# On the existence, uniqueness and stability of Periodic Waves for the
fractional Benjamin-Bona-Mahony equation

Sabrina Amaral  Department of Mathematics, State University of Maringá,
Maringá, PR, Brazil<EMAIL_ADDRESS>,  Handan Borluk  Ozyegin
University, Department of Natural and Mathematical Sciences, Cekmekoy,
Istanbul, Turkey<EMAIL_ADDRESS>,  Gulcin M. Muslu  Istanbul
Technical University, Department of Mathematics, Maslak, Istanbul, Turkey.
<EMAIL_ADDRESS>,  Fábio Natali∗ Department of Mathematics, State
University of Maringá, Maringá, PR, Brazil<EMAIL_ADDRESS>and  Goksu Oruc
Istanbul Technical University, Department of Mathematics, Maslak, Istanbul,
Turkey<EMAIL_ADDRESS>
###### Abstract.

The existence, uniqueness and stability of periodic traveling waves for the
fractional Benjamin-Bona-Mahony equation is considered. In our approach, we
give sufficient conditions to prove a uniqueness result for the single-lobe
solution obtained by a constrained minimization problem. The spectral
stability is then shown by determining that the associated linearized operator
around the wave restricted to the orthogonal of the tangent space related to
the momentum and mass at the periodic wave has no negative eigenvalues. We
propose the Petviashvili’s method to investigate the spectral stability of the
periodic waves for the fractional Benjamin-Bona-Mahony equation, numerically.
Some remarks concerning the orbital stability of periodic traveling waves are
also presented.

###### Key words and phrases:

BBM type-equations, existence and uniqueness of minimizers, spectral
stability, orbital stability, Petviashvili’s method.

###### 2000 Mathematics Subject Classification:

76B25, 35Q51, 35Q53.

∗Corresponding author
Date: June, 2021.

##  1\. Introduction.

In this paper, we show the spectral/orbital stability of zero mean periodic
traveling wave solutions associated to the fractional Benjamin-Bona-Mahony
(fBBM) type equation

$u_{t}+u_{x}+uu_{x}+(D^{\alpha}u)_{t}=0.$ (1.1)

Here $u=u(x,t):\mathbb{R\times\mathbb{R}\rightarrow\mathbb{R}}$ is a
$2\pi$-periodic function at the $x-$variable and $D^{\alpha}$ represents the
fractional differential operator defined as a Fourier multiplier by

$\widehat{D^{\alpha}g}(\xi)=|\xi|^{\alpha}\widehat{g}(\xi),\ \
\xi\in\mathbb{Z},$ (1.2)

where $0<\alpha\leq 2$.
For the case $\alpha=2$, we have the well known Benjamin-Bona-Mahony (BBM)
equation arising as an improvement of the Korteweg-de Vries equation for
modeling long surface gravity waves of small amplitude which propagate
unidirectionally. It also describes the propagation of long waves with a
balance of nonlinear and dissipative effects. In addition, the model appears
in the analysis of the surface waves of long wavelength in liquids,
hydromagnetic waves in cold plasma, acoustic-gravity waves in compressible
fluids, and acoustic waves in harmonic crystals (see [7] and related
references).
If $\alpha=1$, we obtain the regularized Benjamin Ono equation (rBO) which is
a regularized version of the standard Benjamin-Ono equation. The rBO equation
is a model for the time evolution of long-crested waves at the interface
between two immiscible fluids. Some situations in which the equation is useful
are the pycnocline in the deep ocean, and the two-layer system created by the
inflow of fresh water from ariver into the sea [25].

The fBBM equation (1.1) admits the following conserved quantities, smooth in
their domains, as

$\displaystyle
E(u)=\frac{1}{2}\int_{-\pi}^{\pi}\big{[}(D^{\frac{\alpha}{2}}u)^{2}-\frac{1}{3}u^{3}\big{]}dx$
(1.3)

$\displaystyle
P(u)=\frac{1}{2}\int_{-\pi}^{\pi}\big{[}(D^{\frac{\alpha}{2}}u)^{2}+u^{2}\big{]}dx,$
(1.4)

and

$\displaystyle M(u)=\int_{-\pi}^{\pi}udx.$ (1.5)

A traveling wave solution for (1.1) is of the form $u(x,t)=\phi(x-ct)$, where
$\phi:\mathbb{T}\rightarrow\mathbb{R}$, with $\mathbb{T}=[-\pi,\pi]$, is a
smooth $2\pi$-periodic function and $c$ represents the wave speed.
Substituting this form into the fBBM equation (1.1), we obtain

$cD^{\alpha}\phi+(c-1)\phi-\frac{1}{2}\phi^{2}+A=0,$ (1.6)

where $A$ is a constant of integration. If we suppose that
$\phi:\mathbb{T}\rightarrow\mathbb{R}$ is a periodic function with the zero
mean property, then $A=A(c)$ is defined by

$\displaystyle A(c):=\frac{1}{4\pi}\int_{-\pi}^{\pi}\phi^{2}(x)dx.$ (1.7)

Combining (1.6) and (1.7), we see that the equation $(\ref{ode1})$ can be
expressed by the following boundary-value problem

$cD^{\alpha}\phi+(c-1)\phi=\frac{1}{2}\Pi_{0}\phi^{2},\ \ \phi\in
H^{\alpha}_{per}(\mathbb{T}),$ (1.8)

where $\Pi_{0}f:=f-\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)dx$ is the projection
operator restricted to the mean zero $2\pi$-periodic functions.

The conserved quantities (1.3), (1.4) and (1.5) play an important role in our
spectral stability analysis. In fact, they allow us to consider the augmented
Lyapunov functional

$G(u)=E(u)+(c-1)P(u)+AM(u),$ (1.9)

where $G^{\prime}(\phi)=0$, that is, $\phi$ is a critical point of $G$.
Moreover, from (1.9) we obtain the Hessian operator around the wave $\phi$,
which is commonly called in the current literature, as linearized operator
around $\phi$ and given by

$\mathcal{L}:=G^{\prime\prime}(\phi)=cD^{\alpha}+c-1-\phi.$ (1.10)

It is easy to see that $\mathcal{L}$ is a self-adjoint operator defined in
$L^{2}_{per}(\mathbb{T})$ with dense domain $H_{per}^{\alpha}(\mathbb{T})$.

Next, we establish our linearized spectral problem for the fBBM equation.
Indeed, substituting $u(x,t)=\phi(x-ct)+v(x-ct,t)$ into (1.1) and using that
(1.8) is satisfied by $\phi$, we obtain that $v$ is a solution of the
nonlinear equation

$(\partial_{t}-c\partial_{x})(v+D^{\alpha}v)+\partial_{x}(v+\phi
v)+v\partial_{x}v=0.$ (1.11)

Replacing (1.11) by its linearization about $\phi$ we obtain the basic
spectral stability problem

$\partial_{t}(v+D^{\alpha}v)=\partial_{x}\mathcal{L}v$ (1.12)

where $\mathcal{L}$ is given by (1.10). Since $\phi$ depends on $x$ and but
not $t$, the equation (1.12) has a separation of variables of the form
$v(x,t)=e^{\lambda t}\eta(x)$ with some $\lambda\in\mathbb{C}$ and
$\eta:\mathbb{T}\rightarrow\mathbb{C}$. These arguments allow us to consider
the spectral problem

$\partial_{x}\mathcal{L}\eta=\lambda(1+D^{\alpha})\eta.$ (1.13)

We can rewrite the spectral problem in $(\ref{spectproblem})$ as

$J\mathcal{L}\eta=\lambda\eta,$ (1.14)

where $J:=(1+D^{\alpha})^{-1}\partial_{x}$. Let us denote the spectrum of
$J\mathcal{L}$ by $\sigma(J\mathcal{L})$. In a general setting, the periodic
wave $\phi\in H^{\alpha}_{per}(\mathbb{T})\cap X_{0}$ is said to be spectrally
stable if $\sigma(J\mathcal{L})\subset i\mathbb{R}$ in
$L^{2}_{per}(\mathbb{T})$. Otherwise, that is, if $\sigma(J\mathcal{L})$ in
$L^{2}_{per}(\mathbb{T})$ contains a point $\lambda$ with $Re(\lambda)>0$, the
periodic wave $\phi$ is said to be spectrally unstable.
According to the classical spectral stability results as in [18], the problem
given by $(\ref{hamilt1})$ can not be handled in periodic context since
$J=(1+D^{\alpha})^{-1}\partial_{x}$ is not a one-to-one operator. To overcome
this difficulty, we can consider the linearization of $(\ref{rbbm})$ around
$\phi$ restricted to the space of zero mean periodic function to obtain the
modified problem (see [12])

$J\mathcal{L}|_{X_{0}}\chi=\lambda\chi,$ (1.15)

where
$\mathcal{L}|_{X_{0}}=\mathcal{L}+\frac{1}{2\pi}\langle\phi,\cdot\rangle$ is a
restriction of $\mathcal{L}$ on the closed subspace $X_{0}$ of periodic
functions with zero mean,

$\displaystyle X_{0}=\left\\{f\in
L^{2}_{per}(\mathbb{T}):\int_{-\pi}^{\pi}f(x)dx=0\right\\}.$ (1.16)

Clearly, operator $\mathcal{L}|_{X_{0}}$ is defined in the space $X_{0}$ with
domain $H_{per}^{\alpha}(\mathbb{T})\cap X_{0}$. The new spectral problem
$(\ref{hamilt2})$ allows to consider a new definition for the spectral
stability restricted to the periodic space $X_{0}$ as:

######  Definition 1.1.

The periodic wave $\phi\in H^{\alpha}_{per}(\mathbb{T})\cap X_{0}$ is said to
be spectrally stable if $\sigma(J\mathcal{L}|_{X_{0}})\subset i\mathbb{R}$ in
$X_{0}$. Otherwise, that is, if $\sigma(J\mathcal{L}|_{X_{0}})$ in $X_{0}$
contains a point $\lambda$ with $Re(\lambda)>0$, the periodic wave $\phi$ is
said to be spectrally unstable.

######  Remark 1.2.

In Definition 1.1, we consider a smooth zero mean periodic wave $\phi$.
However, it is possible to obtain the spectral stability/instability in the
space $X_{0}$ using the same definition for non-zero mean periodic waves as
reported, for instance, in [12] and [23].

In [23], the existence of minimizers for the energy functional $V:=P-E$ with
fixed momentum $P$ and mass $M$ has been established. The periodic wave
obtained by this minimization problem is smooth in terms of the independent
parameters $A$ and $c$ in $(\ref{ode1})$ and it has been determined that they
are spectrally stable in the sense of Definition 1.1. The orbital stability of
the periodic minimizers is then established by assuming that the 2-by-2
determinant $\\{P,M\\}_{A,c}$ is nonzero (see the precise definition of
orbital stability in the last section of this paper). We also refer the reader
to see [11] for a different approach. However, in the referred work, a
different minimization problem compared to [23] has been established and the
authors obtained a different criterium for the orbital stability in the energy
space.
For the cases $\alpha=1$ and $\alpha=2$, precise results of orbital stability
for the equation $(\ref{ode1})$ with $A=0$ were determined in [5] combined
that $\mathcal{L}$ has only one negative eigenvalue which is simple and zero
is a simple eigenvalue whose associated eigenfunction is $\phi^{\prime}$. In
order to show that $\mathcal{L}$ has only one negative eigenvalue which is
simple and zero is a simple eigenvalue with associated eigenfunction
$\phi^{\prime}$ they have used the Fourier expansion of the explicit solutions
together with the main theorem in [6]. In addition, the explicit solutions
were also used to calculate the positiveness of $\frac{d}{dc}P(\phi)$.
Concerning the case $\alpha=2$ in $(\ref{rbbm})$ with a general power
nonlinearity as

$u_{t}+u_{x}+(p+1)u^{p}u_{x}-u_{xxt}=0,$ (1.17)

where $p\geq 1$ is an integer and traveling $\phi$ waves satisfying the
equation

$-c\phi^{\prime\prime}+(c-1)\phi-\phi^{p+1}=0,$ (1.18)

existence and stability results of small amplitude periodic waves near of the
equilibrium solution $(c-1)^{1/p}$ for $c>1$ have been considered in [21].
These waves are spectrally stable for all speeds $c>1$ when $1\leq p\leq 2$ in
both $L^{2}(\mathbb{R})$ and $C_{b}(\mathbb{R})$. Here, $C_{b}(\mathbb{R})$
stands the space of bounded uniformly continuous functions. For $p\geq 3$,
there exists a critical speed $c_{p}=\frac{p(p+4)}{p^{2}+p+6}$,
$1<c_{p}<\frac{p}{p-3}$, where the waves are stable for
$c\in\left(c_{p},\frac{p}{p-3}\right)$ and spectrally unstable for
$c\in(1,c_{p})\cup\left(\frac{p}{p-3},\infty\right)$ by considering the same
perturbations. For perturbations which have the same period $L$ as the wave,
all small amplitude periodic waves are spectrally stable for all values of
$p$.
For the model $(\ref{rbbm})$ posed on unbounded domains, the author in [3]
showed the existence of solitary waves employing the arguments in [15]. He
used the abstract approach in [18] ([28]) and determined the orbital
(spectral) stability of solitary waves if:
i) $c>1$ and $\alpha\in\left[\frac{1}{2},1\right)$,
ii) $c>c_{0}:=\frac{2+\sqrt{2(3\alpha-1)}}{6\alpha}>1$ and
$\alpha\in\left(\frac{1}{3},\frac{1}{2}\right)$.
If $\alpha\in\left(\frac{1}{3},\frac{1}{2}\right)$ and $c\in(1,c_{0})$, the
solitary wave is said to be spectrally unstable using the arguments in [28].
His proof relies on the scaling argument for the solitary wave $\phi$ which
gives good features about the Hessian matrix $\frac{d}{dc}P(\phi)$. The same
method in [3] can be used to establish the orbital (spectral) stability of
solitary waves for the case $\alpha\in[1,2]$ and $c>1$.
We can cite some additional contributions concerning the spectral/orbital
stability of periodic waves for related equations to $(\ref{rbbm})$ and other
dispersive equations (see [1], [4], [6], [10], [12], [16], [17], [19], [24],
[30], [31], and references therein). In some cases, the study of the stability
was considered in relation to perturbations of the same period of the wave and
other types of perturbations (in the latter case, we can consider: anti-
periodic, bounded and so on).
The main results of our work can be presented according to the following
theorem:

######  Theorem 1.3.

Let $\alpha\in\left(\frac{1}{3},2\right]$ and $\tau>0$ be fixed. For every
$c>\frac{1}{2}$, there exists an even and periodic single-lobe profile
$\phi\in H_{per}^{\alpha}(\mathbb{T})$ which is solution of the constrained
minimization problem,

$\inf_{u\in{H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}}\left\\{\int_{-\pi}^{\pi}c(D^{\frac{\alpha}{2}}u)^{2}+(c-1)u^{2}dx;\
\int_{-\pi}^{\pi}u^{3}dx=\tau,\ \int_{-\pi}^{\pi}udx=0\right\\},$ (1.19)

and a solution of $(\ref{ode2})$. If
$\ker\left(\mathcal{L}|_{X_{0}}\right)=[\phi^{\prime}]$ for all
$c>\frac{1}{2}$, we obtain a smooth curve
$c\in\left(\frac{1}{2},+\infty\right)\mapsto\phi\in
H_{per}^{\alpha}(\mathbb{T})\cap X_{0}$ of even periodic waves with fixed
period and furthermore, denoting $d:=1+2A(c)-c-cA^{\prime}(c)$, we have,

i) for the linearized operator $\mathcal{L}$: a simple negative eigenvalue if
$d<0$ and two negative eigenvalues if $d\geq 0$,
ii) zero is a simple eigenvalue of $\mathcal{L}$ if and only if $d\neq 0$,
iii) $\phi$ is the unique solution of the problem $(\ref{minp})$ for the case
$\alpha\in\left(\frac{1}{2},2\right],$
iv) the periodic wave $\phi$ is spectrally stable if:

  * •

$d\neq 0$ and $A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$,

  * •

$\alpha\in\left(\frac{1}{2},2\right]$ and $d\neq 0$ for all $c>\frac{1}{2}$,

v) the periodic wave $\phi$ is orbitally stable if $\alpha\in\left(1,2\right]$
and $d\neq 0$ for all $c>\frac{1}{2}$.

The first two parts in Theorem 1.3 concern the existence of periodic
minimizers for the problem $(\ref{minp})$ and the existence of smooth curve of
solutions with fixed period. To prove the first part, we need to use the
Poincaré-Wirtinger inequality to find the bifurcation point $c=\frac{1}{2}$
and compactness tools to establish that the periodic waves exist for all
$c>\frac{1}{2}$. The second part is similar to [31, Lemma 3.8] and it can be
determined by using the crucial hypothesis
$\ker\left(\mathcal{L}|_{X_{0}}\right)=[\phi^{\prime}]$ and the implicit
function theorem.
Assumption $\ker\left(\mathcal{L}|_{X_{0}}\right)=[\phi^{\prime}]$ is also
important to give precise informations about items i)-v) in Theorem 1.3. In
fact, item i) is determined since we show that the our periodic waves solve
the minimization problem $(\ref{minp})$. Thus, we combine this information
with convenient index formulas (see [33, Theorem 4.1]) to obtain that
$n(\mathcal{L})=1$, if $d<0$ and $n(\mathcal{L})=2$, if $d\geq 0$. Here,
$n(\mathcal{L})$ stands for the number of negative eigenvalues of
$\mathcal{L}$. The value of $d$ is crucial in our analysis since it determines
the existence of fold points, that is, values of $c$ (and depending on
$\alpha$) such that $d=0$. Folds points are related with the existence of
additional elements in $\ker(\mathcal{L})$, besides $\phi^{\prime}$, and they
give the exact value of $c$ where the number of negative eigenvalues change.
It is worth mentioning that they were first studied in [31] for the case of
fractional Korteweg-de Vries (fKdV) equation.
Second item can be established since the operator $\mathcal{L}$ in
$(\ref{operator})$ satisfies an Oscillation Theorem for fractional linear
operators with smooth periodic potentials according to [15], [23] and [31]. In
fact, we obtain that $\dim(\ker(\mathcal{L}))\in\\{1,2\\}$ and
$\dim(\ker(\mathcal{L}))=1$ if and only if $d\neq 0$.
Regarding item iii). Using the arguments in [15], our work establishes
sufficient conditions for the uniqueness of periodic minimizers associated to
the problem $(\ref{minp})$. In what follows, let
$\alpha_{0}\in(\frac{1}{2},2]$ be fixed. Assume that $(\phi_{0},c_{0})\in
V\times\left(\frac{1}{2},+\infty\right)$ with $\phi_{0}$ being a non-zero
solution of $(\ref{ode2})$ and satisfying assumption $(\ref{hyp-uniq})$ with
$\alpha=\alpha_{0}$ and $c=c_{0}$. The implicit function theorem guarantees
the existence of a $\delta>0$ and an $C^{1}-$map $\alpha\in
I=[\alpha_{0},\alpha_{0}+\delta)\mapsto(\phi_{\alpha},c_{\alpha})$ such that
$(\phi_{\phi},c_{\alpha})$ is the unique local solution of $(\ref{ode2})$ in a
convenient neighbourhood around $(\phi_{0},c_{0})$. In addition, it is
possible to obtain for all $\alpha\in I$ that

$\int_{-\pi}^{\pi}\phi_{\alpha}^{3}dx=\int_{-\pi}^{\pi}\phi_{0}^{3}dx.$ (1.20)

Using compactness arguments, we prove that the maximal branch
$(\phi_{\alpha},c_{\alpha})$ and equality $(\ref{L3})$ extends to the interval
$[\alpha_{0},2)$. Next, since it is well know that for $\alpha=2$ and a fixed
$c>\frac{1}{2}$ the solution with dnoidal profile (see $(\ref{dnsol})$) is
unique, we obtain the uniqueness of minimizers for the problem $(\ref{minp})$
using the similar arguments as in [15, Theorem 2.4].
We determine item iv) employing again the index formula to get

$n(\mathcal{L}|_{\\{1,(D^{\alpha}+1)\phi\\}^{\bot}})=n(\mathcal{L})-n_{0}-z_{0},$

where $n_{0}$ is the number of negative eigenvalues determined by the Hessian
symmetric matrix formed by momentum and mass and $z_{0}$ indicates the
dimension of the kernel of the same Hessian matrix. To simplify the analysis,
$n_{0}$ and $z_{0}$ can be both precisely determined by analysing the
following quantity for the case $d\neq 0$ (see Section 5).

$\det
S(0):=\frac{4\pi^{2}d^{-1}}{c}\left(A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)\right).$
(1.21)

To obtain the precise statements for this item, we need to discuss the
existence (and spectral stability) of periodic small amplitude waves
associated with the equation $(\ref{rbbm})$ for the case $\alpha\in(0,2]$. For
a small amplitude parameter $a$, we put forward the explicit solutions of
$(\ref{ode2})$ as

$\phi(x)=a\cos(x)+\frac{a^{2}}{2(2^{\alpha}-1)}\cos(2x)+O(a^{3}).$ (1.22)

The wave speed $c$ and parameter $A$ are given respectively by

$c=\frac{1}{2}+\frac{a^{2}}{4(2^{\alpha}+1)(2^{\alpha}-1)}+O(a^{4}),\ \ \
\mbox{and}\ \ \ A(c)=\frac{a^{2}}{4}+O(a^{4}).$ (1.23)

Orbital/spectral stability and related topics of small amplitude periodic
waves associated to several evolution models have been exhaustively studied
(see [20], [22], [24], [27], [31], and references therein). In our paper, we
show the existence of two fold points: $\alpha=\tilde{\alpha}\approx 0.2924$
and $\alpha=\frac{1}{2}$. We prove that $n(\mathcal{L})=1$, if
$\alpha\in(0,\tilde{\alpha})\cup(\frac{1}{2},2]$ and $n(\mathcal{L})=2$, if
$\alpha\in(\tilde{\alpha},\frac{1}{2})$. To determine $A^{\prime}(c)$ in
$(\ref{detS0})$, we can use both expressions in $(\ref{exp-speed1})$ to get

$A^{\prime}(c)=(2^{\alpha}+1)(2^{\alpha}-1)+O(a^{3}).$ (1.24)

Since $A^{\prime}(c)>0$ for $\alpha\in(0,2]$, it follows that $n_{0}$ and
$z_{0}$ are determined only by the sign of $d$. In fact, if $d<0$, one has
$n_{0}=1$ and $z_{0}=0$, while $d>0$ gives us that $n_{0}=2$ and $z_{0}=0$. In
both cases, we obtain the spectral stability of the small amplitude periodic
wave $\phi$ $(\ref{exp-sol1})$. The degenerate case $d=0$ is also established
and the wave is said to be spectrally stable. For the case $\alpha=2$ in
$(\ref{rbbm})$, our results are compatible with those ones in [21] when $p=1$
in equation $(\ref{pBBM})$ but, in our case, the small amplitude periodic
waves are near of the equilibrium solution $\phi=0$.
Using the spectral information obtained by the small amplitude periodic waves,
we establish that the number of negative eigenvalues of the linearized
operator $\mathcal{L}$ is always equal to one if and only if
$\dim(\ker({\mathcal{L}}))=1$ for all $c>\frac{1}{2}$. In addition, since
$d<0$ for the periodic wave obtained by the problem $(\ref{minp})$, we
conclude $A(c)>c-\frac{1}{2}$ and this bound allows us to deduce that $\det
S(0)$ in $(\ref{detS0})$ is always negative. This fact gives us the spectral
stability and moreover, it recovers the same result as in [3] without using
scaling argument. In addition, we obtain periodic travelling wave solutions of
the equation (1.1) numerically by using a Petviashvili’s iteration method. The
numerical results also confirm the analytical results stating that the
periodic wave is spectrally stable for $\alpha\in\left(\frac{1}{2},2\right]$.
Next, we discuss the case $\alpha\in(\frac{1}{3},\frac{1}{2}]$. In fact,
numerical experiments will be used to decide the exact sign of the quantities
$d$ and $A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)$ in $(\ref{detS0})$
in terms of $c>\frac{1}{2}$. Our numerical results point out that there exists
a critical wave speed $c^{*}\in(\frac{1}{2},+\infty)$ at which the sign of $d$
changes. For $c\in(c^{*},+\infty)$, we observe numerically that $d<0$ and
$A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$, therefore the periodic
wave is spectrally stable. On the other hand, for $c\in(\frac{1}{2},c^{*})$,
we have $d>0$ and $A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$. Thus
$\det S(0)>0$ which implies $n_{0}=0$ or $n_{0}=2$. Since the first entry of
the matrix $S(0)$ is negative, we obtain $n_{0}=2$ and the spectral stability
of periodic wave is observed numerically for $c\in(\frac{1}{2},c^{*})$.
In addition to the spectral stability, we prove the orbital stability for
$\alpha\in(1,2]$ by employing an adaptation of the recent arguments in [11]
and new ingredients. Indeed, in our paper we are not considering, for the
orbital stability, the old method which minimizes the energy $E$ in
$(\ref{quantconserved})$ with fixed momentum $P$ in $(\ref{momentum})$ and
mass $M$ in $(\ref{mass})$. When this kind of approach is considered, we need
to prove, besides good spectral properties for the linearized operator
$\mathcal{L}$ in $(\ref{operator})$, the positiveness of the Hessian matrix
$\frac{d}{dc}P(\phi)$ (see [1], [5], [6], [11], [18], and references therein).
As far as we can see, even in the case $\alpha=2$, when explicit solutions are
known in terms of the Jacobi elliptic functions, the calculation of the
derivative of $P(\phi)$ in terms of $c$ becomes a hard task (see [5]). Instead
of this, we use the conservation law $V(u)=P(u)-E(u)$ as a constrained
manifold associated to the augmented Lyapunov functional $G$ in
$(\ref{lyapfunct})$. This consideration sheds new light in the stability
theory since we obtain our results without using any additional information of
the wave, only the basic bound $A(c)>c-\frac{1}{2}$ determined to prove the
spectral stability. This fact enables us to conclude item v) in Theorem 1.3.
Our paper is organized as follows. In Section 2, we establish local and global
well posedness results for the Cauchy problem associated to the equation
$(\ref{rbbm})$. In Section 3 we show the existence of periodic minimizers for
the problem $(\ref{minp})$. and a precise information about the number of
negative eigenvalues of $(\ref{operator})$. Also in this section, we show the
uniqueness of periodic minimizers for the problem $(\ref{minp})$ in the
interval $\alpha\in\left(\frac{1}{2},2\right]$. In Section 4, we present our
spectral stability result for the waves obtained in Section 3. Section 5 is
devoted to the numerical investigation of the spectral stability of periodic
waves. Finally, in Section 6 we give some important remarks concerning the
orbital stability of periodic waves.

##  2\. Well-posedness results

In this section, we present a brief comment concerning the global well-
posedness for the Cauchy problem associated to the equation $(\ref{rbbm})$

$\left\\{\begin{array}[]{llll}u_{t}+u_{x}+uu_{x}+(D^{\alpha}u)_{t}=0,\\\ \\\
u(x,0)=u_{0}(x).\end{array}\right.$ (2.1)

In the whole real line, the best result of local well posedness for the Cauchy
problem $(\ref{cprbbm})$ has been determined in [29] for initial data
$u_{0}\in H^{s}(\mathbb{R})$, $s>\frac{3}{2}-\frac{3}{8}\alpha$ and
$\alpha\in(0,1)$. For initial data $u_{0}\in H_{per}^{\alpha/2}(\mathbb{T})$
and $\alpha>1$, we are going to use fixed point arguments applied to the
representation of $(\ref{cprbbm})$ in an integral form as

$u(x,t)=u_{0}(x)+\int_{0}^{t}\left[\mathcal{K}\ast\left(u+\frac{u^{2}}{2}\right)\right](x,s)ds,$
(2.2)

where $\mathcal{K}$ is defined using the periodic Fourier transform

$\widehat{\mathcal{K}v}(k)=\frac{-ik}{1+|k|^{\alpha}}\widehat{v}(k).$

######  Proposition 2.1.

Let $\alpha>1$ be fixed. For each $s\geq\frac{\alpha}{2}$ and $u_{0}\in
H_{per}^{s}(\mathbb{T})$, there exist $T>0$ and a unique solution $u$ of
$(\ref{cprbbm})$ such that $u\in C([-T,T];H_{per}^{s}(\mathbb{T}))$. Moreover,
for all $T^{\star}<T$, there exists a neighborhood $V$ of $u_{0}$ in
$H_{per}^{s}(\mathbb{T})$ such that the data-solution map

$w_{0}\in V\subset H_{per}^{s}(\mathbb{T})\mapsto w\in
C([-T^{\star},T^{\star}];H_{per}^{s}(\mathbb{T})),$

is continuous.

###### Proof.

The proof of this result is classical because of the integral representation
in $(\ref{intform})$ (for the cases $\alpha=1,2$, see [5]). Define,
$Y=C([-T,T];H_{per}^{s}(\mathbb{T}))$ with the norm
$||w||_{Y}=\sup_{t\in[-T,T]}||w(t)||_{H_{per}^{s}}$. Consider the map
$\Upsilon:B_{r}\rightarrow B_{r}$ given by

$\Upsilon(u)=\displaystyle
u_{0}(x)+\int_{0}^{t}\mathcal{K}\ast\left(u+\frac{u^{2}}{2}\right)ds,$ (2.3)

where $r>0$ will be chosen later and $B_{r}=\\{w\in Y;\ ||w||_{Y}\leq r\\}$.
We show that $\Upsilon$ is well defined, in the sense that $\Upsilon(u)\in
B_{r}$ for $u\in B_{r}$, and $\Upsilon$ is a strict contraction.
In fact, consider $v\in H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$. Since
$\alpha>1$, we have

$\begin{array}[]{llll}||\mathcal{K}\ast v||_{H_{per}^{s}}^{2}\leq\displaystyle
2\pi\sum_{k\in\mathbb{Z}}(1+k^{2})^{s}|\hat{v}(k)|^{2}=||v||_{H_{per}^{s}}^{2}.\end{array}$
(2.4)

In addition, if $v\in L_{per}^{2}(\mathbb{T})$ we obtain the basic smoothing
effect given by

$\begin{array}[]{llll}||\mathcal{K}\ast
v||_{H_{per}^{\frac{\alpha}{2}}}^{2}\leq\displaystyle 2\pi
C_{1}\sum_{k\in\mathbb{Z}}|\hat{v}(k)|^{2}=C_{1}||v||_{L_{per}^{2}}^{2},\end{array}$
(2.5)

where $C_{1}>0$ does not depend on $v$.
On the other hand $\alpha>1$ implies that ${H_{per}^{s}(\mathbb{T})}$ is a
Banach algebra for all $s\geq\frac{\alpha}{2}$, and thus

$||u^{2}||_{{H_{per}^{s}}}\leq C_{2}||u||_{{H_{per}^{s}}}^{2},$ (2.6)

where $C_{2}>0$ does not depend on $u$.

Let $u,v\in Y$ be fixed. Gathering $(\ref{est1})$, $(\ref{est3})$ and the
definition of $\Upsilon$, we obtain

$||\Upsilon
u||_{Y}\leq||u_{0}||_{{H_{per}^{s}}}+T\left(||u||_{Y}+C_{3}||u||_{Y}^{2}\right),$
(2.7)

where $C_{3}>0$ is positive constant depending on $C_{2}>0$.
Next,

$\begin{array}[]{lllll}||\mathcal{K}\ast(u^{2}-v^{2})||_{{H_{per}^{s}}}\leq\left(||u||_{Y}+||v||_{Y}\right)||u-v||_{Y}.\end{array}$
(2.8)

Therefore, by using a similar procedure as in $(\ref{est5})$ we obtain that

$||\Upsilon u-\Upsilon v||_{Y}\leq
T\left(1+||u||_{Y}+||v||_{Y}\right)||u-v||_{Y}.$ (2.9)

By $(\ref{est5})$, $(\ref{est7})$ and the fact that $u,v\in Y$, we have

$||\Upsilon u||_{Y}\leq||u_{0}||_{{H_{per}^{s}}}+T\left(r+C_{3}r^{2}\right)\
\mbox{and}\ ||\Upsilon u-\Upsilon v||_{Y}\leq T\left(1+2r\right)||u-v||_{Y}.$
(2.10)

Using $(\ref{est8})$, we can choose $r=2||u_{0}||_{{H_{per}^{s}}}$ and
$T=\frac{1}{2(1+C_{4}r)}$, where $C_{4}=\max\\{2,C_{3}\\}>0$, to obtain

$||\Upsilon u||_{Y}\leq r\ \mbox{and}\ ||\Upsilon u-\Upsilon
v||_{Y}\leq\frac{1}{2}||u-v||_{Y}.$ (2.11)

Inequalities in $(\ref{est9})$ give us that $\Upsilon:B_{r}\rightarrow B_{r}$
is well defined and a strict contraction. From the Banach Fixed Point Theorem,
we obtain the existence of a unique $u\in B_{r}$ such that $\Upsilon
u(t)=u(t)$ for all $t\in[-T,T]$. The uniqueness in the whole space and the
continuous dependence are determined by a direct application of standard
arguments. ∎

######  Proposition 2.2.

Let $\alpha>1$ be fixed. The quantities $E$, $P$ and $M$ defined in
$(\ref{quantconserved})$, $(\ref{momentum})$ and $(\ref{mass})$, respectively
are conservation laws.

###### Proof.

The proof of this result is standard and we skip the details.

∎

Propositions 2.1 and 2.2 give us the following result.

######  Proposition 2.1.

Let $\alpha>1$ be fixed. For each
$u_{0}\in{H_{per}^{\frac{\alpha}{2}}}(\mathbb{T})$, the Cauchy problem
$(\ref{cprbbm})$ is globally well posed in
${H_{per}^{\frac{\alpha}{2}}}(\mathbb{T})$ with $u\in
C(\mathbb{R};{H_{per}^{\frac{\alpha}{2}}}(\mathbb{T}))$.

$\square$

##  3\. Existence and Uniqueness of Minimizers - Spectral Properties.

###  3.1. Existence of Periodic Minimizers

In this subsection, we first give a sufficient condition for the existence of
periodic waves for the equation (1.6) by showing the existence of minimizers
associated to a convenient variational problem. After that, we use the
minimizers to obtain good spectral properties for the linearized operator
$\mathcal{L}$ in $(\ref{operator})$.
Before starting, we need an important definition which characterizes the
solutions of the equation $(\ref{ode2})$.

######  Definition 3.1.

We say that the periodic traveling wave solutions satisfying the equation
$(\ref{ode2})$ has a single-lobe profile $\phi$ if there exist only one
maximum and minimum of $\phi$ on $\mathbb{T}$. Without the loss of generality,
the maximum of $\phi$ is placed at $x=0$.

For a fixed $\tau>0$, let us consider the set

$\displaystyle Y_{0}=\left\\{u\in
H^{\frac{\alpha}{2}}_{per}(\mathbb{T});\int_{-\pi}^{\pi}u^{3}=\tau,\int_{-\pi}^{\pi}u=0\right\\}.$
(3.1)

Our goal is to find a minimizer of the constrained minimization problem

$\displaystyle q_{c}=\inf_{u\in Y_{0}}\mathcal{B}_{c}(u),$ (3.2)

where

$\displaystyle\mathcal{B}_{c}(u)=\frac{1}{2}\int_{-\pi}^{\pi}c(D^{\frac{\alpha}{2}}u)^{2}+(c-1)u^{2}dx.$
(3.3)

The next lemma establishes the first part of Theorem 1.3.

######  Lemma 3.2.

Let $\alpha>1/3$ and $c>\frac{1}{2}$ be fixed. The minimization problem (3.2)
has at least one solution, that is, there exists a $\phi\in Y_{0}$ satisfying

$\displaystyle\mathcal{B}_{c}(\phi)=\inf_{u\in Y_{0}}\mathcal{B}_{c}(u).$
(3.4)

If $\alpha\in(\frac{1}{3},2]$, the periodic minimizer of $(\ref{minB})$ has an
even, single-lobe profile in the sense of Definition 3.1.

###### Proof.

Let $c>\frac{1}{2}$ be fixed. By Poincaré-Wirtinger inequality, we see that

$\displaystyle
2\mathcal{B}_{c}(u)=\int_{-\pi}^{\pi}c(D^{\frac{\alpha}{2}}u)^{2}+(c-1)u^{2}dx\geq(2c-1)\int_{-\pi}^{\pi}u^{2}dx\geq
0.$ (3.5)

Combining $(\ref{poinc})$ with the Garding inequality, we obtain that
$\mathcal{B}_{c}$ is an equivalent norm in
$H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$ yielding $q_{c}\geq 0$. Since
$\mathcal{B}_{c}$ is a smooth functional in
$H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$, let $\\{u_{n}\\}$ be a minimizing
sequence for (3.2), that is, a sequence in $Y_{0}$ satisfying

$\displaystyle\mathcal{B}_{c}(u_{n})\rightarrow\inf_{u\in
Y_{0}}\mathcal{B}_{c}(u),\ \ \textrm{as}\ n\rightarrow+\infty.$

There exist positive constants $M_{0}$ and $M_{1}$ depending on $c$ satisfying

$M_{0}||u_{n}||_{H_{per}^{\frac{\alpha}{2}}}^{2}\leq\mathcal{B}_{c}(u_{n})\leq
M_{1}||u_{n}||_{H_{per}^{\frac{\alpha}{2}}}^{2},\ \ \forall n\in\mathbb{N},$

that is, $\\{u_{n}\\}$ is bounded in $H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$.
Thus, there is $\phi\in H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$ such that, up
to a subsequence,

$u_{n}\rightharpoonup\phi\ \textrm{weakly in}\
H_{per}^{\frac{\alpha}{2}}(\mathbb{T}),\ \textrm{as}\ n\rightarrow+\infty.$

On other hand, since $\alpha>1/3$ one sees that the energy space
$H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$ is compactly embedded in
$L_{per}^{3}(\mathbb{T})$ (see [2, Theorem 4.2]), and $u_{n}\rightarrow\phi\
\textrm{in}\ L_{per}^{3}(\mathbb{T}),\ \textrm{as}\ n\rightarrow+\infty.$ In
addition, using the fact

$\displaystyle\displaystyle\left|\int_{-\pi}^{\pi}(u_{n}^{3}-\phi^{3})dx\right|\leq||u_{n}-\phi||_{L_{per}^{3}}+3||u_{n}-\phi||_{L_{per}^{3}}||\phi||_{L_{per}^{3}}||u_{n}||_{L_{per}^{3}},$

one has that $\int_{-\pi}^{\pi}\phi^{3}dx=\tau$. A similar argument as above
and using the fact that $H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$ is compactly
embedded in $L_{per}^{1}(\mathbb{T})$, give us $\int_{-\pi}^{\pi}\phi\ dx=0.$

Moreover, since the weak lower semi-continuity of $\mathcal{B}_{c}$, we have
$\displaystyle\mathcal{B}_{c}(\phi)\leq\liminf_{n\rightarrow+\infty}\mathcal{B}_{c}(u_{n})=q_{c}.$
The symmetric rearrangements $\phi^{\\#}\in Y_{0}$ associated to the solution
$\phi$111We can take $\phi^{\\#}$ as a symmetric rearrangement in $Y_{0}$
because it leaves $\int_{-\pi}^{\pi}u^{3}dx$ and $\int_{-\pi}^{\pi}udx$
invariant (see [23, Proposition 2.1]) leave the $L_{per}^{2}-$norm of
$\phi^{\\#}$ and $\phi$ invariants and
$\int_{-\pi}^{\pi}(D^{\alpha/2}\phi^{\\#})^{2}dx$ does not increase in
comparison with $\int_{-\pi}^{\pi}(D^{\alpha/2}\phi)^{2}dx$ thanks to the
fractional Polya–Szegö inequality (for further details and similar
applications, see [10, Lemma A.1], [23, Proposition 2.1] and [31, Theorem
2.1]). Therefore, for $c>\frac{1}{2}$ we have
$\mathcal{B}_{c}(\phi^{\\#})=q_{c}$. Invoking the original notation $\phi$
instead of $\phi^{\\#}$ to simplify the comprehension of the reader, we see
that the minimizer $\phi\in Y_{0}$ of $\mathcal{B}_{c}(u)$ must decrease away
symmetrically from the maximum point. Using the translational invariance, the
maximum point can be placed at $x=0$, which yields an even single-lobe profile
for $\phi$. ∎

From Lemma (3.2) and Lagranges’s Multiplier Theorem, there exists $C_{1}$ and
$C_{2}$ such that

$\displaystyle cD^{\alpha}\phi+(c-1)\phi=C_{1}\phi^{2}+C_{2}.$ (3.6)

We see that $\phi$ is a nontrivial single-lobe because $\phi\in Y_{0}$. Since
$\int_{-\pi}^{\pi}\phi dx=0$, we deduce from (3.6) that
$C_{2}=-\frac{C_{1}}{2\pi}\int_{-\pi}^{\pi}\phi^{2}dx$. In addition,
multiplying equation $(\ref{lagrange})$ by $\phi$ and integrating the result
over $[0,L]$, we obtain by the Poincaré-Wirtinger inequality, the fact
$c>\frac{1}{2}$ and since $\tau>0$ that $C_{1}>0$. By the homogeneity of
$C_{1}\left(\phi^{2}-\frac{1}{2\pi}\int_{-\pi}^{\pi}\phi^{2}dx\right)$, we see
that $C_{1}$ can be chosen as $C_{1}=\frac{1}{2}$. Indeed, for all $s>0$, we
obtain

$\begin{array}[]{lllll}\mathcal{B}_{c}(s\phi)&=&\displaystyle
s^{2}\mathcal{B}_{c}(\phi)\\\ \\\
&=&\displaystyle\inf\left\\{\mathcal{B}_{c}(su);\ u\in
H_{per}^{\frac{\alpha}{2}}(\mathbb{T}),\ \ \int_{-\pi}^{\pi}u^{3}dx=\tau,\ \ \
\int_{-\pi}^{\pi}udx=0\right\\}\\\ \\\
&=&\displaystyle\inf\left\\{\mathcal{B}_{c}(u);\ u\in
H_{per}^{\frac{\alpha}{2}}(\mathbb{T}),\ \
\int_{-\pi}^{\pi}u^{3}dx=s^{3}\tau,\ \ \
\int_{-\pi}^{\pi}udx=0\right\\}.\end{array}$ (3.7)

Thus, if $\phi$ solves $(\ref{minB})$ we have that $\psi=s\phi$ is a solution
of the minimization problem $(\ref{minsB})$. As above and by Lagrange’s
Multiplier Theorem, there exists $C_{3}>0$ and
$C_{4}=-\frac{C_{3}}{2\pi}\int_{-\pi}^{\pi}\psi^{2}dx$ such that

$s\left(\displaystyle
cD^{\alpha}\phi+(c-1)\phi\right)=s^{2}C_{3}\left(\phi^{2}-\frac{1}{2\pi}\int_{-\pi}^{\pi}\phi^{2}dx\right).$
(3.8)

Since $s>0$ is arbitrary and $\phi$ is non-trivial, we can choose
$s=\frac{1}{2C_{3}}$ to obtain that $\phi$ solves the equation

$cD^{\alpha}\phi+(c-1)\phi-\frac{1}{2}\phi^{2}+A=0,$ (3.9)

so that $C_{1}$ can be chosen as $C_{1}=\frac{1}{2}$. Moreover, by a standard
bootstrap argument (see [11]), we obtain that $\phi$ is smooth.

######  Remark 3.3.

Let $\alpha\in\left(\frac{1}{3},2\right]$ and $L>0$ be fixed. The periodic
wave $\phi$ obtained in Lemma 3.2 can be considered with a general period $L$
instead of the normalized period $L=2\pi$. In the general case, the wave speed
needs to satisfy (by using Poincaré-Wirtinger inequality for general periods)
the basic bound $c>\frac{1}{1+\left(\frac{2\pi}{L}\right)^{\alpha}}$ for the
existence of periodic waves in the space $X_{0,L}=\\{f\in
L_{per}^{2}(\mathbb{T}_{L});\ \int_{-L/2}^{L/2}fdx=0\\}$, where
$\mathbb{T}_{L}=[-L/2,L/2]$. All results in this section, Sections 4 and 6 can
be obtained with a general period by replacing the normalized bifurcation
point $c=\frac{1}{2}$ by $c=\frac{1}{1+\left(\frac{2\pi}{L}\right)^{\alpha}}$
and slight modifications in the arguments when necessary. In Section 5, where
we present our numerical experiments, it makes necessary to fix the period in
order to obtain the plots which give us, for example, the behaviour of $\phi$,
$d$, $A^{\prime}(c)$ and etc. However, the restriction to consider normalized
periodic solutions does not affect the generality of the proposed results in a
general context.

Next, let us denote
$\mathcal{\tilde{L}}:=\frac{1}{c}\mathcal{L}=D^{\alpha}+(1-\frac{1}{c})-\frac{1}{c}\phi$
the linearized and self-adjoint operator around the periodic wave $\phi$ and
suppose that $\ker(\mathcal{\tilde{L}}|_{X_{0}})=[\phi^{\prime}]$. Thus, a
direct application of the implicit function theorem as in [31, Lemma 3.8]
gives us that $c\in(\frac{1}{2},+\infty)\mapsto\phi\in
H_{per}^{\infty}(\mathbb{T})$ defines a smooth curve of periodic waves all of
them with the same period $2\pi$. Thus, one has that $\frac{d}{dc}\phi\in
D(\tilde{\mathcal{L}})=H_{per}^{\alpha}(\mathbb{T})$,

$\displaystyle\mathcal{\tilde{L}}\left(\frac{d}{dc}\phi\right)=\frac{1}{c^{2}}\left(A(c)-\phi-\frac{1}{2}\phi^{2}-cA^{\prime}(c)\right),\
\ \
\mathcal{\tilde{L}}\left(\frac{1}{c}\right)=\frac{1}{c^{2}}\left(c-1-\phi\right)$
(3.10)

and

$\displaystyle\mathcal{\tilde{L}}\left(\frac{1}{c}\phi\right)=-\frac{1}{c^{2}}\left(\frac{1}{2}\phi^{2}+A(c)\right).$
(3.11)

Equations in (3.10) and (3.11) give us an important relation

$c^{2}\mathcal{\tilde{L}}\left(\frac{d}{dc}\phi-\frac{1}{c}-\frac{1}{c}\phi\right)=\left(1+2A(c)-c-cA^{\prime}(c)\right)=d.$
(3.12)

Since
$\mathcal{\tilde{L}}\phi=-\frac{1}{c}\left(\frac{1}{2}\phi^{2}+A(c)\right)$,
we have $\langle\mathcal{\tilde{L}}\phi,\phi\rangle<0$ and we deduce that
$\mathcal{\tilde{L}}$ has at least one negative eigenvalue. Next result
establishes items i) and ii) of Theorem 1.3 by giving the precise behaviour of
the first eigenvalues associated with the linearized operator
$\mathcal{\tilde{L}}$.

######  Proposition 3.1.

Let $\alpha\in\left(\frac{1}{3},2\right]$ be fixed and assume that
$\ker(\mathcal{\tilde{L}}|_{X_{0}})=[\phi^{\prime}]$. For $c>\frac{1}{2}$
consider $d=1+2A(c)-c-cA^{\prime}(c)$. The linearized operator
$\mathcal{\tilde{L}}=D^{\alpha}+(1-\frac{1}{c})-\frac{1}{c}\phi$ around the
periodic solution $\phi$ satisfies

$\displaystyle
n(\mathcal{\tilde{L}})=\left\\{\begin{array}[]{ll}1,&\textrm{if}\ d<0,\\\
2,&\textrm{if}\ d\geq 0\end{array}\right.$ (3.13)

and

$\displaystyle
z(\mathcal{\tilde{L}})=\left\\{\begin{array}[]{ll}1,&\textrm{if}\ d\neq 0,\\\
2,&\textrm{if}\ d=0.\end{array}\right.$ (3.14)

###### Proof.

In view of (3.2), since $\phi$ locally minimizes $\mathcal{B}_{c}$, we have
$\mathcal{\tilde{L}}|_{\\{1,\phi^{2}\\}^{\bot}}\geq 0,$ that is,
$n(\mathcal{\tilde{L}}|_{\\{1,\phi^{2}\\}^{\bot}})=0$ and
$n(\mathcal{\tilde{L}})\in\\{1,2\\}$, since
$\langle\mathcal{\tilde{L}}\phi,\phi\rangle=-2\mathcal{B}_{c}(\phi)<0$.

For $\lambda\notin\sigma(\mathcal{\tilde{L}})$, let $R(\lambda)$ be the
following matrix given by

$R(\lambda):=\left[\begin{array}[]{c c
}\langle(\mathcal{\tilde{L}}-\lambda)^{-1}1,1\rangle&\langle(\mathcal{\tilde{L}}-\lambda)^{-1}\phi^{2},1\rangle\\\
&\\\
\langle(\mathcal{\tilde{L}}-\lambda)^{-1}\phi^{2},1\rangle&\langle(\mathcal{\tilde{L}}-\lambda)^{-1}\phi^{2},\phi^{2}\rangle\end{array}\right].$
(3.15)

For $d\neq 0$, we compute $R(\lambda)$ at $\lambda=0$. In fact,

$\langle\mathcal{\tilde{L}}^{-1}\phi^{2},\phi^{2}\rangle=\displaystyle-4c^{2}\int_{-\pi}^{\pi}(D^{\frac{\alpha}{2}}\phi)^{2}dx-8\pi
A(c)\left(\frac{A(c)}{c}d^{-1}+2(c^{2}-c)\right)$ (3.16)

$\langle\mathcal{\tilde{L}}^{-1}1,\phi^{2}\rangle=\langle\mathcal{\tilde{L}}^{-1}\phi^{2},1\rangle=\frac{4\pi
A(c)}{c}d^{-1},$ (3.17)

and

$\langle\mathcal{\tilde{L}}^{-1}1,1\rangle=\displaystyle\displaystyle-\frac{2\pi}{c}d^{-1}.$
(3.18)

In $(\ref{op1})$, $(\ref{op2})$ and $(\ref{third-term1})$,
$\mathcal{\tilde{L}}^{-1}\phi^{2}$ and $\mathcal{\tilde{L}}^{-1}1$ are
calculated by combining (3.10) and (3.11) to obtain

$\mathcal{\tilde{L}}\left(d^{-1}\left(\frac{d}{dc}\phi-\frac{1}{c}-\frac{1}{c}\phi\right)\right)=1,$
(3.19)

and

$\mathcal{\tilde{L}}\left(-2A(c)\left(d^{-1}\left(\frac{d}{dc}\phi-\frac{1}{c}-\frac{1}{c}\phi\right)\right)-2c\phi\right)=\phi^{2}.$
(3.20)

The determinant of $R(0)$ is now given by

$\displaystyle\det R(0)=8\pi
d^{-1}\left(c\int_{-\pi}^{\pi}(D^{\frac{\alpha}{2}}\phi)^{2}dx+(c-1)\int_{-\pi}^{\pi}\phi^{2}dx\right)=16\pi
d^{-1}\mathcal{B}_{c}(\phi).$ (3.21)

Since $c>\frac{1}{2}$, we obtain that $\mathcal{B}_{c}(\phi)>0$ and the sign
of $R(0)$ is obtained only by $d^{-1}$.
On the other hand, by [33, Theorem 4.1] we have the following identities

$n(\mathcal{{\tilde{L}}}|_{\\{1,\phi^{2}\\}\bot})=n(\mathcal{\tilde{L}})-n_{0}-z_{0},$
(3.22)

and

$z(\mathcal{{\tilde{L}}}|_{\\{1,\phi^{2}\\}\bot})=z(\mathcal{\tilde{L}})+z_{0}-z_{\infty}$
(3.23)

where $n_{0}$ is the number of negative eigenvalues of $R(0)$, $z_{0}$ denotes
the dimension of the kernel of $R(0)$ and $z_{\infty}$ corresponds the number
of diverging eigenvalues, that is, $z_{\infty}=1$ if $d=0$, and $z_{\infty}=0$
otherwise. We then deduce the result (3.22) from the equation (3.13).
Finally, since $\phi$ is a single-lobe solution, we see that
$\phi^{\prime}\in\ker(\tilde{\mathcal{L}})$ has only two zeroes in the
interval $[-\pi,\pi)$. This means, from the Oscillation Theorem in [23], that
$z(\tilde{\mathcal{L}})\in\\{1,2\\}$. If $d\neq 0$, we deduce from the second
equality in $(\ref{eq15})$, $(\ref{eqmatrix})$ and $(\ref{eqmatrix2})$ that
$\\{1,\phi,\phi^{2}\\}\subset{\rm range}(\tilde{\mathcal{L}})$, so that
$z(\tilde{\mathcal{L}})=1$ by using [23, Proposition 3.1]. If $d=0$, we have
$z_{\infty}=1$ and since $\phi^{\prime}\in\\{1,\phi^{2}\\}^{\bot}$, one has
$z(\mathcal{{\tilde{L}}}|_{\\{1,\phi^{2}\\}\bot})\geq 1$. Thus,
$z(\tilde{\mathcal{L}})=2$ as requested. ∎

######  Remark 3.1.

Let $\alpha\in\left(\frac{1}{3},2\right]$ be fixed. If
$\ker(\mathcal{L}|_{X_{0}})=[\phi^{\prime}]$, by [31, Lemma 3.8] we obtain the
existence of a smooth curve $\omega\in I\mapsto\psi_{\omega}\in
H_{per}^{\infty}(\mathbb{T})\cap X_{0,e}$ of periodic waves for the equation
$(\ref{ode2})$, where $I\subset(\frac{1}{2},+\infty)$ is a convenient open
interval containing $c$. Important to mention that we can not assure if
$\psi_{\omega}$ solves the minimization problem $(\ref{minB})$, neither that
it is a single-lobe, except at $\omega=c$, where $\psi_{\omega}=\phi$.

###  3.2. Uniqueness of periodic minimizers

We give sufficient conditions to determine a result of uniqueness for the
periodic minimizer $\phi$ obtained in Lemma 3.2. To do so, we will follow
similar arguments in [15, Section 5].
Let us consider
$\mathcal{L}|_{X_{0}}=\mathcal{L}+\frac{1}{2\pi}\langle\phi,\cdot\rangle$.
Throughout this section we assume that

$z(\mathcal{L}|_{X_{0}})=1,$ (3.24)

for every $c>\frac{1}{2}$ and $\alpha\in\left(\frac{1}{2},2\right]$. Using
that $\phi\in X_{0}$ we obtain
$\langle\mathcal{L}|_{X_{0}}\phi,\phi\rangle=\langle\mathcal{L}\phi,\phi\rangle=-\frac{1}{2}\int_{-\pi}^{\pi}\phi^{3}dx=-2\mathcal{B}_{c}(\phi)<0$,
so that $n(\mathcal{L}|_{X_{0}})\geq 1$. By [31, Corollary 4.5], we can obtain
automatically the condition $(\ref{hyp-uniq})$ by assuming the additional
hypothesis $n(\mathcal{L})=1$ for every $c>\frac{1}{2}$ and
$\alpha\in\left(\frac{1}{2},2\right]$. Equality in $(\ref{hyp-uniq})$ enables
us to conclude an important property for $\mathcal{L}|_{X_{0}}$ as

$\ker(\mathcal{L}|_{X_{0}})=[\phi^{\prime}].$ (3.25)

To simplify the notation, we define the real Banach space

$V=\\{f\in L_{per}^{3}(\mathbb{T})\cap X_{0};\ f\ \mbox{is even and real-
valued}\\}$ (3.26)

whose norm, since $L_{per}^{3}(\mathbb{T})\hookrightarrow
L_{per}^{2}(\mathbb{T})$, is given by
$||f||_{V}:=||f||_{L_{per}^{3}(\mathbb{T})}.$ For a fixed
$\alpha_{0}\in(\frac{1}{2},2]$, our first goal is to construct a local branch
of solutions $(\phi_{\alpha},c_{\alpha})\in
V\times\left(\frac{1}{2},+\infty\right)$ of $(\ref{ode2})$ parameterized by
the index $\alpha$ in some interval $[\alpha_{0},\alpha_{0}+\delta)$, for some
$\delta>0$ small enough.

######  Proposition 3.2.

Let $\alpha_{0}\in(\frac{1}{2},2]$. Suppose that $(\phi_{0},c_{0})\in
V\times\left(\frac{1}{2},+\infty\right)$ with $\phi_{0}$ being a nonzero
solution of $(\ref{ode2})$ and satisfying assumption $(\ref{hyp-uniq})$ with
$\alpha=\alpha_{0}$ and $c=c_{0}$. Then, for some $\delta>0$, there exists a
$C^{1}-$map $\alpha\in I\rightarrow(\phi_{\alpha},c_{\alpha})\in
V\times\left(\frac{1}{2},+\infty\right)$, defined in the interval
$I=[\alpha_{0},\alpha_{0}+\delta)$, such that the following holds:

  * (i)

$(\phi_{\alpha},c_{\alpha})$ solves equation $(\ref{ode2})$ with
$c=c_{\alpha}$, for all $\alpha\in I$ and the pair
$(\phi_{\alpha},c_{\alpha})$ satisfies $(\ref{hyp-uniq})$.

  * (ii)

There exists $\varepsilon>0$ such that $(\phi_{\alpha},c_{\alpha})$ is the
unique solution of $(\ref{ode2})$ for $\alpha\in I$ in the neighborhood
$\\{(\phi,c)\in V\times\left(\frac{1}{2},+\infty\right);\
||\phi-\phi_{0}||_{V}+|c-c_{0}|<\varepsilon\\}$.

  * (iii)

For all $\alpha\in I$, we have
$\int_{-\pi}^{\pi}\phi_{\alpha}^{3}dx=\int_{-\pi}^{\pi}\phi_{0}^{3}dx.$

###### Proof.

The proof of this result relies on the implicit function theorem. Since
$c>\frac{1}{2}$, we obtain that $(\ref{ode2})$ can be written as
$\phi=\left(cD^{\alpha}+\left(c-1\right)\right)^{-1}\frac{1}{2}\Pi_{0}\phi^{2}.$
For some $\delta>0$ to be chosen later, let us define the mapping

$\mathcal{F}:V\times\left(\frac{1}{2},+\infty\right)\times I\rightarrow
V\times\mathbb{R},$ (3.27)

given by

$\mathcal{F}(\phi,c,\alpha)=\left[\begin{array}[]{ccccc}\displaystyle\phi-\left(cD^{\alpha}+\left(c-1\right)\right)^{-1}\frac{1}{2}\Pi_{0}\phi^{2}\\\
\\\
\displaystyle\int_{-\pi}^{\pi}\phi^{3}dx-\int_{-\pi}^{\pi}\phi_{0}^{3}dx\end{array}\right].$
(3.28)

We see that $\mathcal{F}$ is a well defined $C^{1}$ map and
$\mathcal{F}(\phi_{0},c_{0},\alpha_{0})=0$. Our intention is to show the
invertibility of the Fréchet derivative of $\mathcal{F}$ with respect
$(\phi,c)$ at $(\phi_{0},c_{0},\alpha_{0})$. In fact, we see that

$D_{\phi,c}\mathcal{F}(\phi_{0},c_{0},\alpha_{0})=\left[\begin{array}[]{ccccc}1-(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-1}\Pi_{0}\phi_{0}&&\displaystyle(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-2}(D^{\alpha}+1)\frac{1}{2}\Pi_{0}\phi_{0}^{2}\\\
\\\ \displaystyle 3\langle\phi_{0}^{2},\cdot\rangle&&0\end{array}\right].$

We claim that $D_{\phi,c}\mathcal{F}$ is invertible at
$(\phi_{0},c_{0},\alpha_{0})$. To do so, for every $g\in V$ and
$\lambda\in\mathbb{R}$ given, we need to show the existence of a unique pair
$(f,\beta)\in V\times\mathbb{R}$ such that

$\left[\begin{array}[]{ccccc}\displaystyle
f-(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-1}\Pi_{0}(\phi_{0}f)+\frac{\beta}{2}(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-2}(D^{\alpha}+1)\Pi_{0}\phi_{0}^{2}\\\
\\\ \displaystyle
3\langle\phi_{0}^{2},f\rangle\end{array}\right]=\left[\begin{array}[]{llll}g\\\
\\\ \lambda\end{array}\right].$ (3.29)

In order to simplify the notation, we define
$\mathcal{P}=-(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-1}\Pi_{0}\phi_{0}$ and
$h_{0}=\frac{1}{2}(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-2}(D^{\alpha}+1)\Pi_{0}\phi_{0}^{2}$.
By $(\ref{fr-der-uniq2})$ we see that

$(1+\mathcal{P})f+\beta h_{0}=g,$ (3.30)

$3\int_{-\pi}^{\pi}\phi_{0}^{2}fdx=\lambda.$ (3.31)

Now, $\mathcal{P}$ is a compact operator on $X_{0,e}=\\{f\in X_{0};\ f\
\mbox{is even}\\}$ and from $(\ref{hyp-uniq})$ we obtain that $-1$ is not an
element of the spectrum of $\mathcal{P}$. Thus, $1+\mathcal{P}$ is an
invertible operator on $X_{0,e}$ and since $\mathcal{P}:V\rightarrow V$ (this
fact follows similarly from [15, Lemma E.1]), we obtain that
$(1+\mathcal{P})^{-1}$ exists on $V$. Thus, we can express $f$ uniquely as

$f=-\beta(1+\mathcal{P})^{-1}h_{0}+(1+\mathcal{P})^{-1}g.$ (3.32)

Gathering results in $(\ref{fr-der-uniq4})$ and $(\ref{f-uniq1})$, we deduce

$3\beta\int_{-\pi}^{\pi}\phi_{0}^{2}(1+\mathcal{P})^{-1}h_{0}dx=-\lambda+3\int_{-\pi}^{\pi}\phi_{0}^{2}(1+\mathcal{P})^{-1}gdx.$
(3.33)

By $(\ref{hyp-uniq2})$, we see that $(\mathcal{L}|_{X_{0}})^{-1}$ exists on
$X_{0,e}$ and therefore, on $H_{per}^{\alpha_{0}}(\mathbb{T})\cap X_{0,e}$

$(1+\mathcal{P})^{-1}=(\mathcal{L}|_{X_{0}})^{-1}(c_{0}D^{\alpha_{0}}+(c_{0}-1))$
(3.34)

is well defined. To show that $\beta$ can be explicitly expressed in terms of
$\phi_{0}$, $\mathcal{P}$ and $g$, we need to prove that
$\langle\phi_{0}^{2},(1+\mathcal{P})^{-1}h_{0}\rangle\neq 0$. Indeed, using
$(\ref{eq-inv-uniq})$, the fact that
$\mathcal{L}|_{X_{0}}\phi_{0}=-\frac{1}{2}\Pi_{0}\phi_{0}^{2}$ and since
$h_{0}\in V$, we see from $(\ref{inp-uniq1})$ that

$\displaystyle\langle\phi_{0}^{2},(1+\mathcal{P})^{-1}h_{0}\rangle$
$\displaystyle=$ $\displaystyle\langle
A(c_{0}),(1+\mathcal{P})^{-1}h_{0}\rangle-2\langle\mathcal{L}|_{X_{0}}\phi_{0},(1+\mathcal{P})^{-1}h_{0}\rangle$
$\displaystyle=$
$\displaystyle-2\langle\mathcal{L}|_{X_{0}}\phi_{0},(\mathcal{L}|_{X_{0}})^{-1}(c_{0}D^{\alpha_{0}}+(c_{0}-1))h_{0}\rangle$
$\displaystyle=$
$\displaystyle-\langle(c_{0}D^{\alpha_{0}}+(c_{0}-1))\phi_{0},(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-2}(D^{\alpha}+1)\Pi_{0}\phi_{0}^{2}\rangle$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\langle\Pi_{0}\phi_{0}^{2},(c_{0}D^{\alpha_{0}}+(c_{0}-1))^{-2}(D^{\alpha}+1)\Pi_{0}\phi_{0}^{2}\rangle$
$\displaystyle=$ $\displaystyle-\pi\sum_{k\neq
0}\frac{|k|^{\alpha_{0}}+1}{(c_{0}|k|^{\alpha_{0}}+(c_{0}-1))^{2}}|\widehat{\Pi_{0}\phi_{0}^{2}}(k)|^{2}\neq
0,$

where $\widehat{f}$ indicates the periodic Fourier transform for a function
$f\in L_{per}^{2}(\mathbb{T})$. The rest of the proof can be done by a direct
application of the implicit function theorem. ∎

The next step is to follow similar arguments as found in [15, Subsection 5.2].
In fact, we consider the corresponding maximal extension of the branch
$(\phi_{\alpha},c_{\alpha})$ for $\alpha\in[\alpha_{0},\alpha_{*})$, where
$\alpha_{*}$ is given by

$\begin{array}[]{rrrr}\alpha_{*}:=\sup\\{q;\ \alpha_{0}<q<2,\
(\phi_{\alpha},c_{\alpha})\in
C^{1}([\alpha_{0},q);V\times\left(\frac{1}{2},+\infty\right))\ \mbox{given by
Proposition \ref{prop1-uniq}}\\\ \\\ \mbox{and}\ (\phi_{\alpha},c_{\alpha})\
\mbox{satisfies (\ref{ode2}) for}\ \alpha\in[\alpha_{0},q)\\}.\end{array}$

It is clear that $\alpha_{*}\leq 2$ and we prove that $\alpha_{*}=2$.

######  Proposition 3.3.

Let $\\{\alpha_{n}\\}_{n=1}^{n=+\infty}\subset(\frac{1}{2},\alpha_{*})$ be a
sequence such that $\alpha_{n}\rightarrow\alpha_{*}$. Furthermore, we assume
that $\phi_{\alpha_{n}}\in X_{0,e}$ are the corresponding solutions obtained
in Proposition 3.2 with wave speed $c_{\alpha_{n}}$. Up to a subsequence, it
follows that

$\phi_{\alpha_{n}}\rightarrow\phi_{*}\ \mbox{in}\ V\cap
L_{per}^{4}(\mathbb{T})\ \ \mbox{and}\ \ c_{\alpha_{n}}\rightarrow c_{*}$

where $c_{*}\in\left(\frac{1}{2},+\infty\right)$ and $\phi_{*}$ satisfy
$(\ref{ode2})$. Moreover, the corresponding maximal branch
$(\phi_{\alpha},c_{\alpha})\in
C^{1}([\alpha_{0},\alpha_{*});V\times\left(\frac{1}{2},+\infty\right))$
extends to $\alpha_{*}=2$.

###### Proof.

First, let us suppose that
$\\{c_{\alpha_{n}}\\}_{n=1}^{+\infty}\subset\left(\frac{1}{2},+\infty\right)$
is not a bounded sequence. For all $M>0$ large enough, there exists an index
$m:=n_{M}\in\mathbb{N}$ such that $c_{\alpha_{m}}>M$. Thus, from
$(\ref{ode2})$ and the fact that
$\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{3}dx=\int_{-\pi}^{\pi}\phi_{0}^{3}>0$ for
all $n\in\mathbb{N}$, we obtain

$0\leq\int_{-\pi}^{\pi}(D^{\alpha_{m}/2}\phi_{\alpha_{m}})^{2}dx\leq\frac{1}{2M}\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{3}dx\leq\int_{-\pi}^{\pi}\phi_{0}^{3}dx.$

Since $\alpha_{m}\geq\alpha_{0}$, the Sobolev embedding
$H_{per}^{\alpha_{m}/2}(\mathbb{T})\cap X_{0,e}\hookrightarrow
H_{per}^{\alpha_{0}/2}(\mathbb{T})\cap X_{0,e}$ for all
$\alpha_{0}>\frac{1}{2}$ is valid. Now,
$f\mapsto||D^{s}f||_{L_{per}^{2}(\mathbb{T})}$ is an equivalent norm on the
space $H_{per}^{s}(\mathbb{T})\cap X_{0,e}$ for all $s>0$, and thus
$\\{\phi_{\alpha_{m}}\\}_{n}$ is a bounded sequence on
$H_{per}^{\alpha_{0}/2}(\mathbb{T})\cap X_{0,e}$. Since
$H_{per}^{\alpha_{0}/2}(\mathbb{T})\cap X_{0,e}$ is compactly embedded into
$V\hookrightarrow X_{0,e}$ and into $L_{per}^{4}(\mathbb{T})$, there exists
$\phi_{0}\in V\cap L_{per}^{4}(\mathbb{T})$ such that, up to a subsequence
$\int_{-\pi}^{\pi}\phi_{\alpha_{m}}^{3}dx\rightarrow\int_{-\pi}^{\pi}\phi_{0}^{3}dx>0$,
$\int_{-\pi}^{\pi}\phi_{\alpha_{m}}^{4}dx\rightarrow\int_{-\pi}^{\pi}\phi_{0}^{4}dx>0$
and
$\int_{-\pi}^{\pi}\phi_{\alpha_{m}}^{2}dx\rightarrow\int_{-\pi}^{\pi}\phi_{0}^{2}dx>0$,
as $m\rightarrow+\infty$. By $(\ref{ode2})$ and the Poincaré-Wirtinger
inequality, we deduce

$0\leq(2M-1)\int_{-\pi}^{\pi}\phi_{\alpha_{m}}^{2}dx\leq(2c_{\alpha_{m}}-1)\int_{-\pi}^{\pi}\phi_{\alpha_{m}}^{2}dx\leq\frac{1}{2}\int_{-\pi}^{\pi}\phi_{0}^{3}dx.$
(3.35)

When $m\rightarrow+\infty$, we obtain from $(\ref{est-bound-seq-uniq1})$ a
contradiction, so that $\\{c_{\alpha_{n}}\\}_{n=1}^{+\infty}$ is a bounded
sequence on $\mathbb{R}$. Therefore, there exists a
$c_{*}\in\left[\frac{1}{2},+\infty\right)$ such that, up to a subsequence
$c_{\alpha_{n}}\rightarrow c_{*}.$ On the other hand, the fact that
$c_{\alpha_{n}}\rightarrow c_{*}\geq\frac{1}{2}$ enables us to obtain,

$0\leq(2c_{\alpha_{n}}-1)||\phi_{\alpha_{n}}||_{L_{per}^{2}}^{2}\leq
2\mathcal{B}_{c_{\alpha_{n}}}(\phi_{\alpha_{n}})=\frac{1}{2}\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{3}dx=\frac{1}{2}\int_{-\pi}^{\pi}\phi_{0}^{3}dx,$

using again the Poincaré-Wirtinger inequality. Similarly as determined above,
we guarantee the existence of $\phi_{*}\in V\cap L_{per}^{4}(\mathbb{T})$ such
that

$\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{3}dx\rightarrow\int_{-\pi}^{\pi}\phi_{*}^{3}dx,\
\ \
\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{2}dx\rightarrow\int_{-\pi}^{\pi}\phi_{*}^{2}dx,\
\ \ \mbox{and}\ \ \
\int_{-\pi}^{\pi}\phi_{\alpha_{n}}^{4}dx\rightarrow\int_{-\pi}^{\pi}\phi_{*}^{4}dx.$
(3.36)

Next, from the equation $(\ref{ode2})$, we obtain

$D^{\alpha_{n}}\phi_{\alpha_{n}}=-\left(1-\frac{1}{c_{\alpha_{n}}}\right)\phi_{\alpha_{n}}+\frac{1}{2c_{\alpha_{n}}}\phi_{\alpha_{n}}^{2}-\frac{A(c_{\alpha_{n}})}{c_{\alpha_{n}}},$

so that $\\{\phi_{\alpha_{n}}\\}_{n}$ is bounded in
$H_{per}^{\alpha_{n}}(\mathbb{T})$. Since $\alpha_{n}>\frac{1}{2}$, it follows
that $H_{per}^{\alpha_{n}}(\mathbb{T})$ is a Banach algebra and thus
$\\{\phi_{\alpha_{n}}^{2}\\}_{n}$ is bounded in
$H_{per}^{\alpha_{n}}(\mathbb{T})$. From

$D^{2\alpha_{n}}\phi_{\alpha_{n}}=-\left(1-\frac{1}{c_{\alpha_{n}}}\right)D^{\alpha_{n}}\phi_{\alpha_{n}}+\frac{1}{2c_{\alpha_{n}}}D^{\alpha_{n}}\phi_{\alpha_{n}}^{2},$
(3.37)

one has that $\\{\phi_{\alpha_{n}}\\}_{n}$ is bounded in
$H_{per}^{2\alpha_{n}}(\mathbb{T})$. Following with this inductive process,
there is $r\in\mathbb{N}$ large enough such that $r\alpha_{n}>\alpha_{*}$ and
$\\{\phi_{\alpha_{n}}\\}_{n}$ is bounded in
$H_{per}^{r\alpha_{n}}(\mathbb{T})$. By the compact embedding
$H_{per}^{r\alpha_{n}}(\mathbb{T})\hookrightarrow
H_{per}^{\alpha_{*}}(\mathbb{T})$, one has (modulus to a subsequence)

$\phi_{\alpha_{n}}\rightarrow\phi_{*}\ \mbox{in}\
H_{per}^{\alpha_{*}}(\mathbb{T}),\ \mbox{as}\ n\rightarrow+\infty.$ (3.38)

Therefore,

$\begin{array}[]{llll}||D^{\alpha_{n}}\phi_{\alpha_{n}}-D^{\alpha_{*}}\phi_{{*}}||_{L_{per}^{2}}\leq||D^{\alpha_{*}}(\phi_{\alpha_{n}}-\phi_{{*}})||_{L_{per}^{2}}+||D^{\alpha_{n}}\phi_{*}-D^{\alpha_{*}}\phi_{{*}}||_{L_{per}^{2}}\end{array}$
(3.39)

We obtain from $(\ref{equalitywave2})$, $(\ref{equalitywave3})$ and the fact
that $\alpha_{n}\rightarrow\alpha_{*}$,

$D^{\alpha_{n}}\phi_{\alpha_{n}}\rightarrow D^{\alpha_{*}}\phi_{{*}}\
\mbox{in}\ L_{per}^{2}(\mathbb{T}),\ \mbox{as}\ n\rightarrow+\infty.$ (3.40)

Since $c_{\alpha_{n}}\rightarrow c_{*}$, the second and third convergences in
$(\ref{conv1})$ give us by $(\ref{equalitywave4})$ that $\phi_{*}$ satisfies
the equation

$c_{*}D^{\alpha_{*}}\phi_{*}+(c_{*}-1)\phi_{*}-\frac{1}{2}\phi_{*}^{2}+A(c_{*})=0.$
(3.41)

To prove $\alpha_{*}=2$, since $\phi_{*}$ satisfies $(\ref{ode123})$,
Proposition 3.2 allows to extend the branch $(\phi_{\alpha},c_{\alpha})$
beyond $\alpha_{*}$. This fact contradicts the maximality property of
$\alpha_{*}$. This fact proves that $\alpha_{*}=2$.
It remains to establish $c_{*}>\frac{1}{2}$. First of all, we obtain the
smoothness of $\phi_{*}$ by a standard bootstrapping argument. If
$c_{*}=\frac{1}{2}$, the only possibility for the smooth solution $\phi_{*}\in
H_{per}^{\infty}(\mathbb{T})\cap X_{0,e}$ of $(\ref{ode123})$ with
$\alpha_{*}=2$ is that $\phi_{*}\equiv 0$ (see Remark 3.4 below). This fact
generates a contradiction since $\int_{-\pi}^{\pi}\phi_{*}^{3}dx>0$.

∎

######  Remark 3.4.

The explicit solution for the equation $(\ref{ode123})$ with $\alpha_{*}=2$
depends on the Jacobi elliptic function of dnoidal type and it is given by

$\phi_{*}(x)=a_{*}\left({\rm
dn}^{2}\left(\frac{K(\kappa)}{\pi}x,\kappa\right)-\frac{E(\kappa)}{K(\kappa)}\right),$
(3.42)

where

$a_{*}=12c_{*}\frac{K(\kappa)^{2}}{\pi^{2}}.$

The value of $c_{*}$ can be expressed by

$c_{*}={\frac{{\pi}^{2}}{\left(4\,{\kappa}^{2}-8\right){\it
K}(\kappa)^{2}+12\,{\it K}(\kappa){\it E}\left(\kappa\right)+{\pi}^{2}}},$
(3.43)

where $K$ and $E$ are, respectively, the complete elliptic integral of first
and second kind. Both functions depending on the modulus of the Jacobi
elliptic function $\kappa\in(0,1)$. For this solution, the integration
constant

$A(c_{*})=\frac{1}{16}\frac{(24\kappa^{2}-24)K(\kappa)^{4}+(96-48k^{2})E(\kappa)K(\kappa)^{3}-72K(\kappa)^{2}E(\kappa)^{2}}{\left((\kappa^{2}-2)K(\kappa)^{2}+3E(\kappa)K(\kappa)+\frac{\pi^{2}}{4}\right)^{2}}$
(3.44)

is obtained by using symbolic computations in Maple. By the explicit
expressions given by $(\ref{cvalue})$ and $(\ref{constant})$, respectively, it
is easy to see that $c_{*}$ and $A(c_{*})$ are strictly monotonic functions in
terms of $\kappa\in(0,\kappa_{0})$, where $\kappa_{0}\approx 0.994$ is the
unique zero of the function $p(\kappa)=\left(4\,{\kappa}^{2}-8\right){\it
K}(\kappa)^{2}+12\,{\it K}(\kappa){\it E}\left(\kappa\right)+{\pi}^{2}$.
Furthermore, we have

$c_{*}\rightarrow\frac{1}{2}^{+}\ \mbox{and}\ A(c_{*})\rightarrow 0^{+},\
\mbox{as}\ \kappa\rightarrow 0^{+}.$

From $(\ref{cvalue})$, we see that the limit $c_{*}>\frac{1}{2}$ is
independent of the sequence $\\{\alpha_{n}\\}_{n=1}^{+\infty}$ presented by
the Proposition 3.3. Furthermore, since $\phi_{*}$ is unique for each fixed
$c_{*}$ and satisfying $c_{*}>\frac{1}{2}$, we conclude that the limit
$\phi_{*}$ is also independent of $\\{\alpha_{n}\\}_{n=1}^{+\infty}$.

Collecting all the results enunciated above, we can prove our uniqueness
result to establish the precise statement in Theorem 1.3-iii).

######  Proposition 3.5.

Let $\alpha\in(\frac{1}{2},2]$ be fixed. If $(\ref{hyp-uniq})$ is valid, then
the solution obtained in Lemma 3.2 is unique.

###### Proof.

The proof of this result has the same spirit as determined in [15, Theorem
2.4]. Suppose that $\phi_{0}$ and $\tilde{\phi}_{0}$ are solutions of the
problem $(\ref{minp})$ satisfying $\phi_{0}\not\equiv\tilde{\phi}_{0}$. Since
both $\phi_{0}$ and $\tilde{\phi}_{0}$ are in $Y_{0}$, we obtain that
$\int_{-\pi}^{\pi}\phi_{0}dx=\int_{-\pi}^{\pi}\tilde{\phi}_{0}dx=0$ and

$\int_{-\pi}^{\pi}\phi_{0}^{3}dx=\int_{-\pi}^{\pi}\tilde{\phi}_{0}^{3}dx$
(3.45)

For the case of solitary waves, equality $(\ref{L3uniq})$ is one of the
crucial parts in the uniqueness proof contained in [15] since they use the
classical Pohozaev equality to reach the result. According to our best
knowledge, it is well known that we do not have a similar equality in the
periodic context, so that equality $(\ref{L3uniq})$ is essential for our
purpose. By Proposition 3.3, there exist two smooth branch of solutions
$(\phi_{\alpha},c_{\alpha}),\ (\tilde{\phi}_{\alpha},\tilde{c}_{\alpha})\in
C^{1}([\alpha_{0},2);V\times\left(\frac{1}{2},+\infty\right))$ satisfying
(1.8) with $c_{\alpha},\
\tilde{c}_{\alpha}\in\left(\frac{1}{2},+\infty\right)$,
$(\phi_{\alpha_{0}},c_{\alpha_{0}}):=(\phi_{0},c_{0})$ and
$(\tilde{\phi}_{\alpha_{0}},\tilde{c}_{\alpha_{0}}):=(\tilde{\phi}_{0},c_{0})$.
Notice that by the local uniqueness obtained from Proposition 3.2, the smooth
branches $(\phi_{\alpha},c_{\alpha})$ and
$(\tilde{\phi}_{\alpha},\tilde{c}_{\alpha})$ do not have a common point. Thus,
for each branch $(\phi_{\alpha},c_{\alpha})$ and
$(\tilde{\phi}_{\alpha},\tilde{c}_{\alpha})$, we have by Proposition 3.3 the
existence of $(\phi_{*},c_{*})$ and $(\tilde{\phi}_{*},\tilde{c}_{*})$
satisfying the following ordinary differential equations

$-c_{*}\phi^{\prime\prime}_{*}+(c_{*}-1)\phi_{*}-\frac{1}{2}\phi_{*}^{2}+A(c_{*})=0$
(3.46)

and

$-\tilde{c}_{*}\tilde{\phi}^{\prime\prime}_{*}+(\tilde{c}_{*}-1)\tilde{\phi}_{*}-\frac{1}{2}\tilde{\phi}_{*}^{2}+A(\tilde{c}_{*})=0,$
(3.47)

respectively. Moreover, by Proposition 3.2-(iii), one has

$\displaystyle\int_{-\pi}^{\pi}\phi_{\alpha}^{3}dx=\int_{-\pi}^{\pi}\phi_{0}^{3}dx=\int_{-\pi}^{\pi}\tilde{\phi}_{0}^{3}dx=\int_{-\pi}^{\pi}\tilde{\phi}_{\alpha}^{3}dx.$
(3.48)

Again, using compactness arguments as done in the proof of Proposition 3.3, it
follows that
$\int_{-\pi}^{\pi}\phi_{\alpha}^{3}dx\rightarrow\int_{-\pi}^{\pi}\phi_{*}^{3}dx$
and
$\int_{-\pi}^{\pi}\tilde{\phi}_{\alpha}^{3}dx\rightarrow\int_{-\pi}^{\pi}\tilde{\phi}_{*}^{3}dx$,
as $\alpha\rightarrow 2$. By (3.48) we have

$\int_{-\pi}^{\pi}\phi_{*}^{3}dx=\int_{-\pi}^{\pi}\tilde{\phi}_{*}^{3}dx.$
(3.49)

We claim that $c_{*}=\tilde{c}_{*}$. Indeed, for $\alpha_{*}=2$, let us
consider $\varphi:=\varphi_{\omega}$ a general zero mean solution for the
equation $(\ref{eq-sol})$ with wave speed

$\omega:=\omega_{\kappa}={\frac{{\pi}^{2}}{\left(4\,{\kappa}^{2}-8\right){\it
K}(\kappa)^{2}+12\,{\it K}(\kappa){\it E}\left(\kappa\right)+{\pi}^{2}}},$

instead of $c_{*}$. Deriving equation (3.46) with respect to $\omega$, we
obtain

$\displaystyle-\omega\frac{d}{d\omega}\varphi^{\prime\prime}-\omega\varphi^{\prime\prime}+(\omega-1)\frac{d}{d\omega}\varphi+\varphi-\varphi\frac{d}{d\omega}\varphi+\frac{d}{d\omega}A(\omega)=0.$
(3.50)

Next, multiplying equation (3.50) by $\varphi$, integrating the result over
$[-\pi,\pi]$, we have after an integration by parts that

$\displaystyle\frac{\omega}{2}\frac{d}{d\omega}\left(\int_{-\pi}^{\pi}(\varphi^{\prime})^{2}+\varphi^{2}dx\right)-\frac{1}{2}\frac{d}{d\omega}\int_{-\pi}^{\pi}\varphi^{2}dx+\int_{-\pi}^{\pi}(\varphi^{\prime})^{2}+\varphi^{2}dx-\frac{1}{3}\frac{d}{d\omega}\int_{-\pi}^{\pi}\varphi^{3}dx=0.$
(3.51)

On the other hand, multiplying (3.46) by $\varphi$ and integrating the final
result over $[-\pi,\pi]$, it follows that

$\displaystyle\omega\left(\int_{-\pi}^{\pi}(\varphi^{\prime})^{2}+\varphi^{2}dx\right)-\int_{-\pi}^{\pi}\varphi^{2}dx-\frac{1}{2}\int_{-\pi}^{\pi}\varphi^{3}dx=0.$
(3.52)

Deriving expression (3.52) with respect to $\omega$ and combining the final
result with $(\ref{eqsol3})$, we obtain

$\displaystyle\frac{d}{d\omega}\int_{-\pi}^{\pi}\varphi^{3}dx=6\left(\int_{-\pi}^{\pi}(\varphi^{\prime})^{2}+\varphi^{2}dx\right)>0,$
(3.53)

so that, $\omega\mapsto\int_{-\pi}^{\pi}\varphi^{3}dx$ is one-to-one, and
consequently $c_{*}=\tilde{c}_{*}$.
Finally, by Remark 3.4, we have that $\omega$ given explicitly by

$\omega={\frac{{\pi}^{2}}{\left(4\,{\kappa}^{2}-8\right){\it
K}(\kappa)^{2}+12\,{\it K}(\kappa){\it E}\left(\kappa\right)+{\pi}^{2}}}$
(3.54)

is positive and strictly increasing in terms of $\kappa\in(0,\kappa_{0})$. We
have that $\phi_{*}$ and $\tilde{\phi}_{*}$ are $2\pi-$periodic solutions
given explicitly by $(\ref{dnsol})$ with the same wave speed
$c_{*}=\tilde{c}_{*}$. Since $\omega$ is one-to-one according with
$(\ref{cvalue2})$, it follows that the corresponding modulus of $c_{*}$ and
$\tilde{c}_{*}$ must satisfy $\kappa_{*}=\tilde{\kappa}_{*}$, so that
$\tilde{\phi}\equiv\phi_{*}$. The remainder of the proof is similar to [15,
Theorem 2.4].

∎

######  Remark 3.6.

Suppose that $n(\mathcal{L})=1$ for every $c>\frac{1}{2}$ and
$\alpha\in\left(\frac{1}{2},2\right]$. Since $(\ref{hyp-uniq})$ is valid, let
$\\{\alpha_{n}\\}_{n=1}^{n=+\infty}\subset(\frac{1}{2},\alpha_{*})$ be a
sequence such that $\alpha_{n}\rightarrow\alpha_{*}$. The additional
assumption on the number of negative eigenvalues gives us immediately the
following equality:

$n(\mathcal{L}_{\alpha_{n}})=n(\mathcal{L}_{\alpha_{*}})=1,$

where
$\mathcal{L}_{\alpha_{n}}=c_{\alpha_{n}}D^{\alpha_{n}}+(c_{\alpha_{n}}-1)-\phi_{\alpha_{n}}$
and $\mathcal{L}_{\alpha_{*}}=c_{*}D^{\alpha_{*}}+(c_{*}-1)-\phi_{*}$ indicate
the linearized operators around the periodic waves $\phi_{\alpha_{n}}$ and
$\phi_{\alpha_{*}}$, respectively. It is important to mention that our
uniqueness result remains valid if the quantity of negative eigenvalues for
$\mathcal{L}_{\alpha_{n}}$ is not stable in the sense that we may have
$2=n(\mathcal{L}_{\alpha_{n_{0}}})>n(\mathcal{L}_{\alpha_{*}})=1$ for some
$n_{0}\in\mathbb{N}$. However, our numerical approach in Section 5 attests
that $d<0$ for all $c>\frac{1}{2}$, so that the quantity of negative
eigenvalues of $\mathcal{L}_{\alpha_{n}}$ remains stable along
$c\in\left(\frac{1}{2},+\infty\right)$ for all $n\in\mathbb{N}$.

######  Remark 3.7.

It is worth mentioning that Proposition 3.5 agrees in some sense with the
uniqueness result for positive and periodic waves associated to the fractional
Korteweg-de Vries equation as in [27]. The authors employed Krasnoselskii’s
fixed-point theorem and three additional facts:
i) $n(\mathcal{L})=1$.
ii) $\ker(\mathcal{L})=[\phi^{\prime}]$.
iii) $\alpha\in(0.5849,2]$.

######  Remark 3.8.

Similarly as reported on page 1962 of [31], if $\phi$ is a solution of
$(\ref{ode1})$ obtained from Lemma 3.2, we also conjecture that $(\ref{hyp-
uniq})$ is always valid for all $c>\frac{1}{2}$. Our conjecture and the
results in this section allow to conclude that the uniqueness of minimizers is
always expected.

###  3.3. Existence of the small amplitude solutions

Let $0<\alpha\leq 2$. In this subsection, we determine the existence of small
amplitude periodic wave solutions for the boundary-value problem (1.8) which
is an even profile. We show that this kind of solutions are given by the
corresponding expansion of the wave for $c$ near $\frac{1}{2}$.

For obtaining the small amplitude periodic waves of the boundary-value problem
(1.8), we use the arguments contained in [9, Chapter 8.4], where the authors
have determined the existence of the referred waves via bifurcation theory and
the Lyapunov-Schmidt reduction.

For $\alpha\in(0,2]$, let $F:H^{\alpha}_{per}(\mathbb{T})\cap
X_{0,e}\times(0,+\infty)\rightarrow X_{0,e}$ be the map defined by

$F(g,\omega):=\omega D^{\alpha}g+(\omega-1)g-\frac{1}{2}\Pi_{0}g^{2}.$ (3.55)

We see that $F$ is smooth in all variables. Moreover, $F(g,\omega)=0$ if and
only if, $g$ is a solution of the boundary-value problem (1.8) corresponding
to the wave speed $\omega\in(0,+\infty)$. In particular, it is clear that
$F(0,\omega)=0$ for all $\omega\in(0,+\infty)$.
The Fréchet derivative associated to the function $F$ with respect to $g$ is
given by

$D_{g}F(g,\omega):H^{\alpha}_{per}(\mathbb{T})\cap X_{0,e}\rightarrow
X_{0,e},\ \ f\mapsto(\omega D^{\alpha}+\omega-1-\Pi_{0}g)f.$ (3.56)

Let $c_{0}>0$ be fixed and consider the corresponding linearized operator
$\mathcal{L}$ around the wave $\phi$ defined as in (1.10). The derivative
(3.56) at the point $(\phi,c_{0})$ becomes the well known self-adjoint
projector operator $\mathcal{L}|_{X_{0}}$. Moreover, at the point $(0,c_{0})$,
we have

$D_{g}F(0,c_{0})=c_{0}D^{\alpha}+c_{0}-1.$ (3.57)

Notice that the nontrivial kernel of $D_{g}F(0,c_{0})$ is determined by
functions $h\in H^{\alpha}_{per}(\mathbb{T})\cap X_{0,e}$ satisfying

$\widehat{h}(k)((c_{0}-1)+c_{0}|k|^{\alpha})=0,\ \ k\neq 0.$ (3.58)

Then, $D_{g}F(0,c_{0})$ has the one-dimensional kernel if and only if,
$c_{0}=(1+|k|^{\alpha})^{-1}$ for some $k\in\mathbb{Z}$, in which case it is
given by

$\textrm{ker}D_{g}F(0,c_{0})=[\tilde{\phi}_{k}(x)],$ (3.59)

where $\tilde{\phi}_{k}(x)=\cos(xk)$.
The local bifurcation theory in [9, Chapter 8.4] enables us to guarantee the
existence of an open interval $\mathcal{I}$ containing
$c_{0}=(1+|k|^{\alpha})^{-1}$, an open ball $B_{r}\in
H^{\alpha}_{per}(\mathbb{T})\cap X_{0,e}$ of radius $r>0$ centered at $g=0$
and a unique smooth mapping $c\in\mathcal{I}\mapsto\phi(c)\in
H^{\alpha}_{per}(\mathbb{T})\cap X_{0,e}$ such that $F(\phi(c),c)=0$ for all
$c\in\mathcal{I}$ and $\phi(c_{0})=0$. In other words, every solution of
$F(g,\omega)=0$ has the form $(\phi(c),c)$.

For each integer $k\geq 1$, the point $(0,\tilde{c}_{k})$, where
$\tilde{c}_{k}:=(1+k^{\alpha})^{-1}$, is a bifurcation point. Furthermore,
there exists $a_{0}>0$ and a local bifurcation curve
$a\in(0,a_{0})\mapsto(\phi_{k}(a),c_{k}(a))$ through $(0,\tilde{c}_{k})$
constituted by even $\frac{2\pi}{k}$-periodic solutions of the boundary-value
problem (1.8). Additionally, one has $c_{k}(0)=\tilde{c}_{k}$ and
$D_{a}\phi_{k}(0)=\tilde{\phi}_{k}(x)=\cos(xk)$ and all solutions of
$F(g,\omega)=0$ in a neighbourhood of $(0,\tilde{c}_{k})$ belongs to the above
curve depending on $a$.

We apply the method of Lyapunov-Schmidt reduction to obtain the explicit
expansion of the wave $\phi$ for $c$ near $\frac{1}{2}$. In addition, the
local bifurcation curve extends to a global smooth curve
$c\in((1+k^{\alpha})^{-1},+\infty)\mapsto(\phi_{k}(c),c_{k}(c))$ of solutions
for the equation $F(g,\omega)=0$. For the simple case $k=1$, both results can
be summarized in the next proposition.

######  Proposition 3.4.

Let $\alpha\in(0,2]$ be fixed. There exists $a_{0}>0$ such that for all
$a\in(0,a_{0})$ there is a unique even local periodic solution $\phi$ for the
boundary value problem (1.8) given by the following expansion:

$\phi(x)=a\cos(x)+\frac{a^{2}}{2(2^{\alpha}-1)}\cos(2x)+O(a^{3})$ (3.60)

and

$c=\frac{1}{2}+\frac{a^{2}}{4(2^{\alpha}+1)(2^{\alpha}-1)}+O(a^{4}).$ (3.61)

In addition, constant $A$ is given by

$A(c)=\frac{a^{2}}{4}+O(a^{4}).$ (3.62)

The pair $(\phi,c)\in H^{\alpha}_{per}(\mathbb{T})\cap
X_{0,e}\times(\frac{1}{2},+\infty)$ is global in terms of the parameter
$c>\frac{1}{2}$ and it satisfies $(\ref{ode2})$.

###### Proof.

The proof of this result relies on slight modifications of [8, Theorem 5.4 and
Theorem 5.6]. ∎

In the next result, we see that $q_{c}$ in $(\ref{infB})$ is continuous in $c$
and for $c>\frac{1}{2}$ and that $q_{c}\rightarrow 0^{+}$ as
$c\rightarrow\frac{1}{2}^{+}$. This fact gives us that the small amplitude
periodic waves obtained in the Proposition 3.4 satisfy the minimization
problem $(\ref{minB})$ for $c\rightarrow\frac{1}{2}^{+}$.

######  Proposition 3.5.

Let $\phi\in Y_{0}$ be the solution of the constrained minimization problem
obtained in Lemma 3.2 and $q_{c}=\mathcal{B}_{c}(\phi)$. Then $q_{c}$ is
continuous in $c$ for $c>\frac{1}{2}$ and $q_{c}\to 0$ as $c\to\frac{1}{2}$.

###### Proof.

This result is similar to [31, Lemma 2.3]. ∎

Next result establishes good spectral properties for the linearized operator
$\mathcal{L}$ in $(\ref{operator})$ around the single-lobe solution $\phi$
obtained in Lemma 3.2 by knowing the spectral information for the same
operator around the small amplitude periodic waves obtained by Proposition
3.4.

######  Proposition 3.6.

Let $\alpha\in\left(\frac{1}{2},2\right]$ be fixed. For every $c>\frac{1}{2}$,
we have $n(\mathcal{L})=1$ if and only if $z(\mathcal{L})=1$. Moreover,
$\mathcal{L}$ consists in a countable sequence of positive eigenvalues bounded
away from zero.

###### Proof.

The proof has the same spirit as in [27, Lemma 2.3]. According to the
Proposition 3.1, the number of negative eigenvalues of $\mathcal{L}$ may
change in the parameter continuations in $c$ if and only if, the eigenvalues
pass through zero eigenvalue. If $n(\mathcal{L})=1$ for every $c>\frac{1}{2}$,
we see that $\ker(\mathcal{L}|_{X_{0}})=[\phi^{\prime}]$ by [31, Corollary
4.5]. If $z(\mathcal{L})=2$ for some $c_{1}>\frac{1}{2}$ we see that parameter
$d$ defined in 3.1 satisfies $d=0$ at $c=c_{1}$. This means by Proposition 3.1
that $n(\mathcal{L})=2$ at $c=c_{1}$ which is a contradiction. Suppose that
$z(\mathcal{L})=1$ for every $c>\frac{1}{2}$. For the small amplitude periodic
waves $\phi$ obtained by the Proposition 3.4, we see that $c\mapsto\phi\in
H_{per}^{\infty}(\mathbb{T})$ is smooth. Now, from (3.61), we have the
expansion

$d=1+2A(c)-c-cA^{\prime}(c)=\frac{1}{2}\frac{-{16}^{\alpha}+14\,{2}^{-2+2\,\alpha}-3}{{4}^{\alpha}-1}+O(a^{3}).$
(3.63)

For $\alpha\in\left(\frac{1}{2},2\right]$, we have $d<0$ and this means by
Proposition 3.1 that $n(\mathcal{L})=1$ for the small amplitude periodic
waves. By a continuity argument, we obtain that $n(\mathcal{L})=1$ for the
linearized operator $\mathcal{L}$ around the single-lobe solution $\phi$
determined by Lemma 3.2 for all $c>\frac{1}{2}$. ∎

##  4\. Spectral Stability

In this section, we study the spectral stability of (1.13). First, we see that
the even solution for the boundary-value problem (1.8) obtained in Lemma 3.2
is smooth by using similar arguments as in [31, Proposition 2.4]. Moreover
$\phi$ has a single-lobe profile, namely, there exists only one maximum (at
$x=0$) and minimum of $\phi$ on $\mathbb{T}$. This fact is in accordance with
the approach in [23] to decide about the non-degeneracy of the
$\ker(\mathcal{\tilde{L}})$.
For $\alpha\in(0,2]$ and since $\phi$ is a single-lobe solution, we see by
Oscillation Theorem in [23] that the operator $\mathcal{\tilde{L}}$ has at
most two negative eigenvalues, that is, $n(\mathcal{\tilde{L}})\in\\{1,2\\}$.
Consequently, there may be at most two eigenfunctions of $\mathcal{\tilde{L}}$
for the zero eigenvalue, that is, $z(\mathcal{\tilde{L}})\in\\{1,2\\}$. Recall
that $\ker(\tilde{\mathcal{L}}|_{X_{0}})=[\phi^{\prime}]$ implies
$c\in\left(\frac{1}{2},+\infty\right)\mapsto\phi$ is smooth and $d$ in
$(\ref{vd1})$ can be used to decide about the non-degeneracy of
$\ker(\tilde{\mathcal{L}})$ and the exact quantity of $n(\tilde{\mathcal{L}})$
according to the Proposition 3.1. Indeed, if $d\neq 0$ we conclude by the
first equality in (3.10), (3.11), and (3.12) that $\\{1,\phi,\phi^{2}\\}\in$
range$(\mathcal{\tilde{\mathcal{L}}})$. Proposition 3.1 in [23] is now used to
conclude that ker($\mathcal{\tilde{L}})=$ $[\phi^{\prime}]$. On the other
hand, if $d=0$, we have
$\ker(\mathcal{\tilde{L})}=\left[\phi^{\prime},\frac{d}{dc}\phi-\frac{1}{c}-\frac{1}{c}\phi\right]$.
In addition, $d<0$ implies $n(\tilde{\mathcal{L}})=1$ and $d>0$ gives us that
$n(\tilde{\mathcal{L}})=2$.
First, we prove the spectral stability for the small amplitude periodic waves
associated to the case $\alpha\in(0,2]$.

######  Proposition 4.1.

Let $\alpha\in(0,2]$. The small amplitude periodic waves $\phi\in
H^{\infty}_{per}(\mathbb{T})$ determined by Proposition 3.4 are spectrally
stable in the sense of Definition 1.1.

###### Proof.

Since $\mathcal{L}$ is a Hessian operator for $G(u)$ in (1.10) and
$\mathcal{\tilde{L}}=\frac{1}{c}\mathcal{L}$ for all $c>\frac{1}{2}$, the
spectral stability holds if
$\mathcal{\tilde{L}}|_{\\{1,(D^{\alpha}+1)\phi\\}^{\bot}}\geq 0$, that is, if
$n(\mathcal{\tilde{L}}|_{\\{1,(D^{\alpha}+1)\phi\\}^{\bot}})=0$. On the other
hand, the periodic wave $\phi$ is spectrally unstable if
$n(\mathcal{\tilde{L}}|_{\\{1,(D^{\alpha}+1)\phi\\}^{\bot}})=1$.

For $\lambda\notin\sigma(\mathcal{\tilde{L}})$, let us define the following
symmetric 2-by-2 matrix given by

$\displaystyle S(\lambda)$ $\displaystyle:=$
$\displaystyle\left[\begin{array}[]{cc}\langle(\mathcal{\tilde{L}}-\lambda
I)^{-1}1,1\rangle&\langle(\mathcal{\tilde{L}}-\lambda
I)^{-1}(D^{\alpha}+1)\phi,1\rangle\\\ &\\\ \langle(\mathcal{\tilde{L}}-\lambda
I)^{-1}1,(D^{\alpha}+1)\phi\rangle&\langle(\mathcal{\tilde{L}}-\lambda
I)^{-1}(D^{\alpha}+1)\phi,(D^{\alpha}+1)\phi\rangle\end{array}\right].$

Using (3.63), for $\alpha\in(0,\tilde{\alpha})\cup(\frac{1}{2},+\infty)$ we
obtain $d<0$, whereas for $\tilde{\alpha}<\alpha<\frac{1}{2}$, we have $d>0$.
For $\alpha=\tilde{\alpha}\approx 0.2924$ and $\alpha=\frac{1}{2}$, we have
the existence of fold points for which $d=0$.
Assume $d\neq 0$, we compute at $\lambda=0$. Indeed,

$\displaystyle\langle\mathcal{\tilde{L}}^{-1}(D^{\alpha}+1)\phi,(D^{\alpha}+1)\phi\rangle$
$\displaystyle=$ $\displaystyle\displaystyle-\frac{2\pi
d^{-1}}{c}(A^{\prime}(c))^{2}-2\pi
A^{\prime}(c)-\frac{1}{6}\gamma^{\prime}(c),$ (4.2)

$\langle\mathcal{\tilde{L}}^{-1}(D^{\alpha}+1)\phi,1\rangle=\langle\mathcal{\tilde{L}}^{-1}1,(D^{\alpha}+1)\phi\rangle=\displaystyle\frac{2\pi
d^{-1}}{c}A^{\prime}(c),$ (4.3)

and

$\langle\mathcal{\tilde{L}}^{-1}1,1\rangle=\displaystyle\displaystyle-2\pi\frac{d^{-1}}{c},$
(4.4)

where $\gamma(c)=\int_{-\pi}^{\pi}\phi^{3}dx$ and
$\frac{1}{3}\gamma^{\prime}(c)=\frac{8\pi}{c}A(c)-\frac{1}{c}\gamma(c)$. The
determinant of $S(0)$, for $d\neq 0$, is given by

$\displaystyle\textrm{det}S(0)=\frac{4\pi^{2}d^{-1}}{c}\left(A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)\right).$
(4.5)

By (1.24) we have $A^{\prime}(c)>0$ for all $\alpha\in(0,2]$. Since
$\mathcal{B}_{c}(\phi)>0$, we see that the sign of $\det S(0)$ is determined
only by $d^{-1}$ since the terms between parentheses in (4.5) are strictly
positive.

We denote by $n_{0}$ and $z_{0}$ the number of negative and zero eigenvalues
of $S(0)$, respectively. If $d=0$, then $S(0)$ is singular, in which case we
denote the number of diverging eigenvalues of $D(\lambda)$ as
$\lambda\rightarrow 0$ by $z_{\infty}$. Thus, [33, Theorem 4.1] gives us the
following relations:

$\displaystyle\displaystyle\left\\{\begin{array}[]{ccc}n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})&=&n(\mathcal{\tilde{L}})-n_{0}-z_{0},\\\
z(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})&=&z(\mathcal{\tilde{L}})+z_{0}-z_{\infty}\end{array}\right..$

Assume $d\neq 0$, so that $z_{\infty}=0$. If $0<\alpha<\tilde{\alpha}$ or
$\alpha\in\left(\frac{1}{2},2\right]$, we have $z_{0}=0$ and det$S(0)<0$ which
implies that $n_{0}=1$. On the other hand by Corollary 3.5, we see that the
waves in $(\ref{exp-sol})$ solve the minimization problem $(\ref{infB})$ and
Proposition 3.1 gives us $n(\mathcal{L})=1$. Therefore, we have the spectral
stability in this case because
$n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})=1-1-0=0$.
For the case $\tilde{\alpha}<\alpha<\frac{1}{2}$, we have $n_{0}=0$ or
$n_{0}=2$. Since the trace of $S(0)$ is strictly negative, we conclude that
$n_{0}=2$. Hence, we obtain the spectral stability of $\phi$ since
$n(\tilde{\mathcal{L}})=2$ and
$n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})=2-2-0=0$.
Finally, for the case $\alpha=\tilde{\alpha}$ or $\alpha=\frac{1}{2}$, we
obtain from the relations above and Proposition 3.1 that $z_{\infty}=1$,
$n(\mathcal{\tilde{L}})=2$, so that
$n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})=2-2=0$.
∎

######  Remark 4.2.

The result contained in Proposition 4.1 can be extended for the case
$\alpha>2$.

The next result guarantees sufficient conditions for the spectral stability of
the periodic minimizer $\phi$ and it gives the proof of Theorem 1.3-iv).

######  Proposition 4.3.

Let $\alpha\in(\frac{1}{3},2]$ be fixed. If
$\ker\left(\mathcal{L}|_{X_{0}}\right)=[\phi^{\prime}]$, the periodic
minimizer $\phi$ obtained in Lemma 3.2 is spectrally stable in the sense of
Definition 1.1 if:
i) $d\neq 0$ and $A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$.
ii) $\alpha\in\left(\frac{1}{2},2\right]$ and $d\neq 0$ for all
$c>\frac{1}{2}$.

###### Proof.

We use the same computations as determined in the proof of Proposition 4.1. To
prove i), suppose that $\ker(\mathcal{L}|_{X_{0}})=[\phi^{\prime}]$, $d>0$ and
$A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$. By Proposition 3.1, we
have $n(\mathcal{L})=2$ and by $(\ref{third-term})$ the first entry of the
matrix $S(0)$ in Proposition 4.1 is negative. Thus $n_{0}=2$ and we have the
spectral stability since
$n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})=n(\mathcal{\tilde{L}})-n_{0}-z_{0}=2-2-0=0$.
If $d<0$ and $A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$, we see that
$\det S(0)<0$ and we have the spectral stability since
$n(\mathcal{\tilde{L}}\mid_{\left\\{1,(D^{\alpha}+1)\phi\right\\}^{\bot}})=n(\mathcal{\tilde{L}})-n_{0}-z_{0}=1-1-0=0$.
To prove ii), since $d\neq 0$ for all $c>\frac{1}{2}$, we obtain by
Proposition 3.1 that $z(\mathcal{L})=1$, so that $n(\mathcal{L})=1$ by
Proposition 3.6. We claim that $1+2A(c)-c-cA^{\prime}(c)<0$ for all
$c>\frac{1}{2}$. Indeed, if there exists $c_{2}>\frac{1}{2}$ such that
$1+2A(c_{2})-c_{2}-c_{2}A^{\prime}(c_{2})\geq 0$, we deduce by Proposition 3.1
that $n(\mathcal{L})=2$ at $c=c_{2}$. This fact generates a contradiction and
the claim is satisfied. Next, we obtain after multiplying this inequality by
the factor $\frac{1}{c^{3}}$ that

$\frac{d}{dc}\left(A(c)\frac{1}{c^{2}}\right)>\frac{1}{c^{3}}-\frac{1}{c^{2}}.$
(4.7)

Integrating $(\ref{ineqdiff1})$ over the interval $\left(\frac{1}{2},c\right]$
and using $A(c)\rightarrow 0$ when $c\rightarrow\frac{1}{2}$, we obtain the
important inequality

$A(c)>c-\frac{1}{2}.$ (4.8)

Thus, we have from the Poincaré-Wirtinger inequality, $(\ref{ineqdiff2})$,
inequality $1+2A(c)-c-cA^{\prime}(c)<0$, and a simple computation

$\begin{array}[]{lllll}\displaystyle
A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)\geq\displaystyle\left(c-\frac{1}{2}\right)^{2}>0.\end{array}$
(4.9)

Therefore, if $c>\frac{1}{2}$ one has by $(\ref{ineqdiff3})$ that
$A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$. From
$(\ref{determinant-D})$, we obtain

$\det
S(0)=\frac{4\pi^{2}d^{-1}}{c}\left(A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)\right)<0.$

This together with $n(\mathcal{L})=1$ give us from the index formula contained
in Proposition 4.1 the spectral stability of the single-lobe $\phi$.

∎

######  Remark 4.4.

The numerical approach in the next section will attest for a fixed
$\alpha\in\left(\frac{1}{2},2\right]$ that $d\neq 0$ for all $c>\frac{1}{2}$,
so that the periodic waves obtained in Lemma 3.2 are spectrally stable by
Proposition $\ref{spectmin}-ii)$. For the case
$\alpha\in\left(\frac{1}{3},\frac{1}{2}\right]$ we will have the following
situation: there exists a unique $c^{*}$ such that $d>0$ for all
$c\in\left(\frac{1}{2},c^{*}\right)$ and $d<0$ for all $c\in(c^{*},+\infty)$.
Since in both cases we have
$A^{\prime}(c)+\frac{1}{\pi}\mathcal{B}_{c}(\phi)>0$, we obtain from
$(\ref{determinant-D})$ that $\det S(0)>0$ for the case $d>0$ and $\det
S(0)<0$ if $d<0$. In both cases, the spectral stability is obtained by
Proposition $\ref{spectmin}-i)$. For the case $d=0$, we also have the spectral
stability of $\phi$ using a similar procedure as in Proposition 4.1.

##  5\. Numerical Experiments

In this section we propose a Petviashvili’s iteration method for the numerical
generation of periodic travelling wave solutions of the equation (1.1). The
method is widely used for the generation of travelling wave solutions [13, 14,
27, 32, 34]. The iteration method for the periodic waves, a modification of
standard Petviashvilli’s algorithm [14, 27], is based on the following
solution steps. First, we use the transformation

$\psi=\displaystyle\frac{1}{2c}\bigg{[}\phi-(c-1)+\sqrt{(c-1)^{2}+2A}\bigg{]}$
(5.1)

to convert the equation (1.6) into

${D}^{\alpha}\psi+w\psi-\psi^{2}=0,$ (5.2)

where $w=\displaystyle\frac{1}{c}\sqrt{(c-1)^{2}+2A}$. Thus, we obtain the
equation with a zero integration constant. Integrating (5.1) on $[-\pi,\pi]$
and using the fact that $\phi:\mathbb{T}\rightarrow\mathbb{R}$ is a periodic
function with the zero mean value yield that

$c=\left(1-w+\frac{1}{\pi}\int_{-\pi}^{\pi}\psi dx\right)^{-1}.$ (5.3)

Next, we use the standard Petviashvilli’s method to solve the equation (5.2).
Employing the Fourier transform to the equation (5.2) gives

$\left(|\xi|^{\alpha}+w\right)\widehat{\psi}(\xi)-\widehat{\psi^{2}}(\xi)=0.$
(5.4)

A simple iterative algorithm for numerical calculation of
$\widehat{\psi}(\xi)$ for the above equation can be proposed in the form

$\widehat{\psi}_{n+1}(\xi)=\frac{\widehat{\psi^{2}_{n}}(\xi)}{|\xi|^{\alpha}+w},\hskip
30.0ptn\in\mathbb{N}$ (5.5)

where $\widehat{\psi}_{n}(\xi)$ is the Fourier transform of ${\psi}_{n}$ which
is the $n^{th}$ iteration of the numerical solution. Here the solutions are
constructed under the assumption

$|\xi|^{\alpha}+w\neq 0.$ (5.6)

Since the above algorithm is usually divergent, we finally present the
Petviashvilli’s method as

$\widehat{\psi}_{n+1}(\xi)=\frac{M_{n}^{\nu}}{|\xi|^{\alpha}+w}\widehat{\psi^{2}_{n}}(\xi)$
(5.7)

by introducing the stabilizing factor

$M_{n}=\frac{\langle({D}^{\alpha}+w)\psi_{n},\psi_{n}\rangle}{\langle\psi^{2}_{n},\psi_{n}\rangle},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hskip
20.0pt\psi_{n}\in H^{\alpha}_{per}(\mathbb{T})$ (5.8)

where $\langle\cdot~{},\cdot\rangle$ denotes the standard inner product in
$L^{2}_{per}(\mathbb{T})$. Here, the free parameter $\nu$ is chosen as $2$ for
the fastest convergence (see [34] for details). The iterative process is
controlled by the error between two consecutive iterations given by

$Error(n)=\|\psi_{n}-\psi_{n-1}\|_{L_{per}^{\infty}}$

and the stabilization factor error given by

$|1-M_{n}|.$

The residue of the interaction process is determined by
${RES(n)}=\|{\mathcal{S}}\psi_{n}\|_{L_{per}^{\infty}},$ where

${\mathcal{S}}\psi={D}^{\alpha}\psi+w\psi-\psi^{2}.$

The periodic traveling wave solution $\phi$ of the BBM equation corresponds to
boundary value problem (1.6) with $\alpha=2$ is given in the equation (3.42).

Figure 5.1. The exact and the numerical solutions of the BBM equation with the
wave speed $c=1.2181$ and the variation of $Error(n)$, $|1-M_{n}|$ and $RES$
with the number of iterations in semi-log scale.

In order to test the accuracy of our scheme, we compare the exact solution
(3.42) with the numerical solution obtained by using the expansion
corresponding to (5.2) as the initial guess

$\psi_{0}(x)=1+a\cos(x)+a^{2}\left[\frac{1}{2}-\frac{1}{2(2^{\alpha}-1)}+\frac{\cos(2x)}{2(2^{\alpha}-1)}\right]+a^{3}\frac{\cos(3x)}{2(2^{\alpha}-1)(3^{\alpha}-1)},$
(5.9)

where $a=0.2$ and $\alpha=2$. In this experiment, the space interval is
$[-\pi,\pi]$ and number of grid points is chosen as $N=2^{12}$. In the first
panel of Figure 5.1, we depict the exact and numerical solutions for the wave
speed $c=1.2181$. As it is seen from the figure, the exact and the numerical
solutions coincide. In the other panels of Figure 5.1, the variations of three
different errors with the number of iteration are presented. These results
show that our numerical scheme captures the solution remarkably well.

Figure 5.2. Variation of exact and numerical values of integration constant
$A$ versus $c$ (left panel) and variation of numerical value of $d$ versus $c$
(right panel) for $\alpha=2$.

The left panel of Figure 5.2 shows the exact and numerical variation of the
integration constant $A$ with respect to $c$. The exact relation between $A$
and $c$ is illustrated by using the eqs. (3.43) and (3.44). From the numerical
point of view, first we obtain the numerical solution by using Petviashvili’s
scheme (5.7) for every $w\in(1,+\infty)$. Finally, the numerical value of the
integration constant $A(c)$ is evaluated by

$A(c)=\frac{1}{2}\left[c^{2}w^{2}-(c-1)^{2}\right].$ (5.10)

As it is seen from the figure, the numerical and the exact values of $A(c)$
coincide. In the right panel of the figure, we present the variation of $d$
with $c$. We observe that $d$ is always negative.

Figure 5.3. The exact and the numerical solutions of rBO equation for the wave
speed $c=1.2192$.

Figure 5.4. Variation of exact and numerical values of integration constant
$A$ versus $c$ (left panel) and variation of numerical value of $d$ versus $c$
(right panel) for $\alpha=1$.

The single-lobe periodic solution to the boundary value problem (5.2) for
$\alpha=1$ is given by

$\psi(x)=\frac{\sinh\gamma}{\cosh\gamma-\cos x},$ (5.11)

where parameter $\gamma\in(0,+\infty)$ is given by $\gamma=\coth^{-1}(w)$. The
transformation (5.1) allows one to obtain the single-lobe periodic solution
for the rBO equation as

$\phi(x)=2c\bigg{(}\frac{\sinh\gamma}{\cosh\gamma-\cos x}-1\bigg{)}.$ (5.12)

We compare the exact solution (5.12) with the numerical solution obtained by
using the expansion (5.9) with $a=0.2$ and $\alpha=1$ as the initial guess.
The space interval is $[-\pi,\pi]$ and number of grid points is chosen as
$N=2^{9}$. We present the exact and numerical solutions for the wave speed
$c=1.2192$ in Figure 5.3. Since we have $\int_{0}^{\pi}\psi(x)dx=\pi$ from
(5.11), we obtain $c=(3-w)^{-1}$ by using (5.3). Eliminating $w$ in (5.10), we
compute the integration constant $A(c)=4c^{2}-2c$ explicitly. In the left
panel of Figure 5.4, we compare the exact and numerical variation of the
integration constant $A$ with respect to $c$ for $\alpha=1$. The right panel
shows the variation of $d$ with $c$ for $\alpha=1$. Figures 5.2 and $5.4$ show
that $A(c)$ is strictly increasing for all values of $c$ and $d$ is always
negative. Therefore, numerical results are compatible with Proposition 4.3
stating the spectral stability of single-lobe solution for
$\alpha\in(\frac{1}{2},2]$.

Figure 5.5.  Various numerical wave profiles (left panel) and the modulus of
the Fourier coefficients of the numerical solution (right panel) for
$\alpha=0.55$.

In the next numerical experiment, we choose $\alpha=0.55>\alpha_{0}$ since the
fold point $\alpha_{0}=0.5$ for the fBBM equation. In the left panel of Figure
5.5, we illustrate the periodic wave profiles for several values of $c$. It
can be seen that the amplitude becomes more and more peaked with the
increasing wave speed $c$. The right panel of Figure 5.5 shows that the
modulus of the Fourier coefficients computed via a discrete Fourier
transformation decreases to machine precision for $N=2^{12}$ Fourier modes. In
Figure 5.6, we depict the variation of $A(c)$ and $d$ with $c$. The numerical
results are again compatible with the Proposition 4.3 for
$\alpha>\frac{1}{2}$.

Figure 5.6. Variation of $A$ versus $c$ (left panel) and variation of $d$
versus $c$ (right panel) both evaluated numerically for $\alpha=0.55$.

Figure 5.7. Variation of values of $A$ versus $c$ (left panel) and close-up
look to the gap (right panel) ($\alpha=0.45$). The solid line shows the
numerical result obtained by Petviashvili’s method and the dashed line shows
the relation (5.13) from the small amplitude expansion.

In Figure 5.7, we show the variation of $A(c)$ with $c$ for
$\alpha=0.45<\alpha_{0}$. The picture shows a lack of convergence using
Petviashvili’s scheme in the sense that function $A(c)$ can not reach the
bifurcation point $c=\frac{1}{2}$ as determined for the case
$\alpha\in\left(\frac{1}{2},2\right]$. To do best of our knowledge, this
phenomena gives us (at least numerically) an indication that the number of
negative eigenvalues for the linearized operator is $2$ and/or the existence
of a fold point, that is, a value of $c$ such that
$\dim(\ker(\mathcal{L}))=2$. In both cases, it is well known that the
numerical method does not converge (see [27, 31, 34]) and prevent us to show
that $\phi$ solves the minimization problem $(\ref{minp})$. Due to the lack of
numerical data for $c\in(0.5,1.8)$, we illustrate the curve (dashed line) by
using the relation

$A(c)\approx\left(c-\frac{1}{2}\right)(2^{\alpha}+1)(2^{\alpha}-1),$ (5.13)

obtained by the expressions of $c$ and $A(c)$ in $(\ref{exp-speed1})$. The
right panel gives a closer look to the gap between the branch of the small
amplitude periodic waves and numerical result. In order to fill this gap, we
use the Newton’s method for $c\in(0.5,1.8)$ in Figure 5.8.

Figure 5.8. Variation of numerical values of $A$ versus $c$ with the
combination of Newton and Petviashvili method ($\alpha=0.45$)

Figure 5.9. Variation of $d$ versus $c$ (top left), close-up look near
critical value $c^{*}$ (top right) and the variation of
$A^{\prime}(c)+\frac{1}{\pi}{\mathcal{B}}_{c}(\phi)$ with $c$ (bottom) for
$\alpha=0.45$.

Figure 5.10. Variation of $d$ versus $c$ (top left), close-up look near
critical value $c^{*}$ (top right) and the variation of
$A^{\prime}(c)+\frac{1}{\pi}{\mathcal{B}}_{c}(\phi)$ with $c$ (bottom) for
$\alpha=0.5$.

In Figure 5.9, we present the variation of $d$ and the term
$A^{\prime}(c)+\displaystyle\frac{1}{\pi}{\mathcal{B}}_{c}(\phi)$ with $c$ for
$\alpha=0.45$. We observe that
$A^{\prime}(c)+\displaystyle\frac{1}{\pi}{\mathcal{B}}_{c}(\phi)$ is positive
for all values of $c$. However, $d$ is positive up to a critical speed
$c^{*}\approx 0.953$ and then becomes negative. Therefore, $\mbox{det}S(0)$ is
positive for $c\in(\frac{1}{2},c^{*})$ and negative for $c\in(c^{*},+\infty)$.
We also illustrate the variation of $d$ and the term
$A^{\prime}(c)+\displaystyle\frac{1}{\pi}{\mathcal{B}}_{c}(\phi)$ with $c$ in
Figure 5.10 for the fold point $\alpha_{0}=0.5$. The numerical results are
very similar to the ones for $\alpha=0.45$ with $c^{*}\approx 0.67$. As it is
seen from the figures, $\det S(0)<0$ for $c\in(c^{*},+\infty)$ which indicates
the spectral stability. However, $\det S(0)>0$ for $c\in(\frac{1}{2},c^{*})$
which yields that $n_{0}=0$ or $n_{0}=2$. Since in this case $d>0$, we have by
Proposition 6.1 that $\phi$ is spectrally stable for all
$c\in(\frac{1}{2},c^{*})$.

##  6\. Remarks on the Orbital Stability of Periodic Waves

In this section, we present a brief discussion concerning the orbital
stability of the periodic wave $\phi$ obtained by Lemma 3.2.
Before stating the result, we need some preliminary tools. For functions $u$
and $v$ in $H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$, we define $\rho$ as the
“distance” between $u$ and $v$ given by

$\rho(u,v)=\inf_{y\in\mathbb{R}}||u-v(\cdot+y)||_{{H_{per}^{\frac{\alpha}{2}}}}.$

Our precise definition of orbital stability is given below.

######  Definition 6.1.

We say that the periodic solution $\phi$ is orbitally stable in
${H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}$, by the periodic flow of (1.1), if
for any ${\varepsilon}>0$ there exists $\delta>0$ such that for any $u_{0}\in
H_{per}^{\frac{\alpha}{2}}(\mathbb{T})$ satisfying
$\|u_{0}-\phi\|_{{H_{per}^{\frac{\alpha}{2}}}}<\delta$, the solution $u(t)$ of
(1.1) with initial data $u_{0}$ exists globally and satisfies

$\rho(u(t),\phi)<{\varepsilon},$

for all $t\geq 0$.

######  Remark 6.2.

Our notion of orbital stability prescribes the existence of global solutions.
Thus, from Section 2 we need to assume that $\alpha>1$.

Next, let us introduce the conserved quantity

$V(u)=\frac{1}{2}\int_{-\pi}^{\pi}\left(u^{2}+\frac{u^{3}}{3}\right)dx=P(u)-E(u)$
(6.1)

and the auxiliary functional

$Q(u):=-V(u)+(c-1-A(c))M(u).$

In what follows, we set

$\Upsilon_{0}=\\{u\in{H_{per}^{\frac{\alpha}{2}}(\mathbb{T})};\ \langle
Q^{\prime}(\phi),u\rangle=0\\}.$

Note that $\Upsilon_{0}$ is nothing but the tangent space to
$\\{u\in{H_{per}^{\frac{\alpha}{2}}(\mathbb{T})};Q(u)=Q(\phi)\\}$ at $\phi$.
With these notations, the result in [11, Theorem 2.1] (see also [1]) reads as
follows.

######  Proposition 6.1.

Suppose that $n(\mathcal{L})=1$ and $z(\mathcal{L})=1$. If there exists
$\Phi\in H_{per}^{\alpha}(\mathbb{T})$ such that
$\langle\mathcal{L}\Phi,\Psi\rangle=0$ for all $\Psi\in\Upsilon_{0}$ and
$\langle\mathcal{L}\Phi,\Phi\rangle<0$, then $\phi$ is orbitally stable in
${H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}$ by the periodic flow of
$(\ref{rbbm})$.

$\square$

Finally, we can use the estimate $(\ref{ineqdiff2})$ to obtain the orbital
stability of $\phi$. This last result gives us the proof of Theorem 1.3-v).

######  Proposition 6.2.

Let $\alpha\in(1,2]$ be fixed. If
$\ker\left(\mathcal{L}|_{X_{0}}\right)=[\phi^{\prime}]$ and $d\neq 0$ for all
$c>\frac{1}{2}$, the periodic wave $\phi$ obtained in Lemma 3.2 is orbitally
stable in ${H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}$ in the sense of
Definition 6.1.

###### Proof.

Since $n(\mathcal{L})=z(\mathcal{L})=1$ by Proposition 3.6, we only need to
use Proposition 6.1 for a convenient $\Phi$. Indeed, let us consider
$\Phi=1+\phi$. For

$Q(u)=-V(u)+(c-1-A(c))M(u),$

we obtain $\langle Q^{\prime}(\phi),\phi^{\prime}\rangle=0$ and
$\langle\mathcal{L}(1+\phi),\Psi\rangle=0$ for all $\Psi\in\Upsilon_{0}$. If
$\langle\mathcal{L}(1+\phi),1+\phi\rangle<0$, one has the orbital stability of
$\phi$ in the energy space ${H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}$. To
calculate $\langle\mathcal{L}(1+\phi),1+\phi\rangle$, we employ the Poincaré-
Wirtinger inequality and $(\ref{ineqdiff2})$ to obtain

$\begin{array}[]{lll}\langle\mathcal{L}(1+\phi),1+\phi\rangle&=&\displaystyle
2\pi(c-1)-\displaystyle\int_{-\pi}^{\pi}c(D^{\frac{\alpha}{2}}\phi)^{2}+\left(c+1\right)\phi^{2}dx\\\
\\\ &\leq&2\pi(c-1)-4\pi(2c+1)A(c)\\\ \\\
&\leq&2\pi(c-1)-4\pi(2c+1)\left(c-\frac{1}{2}\right)=-2\pi
c(4c-1)<0.\end{array}$ (6.2)

By Proposition 6.1 one has the orbital stability in the energy space
${H_{per}^{\frac{\alpha}{2}}(\mathbb{T})}$. ∎

## Acknowledgments

The authors are grateful to the two anonymous referees for their valuable
suggestions and comments which greatly improved the presentation of the paper.
S. Amaral was supported by the regular doctorate scholarship from CAPES. F.
Natali is partially supported by CNPq (grant 304240/2018-4), Fundação
Araucária (grant 002/2017) and CAPES MathAmSud (grant 88881.520205/2020-01).

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