,0), (0,0,\varphi,0) \right].
\end{equation}
\indent To evaluate the spectral stability, we need consider the matrix $V$ given in this case by
\begin{equation}
	V = \left( \begin{array}{cc}
		(\mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}} & 0 \\
		0 & (\mathcal{L}_1^{-1}\varphi,\varphi)_{L^2_{per}}
		\end{array}
	\right).
\end{equation}
Applying again Lemmas \ref{lemmaL1-cn} and \ref{LemmaL2-cn}, we have that ${\rm n}(V) = 1$. Since the difference ${\rm n}(\mathcal{L}) - {\rm n}(V) = 4$ is even, we can not conclude the spectral stability.

\textbf{Case IV:} $\gamma = 3\kappa_1$. Here, we have ${\mathcal{L}_1} = \mathcal{L}_3$ and
\begin{equation}
	{\rm n}(\mathcal{L}) = 5 \text{ and } {\rm Ker}(\mathcal{L}) = \left[ (\varphi', 0,0,0), (0,\varphi',0,0),(0,0,\varphi,0)\right].
\end{equation}
Since $z(\mathcal{L})=3$, the matrix $V$ becomes in this case
\begin{equation}
	V = \left( \begin{array}{ccc}
		( \mathcal{L}_2^{-1} \varphi',\varphi')_{L^2_{per}} & 0 & 0 \\
		0 & (\mathcal{L}_1^{-1} \varphi,\varphi)_{L^2_{per}} & 0 \\
		0 & 0 & (\mathcal{L}_4^{-1} \varphi', \varphi')_{L^2_{per}}
	\end{array}
	\right).
\end{equation}
Again, we can apply Lemma \ref{positivity}, Lemma \ref{lemmaL1-cn} and Lemma \ref{LemmaL2-cn} to obtain ${\rm n}(V) = 1$. Since the difference ${\rm n}(\mathcal{L}) -n(V)=4$ is even, we can not decide about the spectral stability.

\subsubsection{Spectral stability of cnoidal waves in the odd sector} We consider the spectral stability for the semi-trivial wave solution $\Psi = (\psi,0,0,0)$ where $\psi$ in defined by
$(\ref{cnoidal-odd})$. The reason for that is to fill the gaps left by our analysis performed in the case of the cnoidal profile $\Phi=(\varphi,0,0,0)$.

So, being $\psi \in H^1_{per,odd}$, we can consider the linearized operator $\mathcal{L}$, $i=1,2,3,4$ restricted to the odd sector of $\mathbb{L}_{per}^2$ as
\begin{equation}
	\mathcal{L}_{odd} : \mathbb{H}^2_{per,odd} \cap \mathbb{L}^2_{per,odd} \subset \mathbb{L}^2_{per,odd} \rightarrow \mathbb{L}^2_{per,odd}
\end{equation}
where $\mathcal{L}$ is given in \eqref{L-semitrivial} and also defined in their respective odd sectors of $L_{per}^2$. Thus, as determined in Subsection \ref{subsection-odd}, we have that
\begin{equation}\label{nL-semitrivial-odd}
	{\rm n}(\mathcal{L}_{1,odd}) = 1, \; \; {\rm n}(\mathcal{L}_{2,odd}) = 0 \; \; \text{ and } \; \; {\rm n}(\mathcal{L}_{4,odd}) = 0.
\end{equation}
In addition, we get
\begin{equation}\label{kerL-semitrivial-odd}
{\rm Ker}(\mathcal{L}_{2,odd}) = [\psi] \; \; \text{ and } \; \; {\rm Ker}(\mathcal{L}_{1,odd}) = {\rm Ker}(\mathcal{L}_{4,odd}) = \{0\}.
\end{equation}

By \eqref{nL-semitrivial-odd}, we have to notice that the number of eigenvalues of the operator $\mathcal{L}_{odd}$ decreased comparing with the full operator $\mathcal{L}$. This fact is useful to use the comparison theorem. We shall describe better our intentions in the four cases ahead:\\
\indent \textbf{Case I:} $\gamma \in (0,\kappa_1)$. In this case, we have the inequality $\mathcal{L}_{1,odd} < \mathcal{L}_{2,odd} < \mathcal{L}_{3,odd} < \mathcal{L}_{4,odd}$. Using the comparison theorem and the informations in \eqref{nL-semitrivial-odd} and \eqref{kerL-semitrivial-odd}, we have that ${\rm n}(\mathcal{L}_{3,odd}) = 0$ and ${\rm Ker}(\mathcal{L}_{3,odd}) = \{0\}$. Thus,
\begin{equation}
	{\rm n}(\mathcal{L}_{odd}) = 1 \; \text{ and } \; {\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\psi,0) \right].
\end{equation}
To evaluate the spectral stability, we need to see that $z(\mathcal{L}_{odd})=1$, so that the matrix $V$ is given by
\begin{equation}
	V = (\mathcal{L}_{1}^{-1} \, \psi,\psi)_{L^2_{per}}.
\end{equation}
By Lemma \ref{lemmaL1-cn}, we have that ${\rm n}(V) = 1$ and the difference ${\rm n}(\mathcal{L}_{odd}) - {\rm n}(V)=0$. Thus, we conclude that the semi-trivial wave solution $\Psi = (\psi,0,0,0)$  is spectrally stable.\\
\indent \textbf{Case II:} $\gamma = \kappa_1$. Here, we have $\mathcal{L}_{2,odd} = \mathcal{L}_{3,odd}$. Using \eqref{nL-semitrivial-odd} and \eqref{kerL-semitrivial-odd}, we get
\begin{equation}
	{\rm n}(\mathcal{L}_{odd}) = 1 \text{ and } {\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,\psi,0,0), (0,0,\psi,0)  \right].
\end{equation}
Since $z(\mathcal{L}_{odd})=2$, the matrix $V$ is given by
\begin{equation}
	V = \left( \begin{array}{cc}
		(\mathcal{L}_{1}^{-1} \, \psi,\psi )_{L^2_{per}} & 0 \\
		0 & (\mathcal{L}_{4}^{-1} \, \psi,\psi)_{L^2_{per}}
	\end{array}
\right).
\end{equation}
Since $\mathcal{L}_{4,odd}$ is positive, we obtain by lemma \ref{positivity} that $(\mathcal{L}_{4}^{-1} \, \psi,\psi)_{L^2_{per}}>0$. In addition, by Lemma \ref{lemmaL1-cn}, we see
$(\mathcal{L}_{1}^{-1} \, \psi,\psi )_{L^2_{per}}<0$, so that ${\rm n}(V) = 1$. Thus, the semi-trivial periodic wave $\Psi = (\psi,0,0,0)$  is spectrally stable.

\textbf{Case III:} $\gamma \in (\kappa_1, 3 \kappa_1)$. Here, we have the inequality $\mathcal{L}_{1,odd} < \mathcal{L}_{3,odd} < \mathcal{L}_{2,odd} < \mathcal{L}_{4,odd}$. Then, by the informations in \eqref{nL-semitrivial-odd}, \eqref{kerL-semitrivial-odd} and the comparison theorem, we obtain ${\rm n}(\mathcal{L}_{3,odd}) = 1$, ${\rm Ker}(\mathcal{L}_{3,odd}) = \{0\}$, ${\rm n}(\mathcal{L}_{odd}) = 2$ and ${\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\psi,0) \right]$. Again, since $z(\mathcal{L}_{odd})=1$, the matrix $V$ is given by $V = (\mathcal{L}_{1}^{-1} \, \psi,\psi)_{L^2_{per}}$. By Lemma \ref{lemmaL1-cn}, we have ${\rm n}(V) = 1$, so that the semi-trivial solution $\Psi = (\psi,0,0,0)$ is spectrally unstable.

\textbf{Case IV:} $\gamma = 3 \kappa_1$. Here, we have $\mathcal{L}_{1,odd} = \mathcal{L}_{3,odd}$ and from \eqref{nL-semitrivial-odd} and \eqref{kerL-semitrivial-odd}, we have that ${\rm n}(\mathcal{L}_{odd}) = 2$ and ${\rm Ker}(\mathcal{L}_{odd}) = \left[ (0,0,\psi,0) \right].$ As we have determined in the last case, we also have ${\rm n}(V) = 1$ and the semi-trivial solution $\Psi = (\psi,0,0,0)$  is spectrally unstable.\\
\indent Summarizing the arguments above, we have proved Theorem $\ref{teooddsemi}$.

\section*{Acknowledgments}
F. Natali is partially supported by CNPq/Brazil (grant 303907/2021-5) and CAPES MathAmSud (grant 88881.520205/2020-01). G. E. Bittencourt Moraes is supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)/Brazil - Finance code 001.

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