# Periodic waves of the modified KdV equation as minimizers of a new
variational problem

Uyen Le  Department of Mathematics and Statistics, McMaster University,
Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>and  Dmitry E. Pelinovsky
Department of Mathematics and Statistics, McMaster University, Hamilton,
Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

Periodic waves of the modified Korteweg-de Vries (mKdV) equation are
identified in the context of a new variational problem with two constraints.
The advantage of this variational problem is that its non-degenerate local
minimizers are stable in the time evolution of the mKdV equation, whereas the
saddle points are unstable. We explore the analytical representation of
periodic waves given by Jacobi elliptic functions and compute numerically
critical points of the constrained variational problem. A broken pitchfork
bifurcation of three smooth solution families is found. Two families represent
(stable) minimizers of the constrained variational problem and one family
represents (unstable) saddle points.

###### Key words and phrases:

Modified Korteweg–de Vries equation, periodic waves, energy minimization

###### 2000 Mathematics Subject Classification:

35Q51, 35Q53, 76B25

##  1\. Introduction

We address travelling periodic waves of the modified Korteweg–de Vries (mKdV)
equation which we take in the normalized form

$u_{t}+6u^{2}u_{x}+u_{xxx}=0.$ (1)

For the sake of clarity, we normalize the wave period to $2\pi$ and denote the
Sobolev spaces of $2\pi$-periodic functions by $H^{k}_{\rm per}$ for
$k\in\mathbb{N}$ with $H^{0}_{\rm per}\equiv L^{2}_{\rm per}$ for $k=0$.

Travelling waves of the form $u(t,x)=\psi(x-ct)$ satisfy the stationary
equation

$-\psi^{\prime\prime}+c\psi+b=2\psi^{3},$ (2)

where $\psi(x):[0,2\pi]\mapsto{\mathbb{R}}$ is the wave profile, $c$ is the
wave speed, and $b$ is the constant of integration. Travelling periodic waves
of the mKdV equation (1) play a fundamental role in physical applications,
e.g. for dynamics of the internal waves in seas and oceans [12, 21, 22].
Period function for periodic solutions of the stationary equation (2) has been
studied in [7, 10, 23]. Stability of periodic waves in the defocusing version
of the mKdV equation has been studied in all details [9]. We consider the
focusing version of the mKdV equation (1).

There exist two families of periodic solutions to the stationary equation (2)
for $b=0$: dnoidal waves with sign-definite profile $\psi$ and cnoidal waves
with sign-indefinite profile $\psi$. Spectral and orbital stability of these
periodic waves has been explored in the recent literature [3, 4, 8]. While
sign-definite dnoidal waves are stable for all speeds, sign-indefinite cnoidal
waves are stable for smaller speeds $c$ and unstable for larger speeds $c$
[8].

Compared to these definite results, the stationary mKdV equation (2) with
$b\neq 0$ has more general families of periodic waves expressed as two
rational functions of Jacobi elliptic functions [6]. Stability of a particular
family of positive periodic waves with $b\neq 0$ has been proven in [2], but
no general results on stability of these periodic waves are available in the
literature to the best of our knowledge. A new variational formulation of the
periodic waves with $b\neq 0$ was developed in our previous works with F.
Natali [17, 18] (see also the follow-up work [1]).

The purpose of this paper is to explore the new variational formulation of
travelling periodic waves and to detect numerically which periodic waves with
$b\neq 0$ are stable and which are unstable in the time evolution of the mKdV
equation (1).

Since the mKdV equation (1) admits the following conserved quantities on the
$2\pi$-periodic domain:

$E(u)=\frac{1}{2}\oint\left[(u^{\prime})^{2}-u^{4}\right]dx,\quad
F(u)=\frac{1}{2}\oint u^{2}dx,\quad M(u)=\oint udx,$ (3)

the stationary equation (2) is the Euler–Lagrange equation for the action
functional

$G_{c,b}(u):=E(u)+cF(u)+bM(u).$ (4)

We refer to $E(u)$, $F(u)$, and $M(u)$ as the energy, momentum, and mass,
respectively.

The standard variational formulation for stability of periodic waves is to
find minimizers of energy $E(u)$ in $H^{1}_{\rm per}$ subject to the fixed
momentum $F(u)$ and mass $M(u)$ [5, 14, 16, 20]. Parameters $c$ and $b$ of the
stationary equation (2) are Lagrange multipliers of the action (4).
Unfortunately, this formulation may suffer from non-smooth dependence of the
minimizers from Lagrange multipliers $(c,b)$ as discussed in [17, 18] after
[15]. This breakdown of the variational theory happens at the bifurcation
points for which the Hessian operator for $G_{c,b}(u)$ admits a zero
eigenvalue, where $\mathcal{L}=G_{c,b}^{\prime\prime}(\psi):H^{2}_{\rm
per}\subset L^{2}_{\rm per}\mapsto L^{2}_{\rm per}$ is given by

$\mathcal{L}=-\partial^{2}_{x}+c-6\psi^{2}.$ (5)

In [18] (based on the previous work [17] in the case of quadratic
nonlinearities), we have proposed a new variational approach to characterize
the periodic waves of the stationary equation (2) as minimizers of the
following constrained variational problem:

$r_{c,m}:=\inf_{u\in H^{1}_{\rm per}}\left\\{\mathcal{B}_{c}(u):\quad\oint
u^{4}dx=1,\quad\frac{1}{2\pi}\oint udx=m\right\\},$ (6)

where

$\mathcal{B}_{c}(u):=\frac{1}{2}\oint\left[(u^{\prime})^{2}+cu^{2}\right]dx$
(7)

It was shown in [18, Appendix B] that the minimizer exists for every
$c\in(-1,\infty)$ and every $m\in[-m_{0},m_{0}]$, where
$m_{0}:=(2\pi)^{-1/4}$. The minimizer has one maximum and one minimum on the
$2\pi$-periodic domain if $m\in(-m_{0},m_{0})$ and is given by the constant
solution if $m=\pm m_{0}$. The minimizer $\chi\in H^{1}_{\rm per}$ such that
$r_{c,m}=\mathcal{B}_{c}(\chi)$ gives the solution of the stationary equation
(2) by using the scaling transformation

$\psi=\chi\frac{\sqrt{\mathcal{B}_{c}(\chi)-\pi
cm^{2}}}{\sqrt{1-m\oint\chi^{3}dx}},$ (8)

and it was shown in [18] that $\mathcal{B}_{c}(\chi)-\pi cm^{2}>0$ and
$1-m\oint\chi^{3}dx>0$. The inverse transformation is given by
$\chi=\psi/\|\psi\|_{L^{4}}$, hence

$m=\frac{1}{2\pi\|\psi\|_{L^{4}}}\oint\psi dx.$ (9)

The family of sign-indefinite cnoidal waves for the stationary equation (2)
with $b=0$ was recovered in [18] from the variational problem (6) for
$c\in(-1,\infty)$ and $m=0$ in the space of odd periodic functions. It was
found that the family is smooth with respect to parameter $c$ but there exists
a bifurcation point $c_{0}\approx 1.425$ such that the sign-indefinite wave is
not a minimizer of the variational problem (6) for $c\in(c_{0},\infty)$ and
$m=0$. The bifurcation point $c_{0}$ coincides with the stability threshold
found in [8].

Similar study in the case of quadratic nonlinearity in [17] also showed that
the family of minimizers of the new variational problem for the travelling
periodic waves with zero mean remains smooth with respect to the wave speed
$c$.

Another example when stability of periodic waves have to be studied outside
the standard variational theory was investigated in [11] within the framework
of the Camassa–Holm equation. An alternative Hamiltonian structure was used in
order to provide smooth continuation of the periodic waves and the stability
conclusion.

Let us now explain the organization of this paper.

In Section 2, we develop the stability theory for the travelling periodic
waves given by non-degenerate local minimizers and saddle points of the
variational problem (6). We derive a precise stability criterion for a non-
degenerate local minimizer and a precise instability criterion for a saddle
point of the variational problem (6).

In Section 3, we perform the numerical search of critical points of the
variational problem (6). We show that the global minimizers remain smooth in
$(c,m)$ for every $c\in(-1,\infty)$ and $m\in(0,m_{0})$. Besides the smooth
family of global minimizers, there exist two other families of periodic waves
in a subset of the region $c\in(-1,\infty)$ and $m\in(0,m_{0})$: one family
contains local minimizers and the other family contains saddle points of the
variational problem (6). The two families disappear at the fold bifurcation
point $c_{*}(m)$. When $m\to 0$, $c_{*}(m)\to c_{0}\approx 1.425$, where the
three families are connected in the pitchfork bifurcation observed in [18]. No
other solution families have been identified in the numerical search.

Computing the stability criterion numerically, we show that the two families
of minimizers are stable in the time evolution of the mKdV equation (1)
whereas the only family of saddle points is unstable. These results generalize
the result of [18] obtained for $m=0$.

Section 4 concludes the paper with a summary and a discussion of further
questions.

##  2\. Stability theory for non-degenerate critical points

Let $\chi\in H^{1}_{\rm per}$ be a minimizer of the variational problem (6)
for $c\in(-1,\infty)$ and $m\in(0,m_{0})$ which always exists by Theorem 2.1
and Proposition 6.5 in [18]. Let $\psi\in H^{1}_{\rm per}$ be obtained by
means of the transformation (8). Then, $\psi$ satisfies the stationary
equation (2) with uniquely defined function

$b=b(c,m):=\frac{1}{\pi}\oint\psi^{3}dx-cm\|\psi\|_{L^{4}}.$ (10)

The profile $\psi$ has exactly one maximum and one minimum point on the
$2\pi$-periodic domain. The main assumption on the minimizer $\chi$ is given
as follows.

Assume that $\chi$ is a non-degenerate minimizer of the variational problem
(6) module to the translational symmetry: $\chi(x)\mapsto\chi(x+x_{0})$ for
every $x_{0}\in\mathbb{R}$.

For the solution $\psi$ of the stationary equation (2) given by (8), this
assumption implies that the Hessian operator $\mathcal{L}$ in (5) restricted
to the orthogonal complement of $\\{1,\psi^{3}\\}$ in $L^{2}_{\rm per}$ is
positive and admits a simple zero eigenvalue with the eigenfunction
$\partial_{x}\psi$. For the sake of notations, we denote the restriction of
$\mathcal{L}$ to $\\{1,\psi^{3}\\}^{\perp}$ in $L^{2}_{\rm per}$ by
$\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}}$. With the standard notations of
$n(\mathcal{L})$ and $z(\mathcal{L})$ for the number of negative eigenvalues
and the multiplicity of the zero eigenvalue of a self-adjoint operator
$\mathcal{L}$, we express the main assumption in the form:

$n(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=0,\quad
z(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=1.$ (11)

By using the implicit function argument (similar to Lemma 2.8 in [18]), it is
easy to prove that the assumption (11) implies smoothness of the family of
minimizers $\chi$ in $(c,m)$. It follows then with the help of (8) and (10)
that the function $b$ is smooth in $(c,m)$. Hence, it can be differentiated in
$(c,m)$, from which we can characterize the range of $\mathcal{L}$ by using

$\displaystyle\mathcal{L}1=c-6\psi^{2},$ (12)
$\displaystyle\mathcal{L}\psi=-b-4\psi^{3},$ (13)
$\displaystyle\mathcal{L}\partial_{c}\psi=-\partial_{c}b-\psi,$ (14)
$\displaystyle\mathcal{L}\partial_{m}\psi=-\partial_{m}b.$ (15)

We recall from [15] (see also Proposition 2.5 in [18]) that ${\rm
Ker}(\mathcal{L})={\rm span}(\partial_{x}\psi)$ if and only if
$\\{1,\psi,\psi^{2}\\}\in\textrm{Range}(\mathcal{L})$. By using this result,
it follows from (12), (14), and (15) that $z(\mathcal{L})=1$ if and only if
$\partial_{m}b\neq 0$.

We also recall that since $\partial_{x}\psi$ has exactly two zeros on the
$2\pi$-periodic domain, Sturm’s nodal theory from [15] (see also Proposition
2.4 in [18]) implies that $\mathcal{L}$ is not positive and admits at least
one negative eigenvalue: $n(\mathcal{L})\geq 1$. Since $\psi$ is related to a
minimizer of the variational problem (6) with two constraints:
$n(\mathcal{L})\leq 2$. Hence, we have $1\leq n(\mathcal{L})\leq 2$.

Next we count $n(\mathcal{L})$ based on the standard count of eigenvalues of a
self-adjoint operator under two orthogonal constraints (see Lemma 2.13 in [18]
and Theorem 4.1 in [19]). We compute the limit $\lambda\to 0$ of the following
matrix:

$P(\lambda):=\left[\begin{array}[]{cc}\langle(\mathcal{L}-\lambda
I)^{-1}\psi^{3},\psi^{3}\rangle&\langle(\mathcal{L}-\lambda
I)^{-1}\psi^{3},1\rangle\\\ \langle(\mathcal{L}-\lambda
I)^{-1}1,\psi^{3}\rangle&\langle(\mathcal{L}-\lambda
I)^{-1}1,1\rangle\end{array}\right],\quad\lambda\notin\sigma(\mathcal{L}).$
(16)

If $\partial_{m}b\neq 0$, it follows from (13) and (15) that

$\displaystyle\langle\mathcal{L}^{-1}1,1\rangle=-\frac{1}{\partial_{m}b}\partial_{m}\left(\oint\psi
dx\right),$
$\displaystyle\langle\mathcal{L}^{-1}1,\psi^{3}\rangle=-\frac{1}{4\partial_{m}b}\partial_{m}\left(\oint\psi^{4}dx\right),$
$\displaystyle\langle\mathcal{L}^{-1}\psi^{3},1\rangle=-\frac{1}{4}\oint\psi
dx-\frac{b}{4}\langle\mathcal{L}^{-1}1,1\rangle,$
$\displaystyle\langle\mathcal{L}^{-1}\psi^{3},\psi^{3}\rangle=-\frac{1}{4}\oint\psi^{4}dx-\frac{b}{4}\langle\mathcal{L}^{-1}1,\psi^{3}\rangle,$

which yields

$\displaystyle\lim_{\lambda\to 0}\det(P(\lambda))$
$\displaystyle=\langle\mathcal{L}^{-1}\psi^{3},\psi^{3}\rangle\;\langle\mathcal{L}^{-1}1,1\rangle-\langle\mathcal{L}^{-1}\psi^{3},1\rangle\;\langle\mathcal{L}^{-1}1,\psi^{3}\rangle,$
$\displaystyle=\frac{1}{4\partial_{m}b}\left[\left(\oint\psi^{4}dx\right)\partial_{m}\left(\oint\psi
dx\right)-\frac{1}{4}\left(\oint\psi
dx\right)\partial_{m}\left(\oint\psi^{4}dx\right)\right],$
$\displaystyle=\frac{1}{4\partial_{m}b}\left(\oint\psi^{4}dx\right)^{\frac{5}{4}}\partial_{m}\left(\frac{\oint\psi
dx}{\left(\oint\psi^{4}dx\right)^{\frac{1}{4}}}\right),$
$\displaystyle=\frac{\pi}{2\partial_{m}b}\|\psi\|_{L^{4}}^{5},$

where the relation (9) has been used. Thus, we conclude that the sign of
$\lim\limits_{\lambda\to 0}\det(P(\lambda))$ coincides with the sign of
$\partial_{m}b$.

Recall the counting formulas (see Proposition 2.12 in [18]):

$\left\\{\begin{array}[]{l}n(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=n(\mathcal{L})-n_{0}-z_{0}=0,\\\
z(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=z(\mathcal{L})+z_{0}-z_{\infty}=1,\end{array}\right.$
(17)

where $z_{0}$ is multiplicity of zero eigenvalue of $\lim\limits_{\lambda\to
0}P(\lambda)$, $n_{0}$ is the number of negative eigenvalues of
$\lim\limits_{\lambda\to 0}P(\lambda)$, and $z_{\infty}$ is the number of
eigenvalues $P(\lambda)$ diverging to infinity as $\lambda\to 0$.

  * •

If $\partial_{m}b<0$, then $\lim\limits_{\lambda\to 0}P(\lambda)$ has one
negative eigenvalue so that $n_{0}=1$, $z_{0}=z_{\infty}=0$ implying from (17)
that $n(\mathcal{L})=1$ and $z(\mathcal{L})=1$.

  * •

If $\partial_{m}b>0$, then $\lim\limits_{\lambda\to 0}P(\lambda)$ is either
positive or negative, but the former case would imply from (17) that
$\mathcal{L}$ is positive, in contradiction with $n(\mathcal{L})\geq 1$. Hence
$\lim\limits_{\lambda\to 0}P(\lambda)$ is negative with $n_{0}=2$,
$z_{0}=z_{\infty}=0$ implying from (17) that $n(\mathcal{L})=2$ and
$z(\mathcal{L})=1$.

  * •

If $\partial_{m}b=0$, then $z_{\infty}=1$, $z_{0}=0$ implying from (17) that
$z(\mathcal{L})=2$. Thus, one of the two negative eigenvalues of $\mathcal{L}$
for $\partial_{m}b>0$ passes zero at $\partial_{m}b=0$ and becomes a positive
eigenvalue of $\mathcal{L}$ for $\partial_{m}b<0$.

These computations are summarized as follows:

$n(\mathcal{L})=\left\\{\begin{array}[]{ll}2,&\mbox{\rm if
}\partial_{m}b>0,\\\ 1,&\mbox{\rm if }\partial_{m}b\leq
0,\end{array}\right.\qquad
z(\mathcal{L})=\left\\{\begin{array}[]{ll}2,&\mbox{\rm if }\partial_{m}b=0,\\\
1,&\mbox{\rm if }\partial_{m}b\neq 0.\end{array}\right.$ (18)

Next we determine the stability of minimizers in the time evolution of the
mKdV equation (1) by computing $n(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})$ and
$z(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})$ and by using the stability criteria
from [14]:

  * •

The periodic wave with profile $\psi$ is stable if

$n(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=0\qquad\mbox{\rm and}\qquad
z(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=1$ (19)

  * •

The periodic wave with profile $\psi$ is unstable if

$n(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=1\qquad\mbox{\rm and}\qquad
z(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=1.$ (20)

Similar to Theorem 2.14 in [18], we compute the limit $\lambda\to 0$ of the
following matrix:

$D(\lambda):=\left[\begin{array}[]{cc}\langle(\mathcal{L}-\lambda
I)^{-1}\psi,\psi\rangle&\langle(\mathcal{L}-\lambda I)^{-1}\psi,1\rangle\\\
\langle(\mathcal{L}-\lambda I)^{-1}1,\psi\rangle&\langle(\mathcal{L}-\lambda
I)^{-1}1,1\rangle\end{array}\right],\quad\lambda\notin\sigma(\mathcal{L}).$
(21)

If $\partial_{m}b\neq 0$, it follows from (14) and (15) that

$\displaystyle\langle\mathcal{L}^{-1}1,1\rangle$
$\displaystyle=-\frac{1}{\partial_{m}b}\partial_{m}\left(\oint\psi dx\right),$
$\displaystyle\langle\mathcal{L}^{-1}1,\psi\rangle$
$\displaystyle=-\frac{1}{2\partial_{m}b}\partial_{m}\left(\oint\psi^{2}dx\right),$
$\displaystyle\langle\mathcal{L}^{-1}\psi,1\rangle$
$\displaystyle=-\partial_{c}\left(\oint\psi
dx\right)-\partial_{c}b\;\langle\mathcal{L}^{-1}1,1\rangle,$
$\displaystyle\langle\mathcal{L}^{-1}\psi,\psi\rangle$
$\displaystyle=-\frac{1}{2}\partial_{c}\left(\oint\psi^{2}dx\right)-\partial_{c}b\;\langle\mathcal{L}^{-1}1,\psi\rangle,$

which yields

$\displaystyle\lim_{\lambda\to 0}\det(D(\lambda))$
$\displaystyle=\langle\mathcal{L}^{-1}\psi,\psi\rangle\;\langle\mathcal{L}^{-1}1,1\rangle-\langle\mathcal{L}^{-1}\psi,1\rangle\;\langle\mathcal{L}^{-1}1,\psi\rangle,$
$\displaystyle=\frac{1}{2\partial_{m}b}\left[\partial_{c}\left(\oint\psi^{2}dx\right)\partial_{m}\left(\oint\psi
dx\right)-\partial_{m}\left(\oint\psi^{2}dx\right)\partial_{c}\left(\oint\psi
dx\right)\right]$
$\displaystyle=\frac{1}{2\partial_{m}b}\begin{vmatrix}\partial_{c}\mathcal{F}(c,m)&\partial_{m}\mathcal{F}(c,m)\\\
\partial_{c}\mathcal{M}(c,m)&\partial_{m}\mathcal{M}(c,m)\end{vmatrix},$

where $\mathcal{F}(c,m):=F(\psi)$ and $\mathcal{M}(c,m):=M(\psi)$ are computed
from the two conserved quantities in (3) at the family of periodic waves with
the profile $\psi$ that depends on parameters $(c,m)$. Note that the
determinant in the last expression is the Jacobian of the transformation
$(c,m)\mapsto(\mathcal{F},\mathcal{M})$.

We shall now use the counting formulas:

$\left\\{\begin{array}[]{l}n(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=n(\mathcal{L})-n_{0}-z_{0},\\\
z(\mathcal{L}|_{\\{1,\psi\\}^{\perp}})=z(\mathcal{L})+z_{0}-z_{\infty},\end{array}\right.$
(22)

where $z_{0}$, $n_{0}$, and $z_{\infty}$ have the same meaning as in (17) but
for the matrix $D(\lambda)$.

  * •

If $\partial_{m}b<0$, then it follows from (18) that $n(\mathcal{L})=1$ and
$z(\mathcal{L})=1$ so that the stability criterion (19) is satisfied if and
only if $n_{0}=1$, $z_{0}=z_{\infty}=0$, which is true if and only if the
Jacobian of the transformation $(c,m)\mapsto(\mathcal{F},\mathcal{M})$ is
strictly positive.

  * •

If $\partial_{m}b>0$, then it follows from (18) that $n(\mathcal{L})=2$ and
$z(\mathcal{L})=1$ so that the stability criterion (19) is satisfied if and
only if $n_{0}=2$, $z_{0}=z_{\infty}=0$, that is,
$\langle\mathcal{L}^{-1}1,1\rangle<0$ and the Jacobian of the transformation
$(c,m)\mapsto(\mathcal{F},\mathcal{M})$ is strictly positive. Since
$\langle\mathcal{L}^{-1}1,1\rangle$ is the same diagonal term of both
$\lim\limits_{\lambda\to 0}P(\lambda)$ and $\lim\limits_{\lambda\to
0}D(\lambda)$ whereas the former is strictly negative, the first condition of
$\langle\mathcal{L}^{-1}1,1\rangle<0$ is satisfied.

  * •

If $\partial_{m}b=0$, then it follows from (18) that $n(\mathcal{L})=1$ and
$z(\mathcal{L})=2$ so that $\det D(\lambda)$ is singular in the limit
$\lambda\to 0$. Hence $z_{\infty}=1$ and one of the two negative eigenvalues
of $\lim\limits_{\lambda\to 0}D(\lambda)$ for $\partial_{m}b>0$ diverges to
infinity as $\partial_{m}b\to 0$, whereas the other eigenvalue remains
negative if and only if the Jacobian of the transformation
$(c,m)\mapsto(\mathcal{F},\mathcal{M})$ is strictly positive. This yields
$n_{0}=1$, $z_{0}=0$, and hence the stability criterion (19).

The stability criterion in all three cases can be summarized as follows.

Let $\psi\in H^{1}_{\rm per}$ be a solution of the stationary equation (2)
satisfying (11). The periodic wave with the profile $\psi$ is stable in the
time evolution of the mKdV equation (1) if
$\begin{vmatrix}\partial_{c}\mathcal{F}(c,m)&\partial_{m}\mathcal{F}(c,m)\\\
\partial_{c}\mathcal{M}(c,m)&\partial_{m}\mathcal{M}(c,m)\end{vmatrix}>0.$
(23)

Note that the assumption (11) is satisfied for the solution $\psi\in
H^{1}_{\rm per}$ related to both the global and local non-degenerate
minimizers $\chi\in H^{1}_{\rm per}$ of the variational problem (6). Let us
now consider the non-degenerate saddle points.

Assume that $\chi$ is a non-degenerate saddle point of the variational problem
(6) module to the translational symmetry: $\chi(x)\mapsto\chi(x+x_{0})$ for
every $x_{0}\in\mathbb{R}$ with exactly one negative direction in $H^{1}_{\rm
per}$ under the two constraints.

The main assumption for the corresponding solution $\psi\in H^{1}_{\rm per}$
of the stationary equation (2) can be expressed in the form:

$n(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=1,\quad
z(\mathcal{L}|_{\\{1,\psi^{3}\\}^{\perp}})=1.$ (24)

The non-degeneracy of saddle point implies smoothness of the function $b$ in
$(c,m)$. As a result, the same count of $n(\mathcal{L})$ can be performed
based on $\lim\limits P(\lambda)$ with exactly the same expression for
$\lim\limits_{\lambda\to 0}\det(P(\lambda))$ if $\partial_{m}b=0$. Since
$\lim\limits_{\lambda\to 0}\det(P(\lambda))\neq 0$, it follows from (17) with
$z_{0}=z_{\infty}=0$ that $n(\mathcal{L})=2$ if $\partial_{m}b<0$ for which
$n_{0}=1$. Since $1\leq n(\mathcal{L})\leq 2$ is no longer true for the saddle
points, the case with $\partial_{m}b>0$ may either give $n(\mathcal{L})=1$ or
$n(\mathcal{L})=3$. To avoid ambiguity, we only consider the saddle points
with $\partial_{m}b<0$.

It follows from (22) that if $\partial_{m}b<0$, then the instability criterion
(20) is satisfied if and only if the Jacobian of the transformation
$(c,m)\mapsto(\mathcal{F},\mathcal{M})$ is strictly positive for which
$n_{0}=1$, $z_{0}=z_{\infty}=0$. The instability criterion can be summarized
as follows.

Let $\psi\in H^{1}_{\rm per}$ be a solution of the stationary equation (2)
satisfying (24). The periodic wave with the profile $\psi$ is unstable in the
time evolution of the mKdV equation (1) if $\partial_{m}b<0$ and
$\begin{vmatrix}\partial_{c}\mathcal{F}(c,m)&\partial_{m}\mathcal{F}(c,m)\\\
\partial_{c}\mathcal{M}(c,m)&\partial_{m}\mathcal{M}(c,m)\end{vmatrix}>0.$
(25)

Although the stability and instability criteria (23) and (25) are only
sufficient conditions, we show with the help of numerical approximations that
these criteria cover all critical points of the variational problem (6).

##  3\. Numerical search of critical points of the variational problem

To perform the numerical search, we use the analytical representation of
solutions to the stationary equation (2) in terms of the Jacobi elliptic
functions. Such representations are known in the literature; we refer to [6]
for precise details.

One family of exact solutions is given by

$\psi(x)=u_{4}+\frac{(u_{1}-u_{4})(u_{2}-u_{4})}{(u_{2}-u_{4})+(u_{1}-u_{2}){\rm\,sn}^{2}(\nu
x;k)},$ (26)

where the turning points $u_{1}$, $u_{2}$, $u_{3}$, $u_{4}$ satisfy the
constraint

$u_{1}+u_{2}+u_{3}+u_{4}=0$ (27)

and define parameters $c$ and $b$ of the stationary equation (2) by

$\displaystyle\begin{cases}c=-(u_{1}u_{2}+u_{1}u_{3}+u_{1}u_{4}+u_{2}u_{3}+u_{2}u_{4}+u_{3}u_{4}),\\\
b=\frac{1}{2}(u_{1}u_{2}u_{3}+u_{1}u_{2}u_{4}+u_{1}u_{3}u_{4}+u_{2}u_{3}u_{4}).\end{cases}$
(28)

Parameters $\nu$ and $k$ of the solution (26) are expressed by the relations:

$\nu=\frac{1}{2}\sqrt{(u_{1}-u_{3})(u_{2}-u_{4})},\quad
k=\frac{\sqrt{(u_{1}-u_{2})(u_{3}-u_{4})}}{\sqrt{(u_{1}-u_{3})(u_{2}-u_{4})}}.$

When a given value of $c\in(-1,\infty)$ is substituted into the first equation
in (28) and the additional constraint (27) is used, the family of solutions
(26) has two arbitrary parameters among the four turning points, which we
choose to be $u_{1}$ and $u_{2}$. These two parameters are defined from two
additional constraints: the period of $\psi$ must be normalized to $2\pi$ by
$2K(k)=2\pi\nu$, where $K(k)$ is the complete elliptic integral, and the mean
value and the $L^{4}$ norm of the solution $\psi$ must be related to a given
value of $m\in(-m_{0},m_{0})$ by (9). Newton’s method is used to solve these
two constraints.

Another family of exact solutions is given by

$\psi(x)=u_{1}+\frac{(u_{2}-u_{1})(1-{\rm\,cn}(\nu x;k))}{1-{\rm\,cn}(\nu
x;k)+\delta(1+{\rm\,cn}(\nu x;k))},$ (29)

with the turning points $u_{1}$, $u_{2}$, $u_{3}=\alpha+i\beta$,
$u_{4}=\alpha-i\beta$. The turning points satisfy the constraint

$u_{1}+u_{2}+2\alpha=0.$ (30)

and define parameters $c$ and $b$ of the stationary equation (2) by

$\displaystyle\begin{cases}c=-(u_{1}u_{2}+2\alpha(u_{1}+u_{2})+\alpha^{2}+\beta^{2}),\\\
b=\alpha
u_{1}u_{2}+\frac{1}{2}(u_{1}+u_{2})(\alpha^{2}+\beta^{2}).\end{cases}$ (31)

Parameters $\nu$, $k$, and $\delta$ of the solution (29) are expressed by the
relations

$\delta=\frac{\sqrt{(u_{2}-\alpha)^{2}+\beta^{2}}}{\sqrt{(u_{1}-\alpha)^{2}+\beta^{2}}},\quad\nu=\sqrt[4]{\left[(u_{1}-\alpha)^{2}+\beta^{2}\right]\left[(u_{2}-\alpha)^{2}+\beta^{2}\right]},$

and

$k=\frac{1}{\sqrt{2}}\sqrt{1-\frac{(u_{1}-\alpha)(u_{2}-\alpha)+\beta^{2}}{\sqrt{(u_{1}-\alpha)^{2}+\beta^{2}}\sqrt{(u_{2}-\alpha)^{2}+\beta^{2}}}},$

Again we use (30) to define $\alpha$ and the first equation in (31) to define
$\beta$ from $c\in(-1,\infty)$. The remaining parameters $u_{1}$ and $u_{2}$
are computed from two additional constraints: the period of $\psi$ must be
normalized to $2\pi$ by $4K(k)=2\pi\nu$ and the mean value and the $L^{4}$
norm of the solution $\psi$ must be related to a given value of
$m\in(-m_{0},m_{0})$ by (9). Newton’s method is used to solve these two
constraints.

(a) $m=0$

(b) $m=0.01$

(c) $m=0.1$

(d) $m=0.2$

Figure 1. Three solution families on the $(b,c)$ diagram for fixed values of
$m$.

Figure 1 presents the main result in obtaining numerical solutions from the
exact solutions (26) and (29) with parameters found from Newton’s method for
$c\in(-1,\infty)$ and $m\in[0,m_{0})$. Three solution families are shown for
$b$ versus $c$ for fixed values of $m$. The solid curves represent solutions
of the form (29) and the dotted curves are solutions of the form (26). The
black and blue dots demark the points at which the two solution forms connect.
Continuing one solution form across these points is impossible because the
elliptic modulus $k$ becomes complex-valued. The red dot shows pitchfork
bifurcation (for $m=0$) and fold bifurcation (for $m\neq 0$) when the solution
families coalesce.

Figure 1(A) agrees with Figure 2 (middle left panel) in [18] obtained from
numerical solutions of the stationary equation (2). For $m=0$, there exists
$c_{0}\approx 1.425$ (red dot) at which the pitchfork bifurcation occurs. For
$m\neq 0$ in Figures 1(B-D), the symmetry is broken, the bifurcation point
$c_{*}(m)$ such that $c_{*}(m)\to c_{0}$ as $m\to 0$ detaches from the upper
branch but remains the connection point for the middle and lower solution
families. As $m$ increases, $c_{*}(m)$ at the bifurcation point increases
rapidly and the two solution families move to larger values of $c$.

(a)

(b)

Figure 2. (A) The same as Figure 1 but for the upper solution family only for
$m=0.2,0.3,0.4,0.5$. (B) Comparison between the upper solution family for
$m=0.6$ and the exact solution for $m=m_{0}$.

(a) $m=0.01$

(b) $m=0.1$

Figure 3. The value of $\mathcal{B}_{c}(\chi)$ for the three critical points
of the variational problem (6) versus $c$ for two values of $m$.

Figure 2(A) shows only the upper solution family on the $(b,c)$ plane but for
larger values of $m$ compared to Fig. 1. Figure 2(B) compares the numerical
result for $m=0.6$ with the analytical result for $m=m_{0}\approx 0.6316$ for
which the solution family for the constant solutions is given in the
parametric form

$c=6u_{1}^{2}-1,\quad b=u_{1}-4u_{1}^{3},\quad u_{1}\in(0,\infty).$ (32)

This exact solution for $m=m_{0}$ follows from either (26) or (29) with
$u_{2}=u_{1}$. Although we do not distinguish between the two solution forms
(26) and (29) in the same solid lines on Figure 2, the connection point
between the two solutions is shown and it moves to smaller values of $c$ as
$m$ increases.

Figure 3 clarifies the meaning of each of the three solution families among
the critical points of the variational problem (6). It shows the values of
$\mathcal{B}_{c}(\chi)$ defined in (7) versus $c$ for fixed values of $m=0.01$
(left) and $m=0.1$ (right), where $\chi$ is computed from $\psi$ by using
$\chi=\psi/\|\psi\|_{L^{4}}$. To compute $\mathcal{B}_{c}(\chi)$, we use
forward finite difference to approximate $\chi^{\prime}$ and then complete the
quadrature using trapezoidal rule. In both panels, the blue, red and black
curves correspond respectively to the upper, middle and lower families of
solutions at the bifurcation diagrams of Figure 1. We observe that the blue
curves represent the global minimizers, the black curves represent the local
minimizers, and the red curves represent the saddle points. The numerical
search shows that no other solutions of the stationary equation (2) yield
critical points of the variational problem (6).

In order to apply the stability criterion (23) for non-degenerate minimizers
of the variational problem (6), we glue individual computations together and
represent the solution surface of $b=b(c,m)$ versus $(c,m)$.

(a)

(b)

Figure 4. (A) The solution surface of $b(c,m)$ for the global minimizers of
the variational problem (6). (B) Two solution surfaces of $b(c,m)$ for the
local minimizers and saddle points of the variational problem (6) which
connect at the fold bifurcation.

Figure 4(A) shows the smooth solution surface for the global minimizers of the
variational problem (6) given by the upper solution family on Figure 1 for
$c\in(-1,\infty)$ and $m\in(0,m_{0})$. It suggests non-degeneracy of the
global minimizers except at the point of the fold bifurcation for $m=0$ and
$c=c_{0}\approx 1.425$. The red curve on the solution surface in Figure 4
denotes the connection line between the two solution forms (26) and (29).

Figure 4(B) shows the solution surface $b(m,c)$ for the other two critical
points of the variational problem (6). The top (red) part of the surface
corresponds to the saddle points and the bottom (black) part of the surface
relates to the local minimizers. The numerical result also suggests that the
surface is smooth except at the points of the fold bifurcation where the
saddle points connect with the local minimizers. The black line denotes the
connection line between the two solution forms (26) and (29).

###  3.1. Stability of the global minimizers

It follows from (18) that the Morse index $n(\mathcal{L})$ and the degeneracy
index $z(\mathcal{L})$ depend on the derivative $\partial_{m}b$. Figure 5
shows $b$ versus $m$ for fixed values of $c$. For $m=0$, the derivative
$\partial_{m}b$ changes sign from positive to negative at $c_{1}\approx 3.1$.
According to (18), it corresponds to the change in the Morse index
$\mathcal{L}$ from $n(\mathcal{L})=2$ for $c\in(-1,c_{1})$ to
$n(\mathcal{L})=1$ for $c\in(c_{1},\infty)$. This agrees with Figure 2 (bottom
left panel) in [18].

Figure 5. $b$ versus $m$ at the solution surface of Fig. 4(A) for various
values of $c=-0.99$ (top left), $c=0.51$ (top right), $c=1.26$ (bottom left)
and $c=3.53$ (bottom right).

For fixed $c\in(-1,\infty)$, it follows from Figure 5 that there exists
$m_{1}(c)$ for $c\in(-1,c_{1})$ such that $n(\mathcal{L})=2$ for
$m\in(0,m_{1}(c))$ and $n(\mathcal{L})=1$ for $m\in(m_{1}(c),m_{0})$. Because
$z(\mathcal{L})=2$ at $m=m_{1}(c)$, the non-degeneracy assumption used in the
conventional stability theory for periodic waves (see [15, 17] and references
therein) is not satisfied at $m=m_{1}(c)$. In particular, minimizers of energy
$E(u)$ for fixed momentum $F(u)$ and mass $M(u)$ are not smooth with respect
to $(c,b)$ at the degeneracy point $m=m_{1}(c)$. This drawback of the
conventional stability theory is not present for the minimizers of the new
variational problem (6).

Figure 6. Top: Mass $\mathcal{M}(c,m)$ versus $c$ for various values of $m$.
Bottom: Mass $\mathcal{M}(c,m)$ versus $m$ for various values of $c$.

Figure 7. Top: Momentum $\mathcal{F}(c,m)$ versus $c$ for various values of
$m$. Bottom: Momentum $\mathcal{F}(c,m)$ versus $m$ for various values of $c$.

Figures 6 and 7 show respectively the mass $\mathcal{M}(c,m)$ and the momentum
$\mathcal{F}(c,m)$ versus $c$ at different values of $m$ (top) and versus $m$
at different values of $c$ (bottom). It is clear that the mass
$\mathcal{M}(c,m)$ is monotonically increasing in both $c$ and $m$, whereas
the momentum $\mathcal{F}(c,m)$ is monotonically increasing in $c$ for every
$m\in(0,m_{0})$ and monotonically decreasing in $m$ for every $c\in(0,c_{0})$,
where $c_{0}$ is the same bifurcation value of $c$ for the pitchfork
bifurcation at $m=0$. With these signs of partial derivatives, the stability
condition (23) is always satisfied for $c\in(0,c_{0})$ and $m\in(0,m_{0})$.
Note that $\partial_{m}\mathcal{F}(c,0)=0$ and
$\partial_{m}\mathcal{M}(c,0)>0$ so that the stability criterion (23) reduces
to $\partial_{c}\mathcal{F}(c,0)>0$, monotonicity of the mapping
$c\mapsto\mathcal{F}(c,0)$, which was the main stability criterion used in
[17] and [18].

It follows from Figure 7 that for $c>c_{0}$, there exists
$m_{*}(c)\in(0,m_{0})$ such that the momentum $\mathcal{F}(c,m)$ is
monotonically decreasing in $m$ for $m\in(0,m_{*}(c))$ and monotonically
increasing in $m$ for $m\in(m_{*}(c),m_{0})$. It is not obvious in the latter
case if the stability criterion (23) is satisfied. Figure 8 shows the contour
plot for the Jacobian in (23) for all $c\in(-1,5)$ and $m\in(0,m_{0})$, which
is strictly positive with the minimal value of $0.0145$ attained at the corner
point shown by the red dot. Therefore, the stability criterion (23) is
satisfied for every non-degenerate minimizer of the variational problem (6).

Figure 8. Contour plot of the Jacobian in the stability criterion (23) for the
global minimizers.

###  3.2. Instability of saddle points

Saddle points of the variational problem (6) corrrespond to the middle
solution family on Figure 1. The solution surface for the saddle point
connects to the solution surface for the local minimizers according to Figure
4(B).

The instability criterion for the saddle points (25) was derived under the
assumption of $\partial_{m}b<0$. Figure 9 shows $b$ versus $m$ for two values
of $c$, which suggests that $\partial_{m}b<0$ is satisfied for all saddle
points of the variational problem (6).

Figure 9. $b$ versus $m$ for the solution surface of Fig. 4(B) for $c=5$ and
$c=3$.

We have found numerically that the mass $\mathcal{M}(c,m)$ of the saddle
points is monotonically increasing in both $m$ and $c$ similar to Figure 6 for
the global minimizers. We also found that the momentum $\mathcal{F}(c,m)$ is
monotonically increasing in $c$ but there exist $c_{1}$ and $c_{2}$ satisfying
$3<c_{1}<c_{2}<9$ such that $\mathcal{F}(c,m)$ is monotonically decreasing in
$m$ for $c<c_{1}$ and monotonically increasing in $m$ for $c>c_{2}$ similar to
Figure 7. Although the sign of the Jacobian in (25) is not obvious in the
latter case, we have computed it numerically and confirmed that the Jacobian
is strictly positive for the entire solution surface. Thus, the saddle points
of the variational problem (6) are unstable in the time evolution of the mKdV
equation (1) according to the instability criterion (25).

###  3.3. Stability of local minimizers

Local minimizers of the variational problem (6) corrrespond to the lower
solution family on Figure 1. We have checked numerically that the plots of $b$
versus $m$, $\mathcal{M}(c,m)$ and $\mathcal{F}(c,m)$ versus both $c$ and $m$
are qualitatively similar to Figures 5, 6, and 7. This is not surprising since
the lower and upper solution families are equivalent to each other for $m=0$.

We have also detected numerically that the Jacobian in the stability criterion
(23) remains positive for the entire solution surface. Thus, the local
minimizers of the variational problem (6) are stable in the time evolution of
the mKdV equation (1) according to the stability criterion (23).

##  4\. Conclusion

The new variational characterization of periodic waves as non-degenerate
minimizers of the variational problem (6) has several advantages compared to
the previous variational theory, where the energy $E(u)$ is minimized for
fixed momentum $F(u)$ and mass $M(u)$. First, the stability criterion is
independent of whether the Morse index $n(\mathcal{L})$ is one or two and
whether the linear operator $\mathcal{L}$ is degenerate with
$z(\mathcal{L})=2$. Second, with the exception of the pitchfork bifurcation
point $(c,m)=(c_{0},0)$, minimizers of the variational problem (6) are always
non-degenerate.

The new variational characterization has also advantages compared to other
(partial) characterizations of periodic waves in the mKdV equation such as
minimization of energy $E(u)$ for fixed momentum $F(u)$ in [13] or
minimization of $B_{c}(u)$ for fixed $L^{4}$ norm in the space of even
functions also considered in [18]. The former minimization only allows to
identify a subset of stable periodic waves as it only applies to the periodic
waves with $n(\mathcal{L})=1$. The latter minimization requires to proceed
with an additional Galilean transformation in order to identify the stability
criterion for the periodic waves and leads to computational formulas which are
not related to the dependence of mass $M(u)$ or momentum $F(u)$ on $c$.

Although our computations are only based on numerical approximations, the
numerical results are rather accurate since we use the exact analytical
representations of the periodic wave solutions. We have confirmed that the
periodic waves that correspond to the global and local minimizers of the
variational problem (6) are stable in the time evolution of the mKdV equation
(1), whereas the periodic waves for the saddle points are unstable.

The main direction to be addressed in further work is to prove analytically
that minimizers of the variational problem (6) are non-degenerate in the
entire existence interval with the exception of the pitchfork bifurcation
point at $(c,m)=(c_{0},0)$. Extensions of these numerical results to the
modified Benjamin–Ono equation or the fractional mKdV equation are also of the
highest priority. Finally, one can apply the same new variational problem to
the generalized fractional KdV equations with powers different from the
quadratic and cubic powers considered in [17] and [18] respectively.

Acknowledgments. U. Le acknowledges the support of McMaster graduate
scholarship. D.E. Pelinovsky acknowledges the support of the NSERC Discovery
grant.

## References

  * [1] S. Amaral, H. Borluk, G. M. Muslu, F. Natali, and G. Oruc, On the existence, uniqueness, and stability of periodic waves for the fractional Benjamin–Bona–Mahony equation, Stud. Appl. Math. 148 (2022), in press, https://doi.org/10.1111/sapm.12428
  * [2] T.P. Andrade and A. Pastor, Orbital stability of one-parameter periodic traveling waves for dispersive equations and applications, J. Math. Anal. Appl. 475 (2019), 1242–1275.
  * [3] J. Angulo, Non-linear stability of periodic travelling-wave solutions for the Schrödinger and modified Korteweg-de Vries equation, J. Diff. Equat. 235 (2007), 1–30.
  * [4] J. Angulo and F. Natali, On the instability of periodic waves for dispersive equations, Diff. Int. Equat. 29 (2016), 837–874.
  * [5] J. C. Bronski, M. A. Johnson, and T. Kapitula, An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 1141–1173
  * [6] J. Chen and D.E. Pelinovsky, Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background, J. Nonlin. Sci. 29 (2019), 2797–2843.
  * [7] S.N. Chow and J.A. Sanders, On the number of critical points of the period, J. Diff. Eqs. 64 (1986), 51–66.
  * [8] B. Deconinck and T. Kapitula, On the spectral and orbital stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations, in Hamiltonian Partial Diff. Eq. Appl. 75 (Fields Institute Communications, Springer, New York, 2015), 285–322.
  * [9] B. Deconinck and M. Nivala, M. The stability analysis of the periodic traveling wave solutions of the mKdV equation, Stud. Appl. Math. 126 (2011) 17–48.
  * [10] L. Gavrilov, Remark on the number of critical points on the period, J. Diff. Eqs. 101 (1993), 58–65.
  * [11] A. Geyer, R. H. Martins, F. Natali, and D. E. Pelinovsky, Stability of smooth periodic travelling waves in the Camassa–Holm equation, Stud. Appl. Math. 148 (2022), in press, DOI: 10.1111/sapm.12430
  * [12] R. Grimshaw, Nonlinear wave equations for the oceanic internal solitary waves, Stud. Appl. Math. 136 (2016), 214–237.
  * [13] S. Hakkaev and A.G. Stefanov, Stability of periodic waves for the fractional KdV and NLS equations, Proc. R. Soc. Edinburgh A 151 (2021), 1171–1203.
  * [14] M. H$\check{a}$r$\check{a}$guş and T. Kapitula, On the spectra of periodic waves for infinite-dimensional Hamiltonian systems, Phys. D 237 (2008), 2649–2671.
  * [15] V. M. Hur and M. Johnson, Stability of periodic traveling waves for nonlinear dispersive equations, SIAM J. Math. Anal. 47 (2015) 3528–3554.
  * [16] M.A. Johnson, Nonlinear stability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation, SIAM J. Math. Anal. 41 (2009), 1921–1947.
  * [17] F. Natali, D. Pelinovsky and U. Le, New variational characterization of periodic waves in the fractional Korteweg–de Vries equation, Nonlinearity 33 (2020), 1956–1986.
  * [18] F. Natali, U. Le, and D.E. Pelinovsky, Periodic waves in the modified fractional Korteweg–de Vries equation, J. Dyn. Diff. Equat. (2021). https://doi.org/10.1007/s10884-021-10000-w
  * [19] D. E. Pelinovsky, Localization in Periodic Potentials: From Schrödinger Operators to the Gross–Pitaevskii Equation, LMS Lecture Note Series 390 (Cambridge University Press, Cambridge, 2011).
  * [20] D. E. Pelinovsky, Spectral stability of nonlinear waves in KdV-type evolution equations, In Nonlinear Physical Systems: Spectral Analysis, Stability, and Bifurcations (Edited by O.N. Kirillov and D.E. Pelinovsky) (Wiley-ISTE, NJ, 2014), 377–400.
  * [21] E. Pelinovsky, T. Talipova, I. Didenkulova, and E. Didenkulova, Interfacial long traveling waves in a two-layer fluid with variable depth, Stud. Appl. Math. 142 (2019), 513–527.
  * [22] B. R. Sutherland, Internal Gravity Waves (Cambridge University Press, Cambridge, 2010).
  * [23] K. Yagasaki, Monotonicity of the period function for $u^{\prime\prime}-u+u^{p}=0$ with $p\in\mathbb{R}$ and $p>1$, J. Diff. Eqs. 255 (2013), 1988–2001.