∎

11institutetext: Guowei Dai 22institutetext: School of Mathematical Sciences,
Dalian University of Technology, Dalian, 116024, People’s Republic of China
22email<EMAIL_ADDRESS>33institutetext: Fei Xu 44institutetext:
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013,
People’s Republic of China
44email<EMAIL_ADDRESS>55institutetext: Yong Zhang (Corresponding author)
66institutetext: Institute of Applied System Analysis, Jiangsu University,
Zhenjiang 212013, People’s Republic of China
66email<EMAIL_ADDRESS>
# Periodic travelling interfacial electrohydrodynamic waves: bifurcation and
secondary bifurcation

Guowei Dai  Fei Xu  Yong Zhang

(Received: date / Accepted: date)

###### Abstract

In this paper, two-dimensional periodic capillary-gravity waves travelling
under the effect of a vertical electric field are considered. The full system
is a nonlinear, two-layered and free boundary problem. The interface dynamics
arises from the coupling between the Euler equations for the lower fluid layer
and an electric contribution from the upper gas layer. To investigate the
electrohydrodynamic wave interactions, we first introduce the naive flattening
technique to transform the free boundary problem into a fixed boundary
problem. Then we prove the existence of the small-amplitude
electrohydrodynamic waves with constant vorticity $\gamma$ by using local
bifurcation theory. Moreover, we prove that these electrohydrodynamic waves
are formally stable in linearized sense. Furthermore, we obtain a secondary
bifurcation curve that emerges from the primary branch at a nonlaminar
solution as $E_{0}$ being close to some special value. This secondary
bifurcation curve consists of ripples solutions on the interface of a
conducting fluid under normal electric fields. As far as we know, this new
phenomenon in electrohydrodynamics (EHD) is first established mathematically.
It is worth noting that the electric field $E_{0}$ plays a key role to control
the shapes and types of waves on the interface.

###### Keywords:

Electrohydrodynamics, Capillary-gravity waves, Bifurcation, Stability, Ripples

###### MSC:

76B15, 35Q35, 76B03.

††journal: myjournal

##  1 Introduction

Water waves propagating on the interface between two fluids have been studied
intensively via using either analytical or numerical methods. Many different
mathematical methods have been introduced to study the steady or time-
dependent solutions both in shallow and deep waters Vanden . These waves are
usually created by the presence of two different layers of different density
combined with a certain configuration of current. As far as we know, the first
small-amplitude interfacial periodic traveling capillary-gravity waves on
finite depth was constructed by Amick and Turner AmickT , where the vorticity
of fluid was ignored. When vorticity is included, it was unclear for long time
how to leave the regime of small perturbations of configurations with a flat
surface. The breakthrough in this direction is due to Constantin and Strauss
Constantin ; ConstantinS , who utilised the semi-hodograph transformation of
Dubreil-Jacotin DubreilJ . Based on this, the work WalshOS  allowed for the
vorticity in the interfacial wind waves without considering stagnation pionts.
Recently, Ambrose, Strauss, and Wright AmbroseSW  considered the new
coordinates and applied the Rabinowitz’s global bifurcation theorem to prove
the existence of pure capillary interfacial waves and gravity-capillary
interfacial waves. We refer to ChuDE ; Sinambela ; Wheeler  for more
information on two-layer or many-layer density stratified water waves. We also
would like to mention the work JonesT85 ; MartinM , where they studied a
pseudodifferential equation by finding the secondary bifurcation branches
emerging from the primary curves for capillary-gravity water waves without or
with vorticity.

In the presence of electric fields, this topic is called Electrohydrodynamics
(EHD), which has recently attracted much attention. It is easy to artificially
manipulate considerably strong electric fields with modern engineering
techniques to cause significant changes in fluid motion, usually manifested
via modifications of the gas-liquid interface dynamics. As a result, EHD
enjoys wide industrial applications in chemical engineering, e.g., coating
processes, cooling systems of conducting fluids, electrospray technology, etc.
(see MelcherT ; Papageorgiou  for a comprehensive overview). Due to the
important role that EHD plays in the engineering community, an in-depth
understanding of the mathematics behind the scene remains essential. As far as
we know, the travelling waves propagating in two-dimensional space under the
effects of gravity, surface tension, and electric field have been studied
intensively by different authors, either with weakly nonlinear models Gleeson
; HuntD ; Wang  or with numerical methods Doak ; DoakG ; Gao ; LinZW .
However, there have been, so far, few studies fully nonlinear water waves with
vorticity under the influence of electric fields by using the analytical
method except the work Smit , where authors considered singularities in the
Electrohydrodynamic equations with ignoring surface tension for simplicity.

In this work, we investigate the resonances mentioned above in the electrified
Euler equation with surface tension by using the bifurcation theory. The
novelty here is however that we remove the assumption of irrotationality in
the case of electrohydrodynamics with an assumption of globally constant
vorticity in the lower fluid. The other contribution of the paper is to reveal
the effect of electric field $E_{0}$ on the appearance of different types of
electrohydrodynamic waves. Especially, we prove that there would be periodic
electrohydrodynamic waves if (3.24) holds and establish the existence of
ripples for the first time on the interface as $E_{0}$ attains some value,
which may be useful in physical and industrial applications. Our results are
true for electrohydrodynamic capillary as well. A very delicate issue is to
reformulate the problem such that it becomes amenable to a certain local
bifurcation theorem in order to construct these waves of small amplitude. To
this end and what may seem surprisingly, the flattening transform plays a key
role. It is known that this transformation can admit the occurrence of
stagnation points and critical layers inside fluid EhrnstromEW ; HenryBM ;
Wahlen09 . However, this is an interesting and important question but beyond
the scope of the preliminary investigations here.

The rest of the paper is structured as follows. The problem is mathematically
formulated in Section 2. We prove the existence and stability of
electrohydrodynamic periodic capillary-gravity waves by using the celebrated
Crandall-Rabinowitz theorems CrandallR ; CrandallR1  in Section 3. Finally,
the ripples are shown by applying the secondary bifurcation theorem Shearer
in Section 4.

##  2 Preliminaries

###  2.1 Introduction of the problem

We consider a two-dimensional steady, inviscid, incompressible, and perfectly
conducting fluid of constant density $(\rho=1)$ and finite depth bounded below
by a flat electrode. We first introduce a Cartesian coordinates $(x,y)$ such
that $x$-axis points to the horizontal and $y$-axis points to the vertical.
Assume that the flat electrode is given by $y=-d$ with $d>0$, the free
interface is given by $y=\eta(x)$ and find the function $\psi(x,y)$ called the
stream function, providing the velocity field $(\psi_{y},-\psi_{x})$ of the
fluid, which satisfies the following equations and boundary conditions:

$\displaystyle\left\\{\begin{array}[]{llll}{\Delta\psi=\gamma}&{\text{ for
}-d<y<\eta(x)},\\\ {\psi=0}&{\text{ on }~{}y=\eta(x)},\\\ {\psi=m}&{\text{ on
}~{}y=-d},\end{array}\right.$ (2.4)

where $\gamma$ is the constant vorticity meaning that the flow is to be
rotational and the constant $m$ is the relative mass flux defined by
$m=\int_{-d}^{\eta(x)}-\psi_{y}\,dy$.

In addition, we assume that the fluid is perfectly conducting so that the
electric strength is zero within the fluid. The surrounding medium, which
occupies the region above the liquid, is assumed to be dielectric with
permittivity $\epsilon_{0}$. Its density is very small and negligible in
comparison to that of the conducting fluid. The upper layer is also bounded by
a flat electrode $y=d$. The electrostatic limit of Maxwell’s equation implies
that the induced magnetic fields are negligible, and it then follows that the
electric field is irrotational due to Faraday’s law. Therefore, we can
introduce the voltage potential $V$, such that the vertical electric field
above the liquid surface satisfies $\mathbf{E}=\nabla V$ with
$\mathbf{E}=(0,E_{0})$ near the above electrode $y=d$ for some constant
$E_{0}$, and hence the voltage potential satisfies

$\Delta V=0\quad\text{for}\quad\eta(x)<y<d.$ (2.5)

Due to the conducting nature of the fluid, the voltage potential $V$ is
invariant in the fluid domain and chosen to be zero without loss of
generality. It is easy to see that $V=V_{0}$ at the top electrode, where
$V_{0}=E_{0}d$ is a constant. Then the boundary conditions for $V$ are imposed
by

$\displaystyle\left\\{\begin{array}[]{llll}{V=0}&{\text{ for }y=\eta(x)},\\\
{V=V_{0}}&{\text{ for }~{}y=d}.\end{array}\right.$ (2.8)

Although the voltage potential vanishes on the free surface, the normal
component of the electric field gives rise to normal stress. Therefore, the
electric field and the fluid motion are coupled through the Maxwell stress
tensor leading to the following modified Bernoulli law

$\displaystyle|\nabla\psi|^{2}+2g\eta-\epsilon_{0}|\nabla
V|^{2}-2\sigma\frac{\eta_{xx}}{\left(1+\eta_{x}^{2}\right)^{\frac{3}{2}}}=Q~{}~{}\text{on}~{}~{}y=\eta(x),$
(2.9)

where $g$ is the gravitational constant of acceleration, $\sigma$ is the
coefficient of interface tension and $Q$ is the Bernoulli’s constant. A
schematic of the problem is presented in Fig. 1.

Figure 1: The schematic of the problem.

As a first step, we introduce a constant $\lambda$ into the problem which will
allow us to describe the trivial solutions of (2.4)-(2.9), that is to say, the
solutions with a flat wave profile located at $y=0$. Later on, we use this
constant as a bifurcation parameter to find nontrivial solutions of
(2.4)-(2.9). To this end, when $\eta(x)=0$ and $(V,\psi)$ is independent of
$x$, we observe that the functions $(\overline{V},\overline{\psi})$ solving
(2.4)-(2.9) are given by

$\displaystyle(\overline{V},\overline{\psi})=(E_{0}y,\frac{\gamma
y^{2}}{2}+\lambda y)$ (2.10)

provided the constants $m$ and $Q$ satisfy

$m=m(\lambda)=\frac{\gamma d^{2}}{2}-\lambda d,\quad
Q=Q(\lambda)=\lambda^{2}-\epsilon_{0}E_{0}^{2}.$

The functions $(\overline{V},\overline{\psi})$ obtained in (2.10) will be
taken as the trivial solutions with velocity field
$(\overline{\psi}_{y},-\overline{\psi}_{x})=(\gamma y+\lambda,0)$. Thus, the
bifurcation parameter $\lambda$ represents physically the horizontal wave
speed of the electrohydrodynamic waves on the interface $y=0$.

###  2.2 Reformulation via the naive flattening transform

The main difficulties associated with the system (2.4)-(2.9) are its nonlinear
character and the fact that the interface $y=\eta(x)$ is unknown. The latter
difficulty can be overcome by introducing a suitable naive flattening
transform. Let’s first define the unknown domains by

$\Omega_{a}:=\\{(x,y)\in\mathbb{R}^{2}:\eta(x)<y<d\\},\quad\Omega_{b}:=\\{(x,y)\in\mathbb{R}^{2}:-d<y<\eta(x)\\}$

and the following horizontal strips by

$D_{a}:=\\{(q,p)\in\mathbb{R}^{2}:0<p<d\\},\quad
D_{b}:=\\{(q,p)\in\mathbb{R}^{2}:-d<p<0\\}$

The main idea is to find a transform to map the unknown domains $\Omega_{a}$
and $\Omega_{b}$ into the strips $D_{a}$ and $D_{b}$.

Figure 2: The flatness of the fluid domains.

Here we introduce the naive flatting transform by

$\displaystyle
T_{\eta}:(x,y)\mapsto(q,p)=\left\\{\begin{array}[]{llll}{\left(x,y-\frac{\eta(x)(d-y)}{d-\eta(x)}\right)}&{\text{
for }~{}\eta(x)\leq y<d},\\\ {}\\\
{\left(x,y-\frac{\eta(x)(d+y)}{d+\eta(x)}\right)}&{\text{ for
}~{}-d<y\leq\eta(x)}.\end{array}\right.$ (2.14)

Assume $|\eta(x)|<d$, it is easy to check that the naive flatting transform
$T_{\eta}$ is a diffeomorphism and maps $\Omega_{a}$ and $\Omega_{b}$ into
$D_{a}$ and $D_{b}$ as in Fig 2.

In the coordinates $(q,p)$, then the equation (2.4) of $\psi$ would become

$\left\\{\begin{array}[]{llll}{\psi_{qq}-\frac{2(d+p)\eta^{\prime}}{d+\eta}\psi_{qp}+\frac{(d+p)^{2}\eta^{\prime
2}+d^{2}}{(d+\eta)^{2}}\psi_{pp}-\frac{(d+p)\left((d+\eta)\eta^{\prime\prime}-2\eta^{\prime
2}\right)}{(d+\eta)^{2}}\psi_{p}=\gamma}&~{}{p\in(-d,0)},\\\
{\psi=0}&~{}\text{on}~{}~{}{p=0},\\\ {\psi=m}&~{}\text{on}~{}~{}{p=-d},\\\
\end{array}\right.$ (2.15)

the equations (2.5)-(2.8) of $V$ would be

$\left\\{\begin{array}[]{llll}{V_{qq}-\frac{2(d-p)\eta^{\prime}}{d-\eta}V_{qp}+\frac{(d-p)^{2}\eta^{\prime
2}+d^{2}}{(d-\eta)^{2}}V_{pp}-\frac{(d-p)\left((d-\eta)\eta^{\prime\prime}+2\eta^{\prime
2}\right)}{(d-\eta)^{2}}V_{p}=0}&~{}{p\in(0,d)},\\\
{V=0}&~{}\text{on}~{}~{}{p=0},\\\ {V=E_{0}d}&~{}\text{on}~{}~{}{p=d}\\\
\end{array}\right.$ (2.16)

and the equation (2.9) on the interface will take the form

$\displaystyle\psi_{q}^{2}+\frac{d^{2}(\eta^{\prime
2}+1)}{(d+\eta)^{2}}\psi_{p}^{2}-\frac{2d\eta^{\prime}}{d+\eta}\psi_{q}\psi_{p}+2g\eta-2\sigma\frac{\eta^{\prime\prime}}{\left(1+\eta^{\prime
2}\right)^{\frac{3}{2}}}$
$\displaystyle-\epsilon_{0}\left(V_{q}^{2}+\frac{d^{2}(\eta^{\prime
2}+1)}{(d-\eta)^{2}}V_{p}^{2}-\frac{2d\eta^{\prime}}{d-\eta}V_{q}V_{p}\right)-Q=0~{}~{}\text{on}~{}~{}p=0.$
(2.17)

##  3 The dynamics of periodic electrohydrodynamic waves

In this section, we aim to construct the nontrivial solutions of (2.15)-(2.2)
by applying bifurcation theory. Thus it is necessary to transform the problem
(2.15)-(2.2) into a suitable abstract operator equation.

We first introduce a functional analytic setup. Without loss of generality,
here we choose the constant $d=1$. Define the function space for an arbitrary
$\alpha\in(0,1)$ by

$X:=\\{\eta\in C^{2,\alpha}_{per,e,0}(\mathbb{R}):|\eta|<1\\},$

where $C^{k,\alpha}_{per}$ denotes the space of functions in Hölder class with
the variable $q$ being periodic, even and having zero average. Similarly, we
introduce the space

$Y:=\\{\eta\in C^{0,\alpha}_{per,e,0}(\mathbb{R}):|\eta|<1\\},$

It follows from (Gilbarg, , Theorem 6.14) that there exist a unique solution
$\psi\in C^{2,\alpha}_{per,e}(\overline{D_{b}})$ to the problem (2.15) and a
unique solution $V\in C^{2,\alpha}_{per,e}(\overline{D_{a}})$ to the problem
(2.16) for any $\eta\in X$. Therefore, for a fixed $\eta$, we have that

$V:=V(E_{0},\eta),\quad\psi:=\psi(\lambda,\eta).$

In addition, $V(E_{0},\eta)$ and $\psi(\lambda,\eta)$ are differential with
respect to $\eta$ by the implicit function theorem. In fact, it follows
(HenryBM, , Lemma 4.1) that $V\in C^{2}(\mathbb{R}\times
X,C^{2,\alpha}_{per,e}(\overline{D_{a}}))$ and $\psi\in C^{2}(\mathbb{R}\times
X,C^{2,\alpha}_{per,e}(\overline{D_{b}}))$. Taking $V(E_{0},\eta)$ and
$\psi(\lambda,\eta)$ into the equation (2.2) on the interface allows us to
define a new abstract equation as our bifurcation equation by

$\displaystyle F(\lambda,E_{0},\eta):$
$\displaystyle=\psi_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1+\eta)^{2}}\psi_{p}^{2}-\frac{2\eta^{\prime}}{1+\eta}\psi_{q}\psi_{p}+2g\eta-2\sigma\frac{\eta^{\prime\prime}}{\left(1+\eta^{\prime
2}\right)^{\frac{3}{2}}}$ (3.1)
$\displaystyle-\epsilon_{0}\left(V_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1-\eta)^{2}}V_{p}^{2}-\frac{2\eta^{\prime}}{1-\eta}V_{q}V_{p}\right)\bigg{|}_{p=0}-Q=0,$

where $F:\mathbb{R}^{2}\times X\rightarrow Y$ and
$V=V(E_{0},\eta),\psi=\psi(\lambda,\eta)$.

###  3.1 Spectrum of the linearized operator as a Fourier multiplier

In order to prove the occurrence of local bifuraction, we will deduce spectral
properties of the linearized operator $\partial_{\eta}F(\lambda,E_{0},0)$.
Indeed, we can prove that $\partial_{\eta}F(\lambda,E_{0},0)$ is a Fourier
multiplier, which allows us to analyse its Fredholm properties by examining
the behaviour of its symbols $D(\lambda,E_{0})$. Recalling that
$V(E_{0},0)=\overline{V}$ and $\psi(\lambda,0)=\overline{\psi}$ where
$\overline{V}$ and $\overline{\psi}$ are given by (2.10), we have that

$\displaystyle F_{\eta}(\lambda,E_{0},0)[\eta]$
$\displaystyle=\partial_{\eta}F(\lambda,E_{0},0)[\eta]+\partial_{\psi}F(\lambda,E_{0},0)[\partial_{\eta}\psi(\lambda,0)[\eta]]$
(3.2)
$\displaystyle+\partial_{V}F(\lambda,E_{0},0)[\partial_{\eta}V(\lambda,0)[\eta]]\bigg{|}_{p=0},$

for all $\eta\in X$. Define

$\displaystyle w(q,p):=\partial_{\eta}\psi(\lambda,0)[\eta],\qquad
h(q,p):=\partial_{\eta}V(\lambda,0)[\eta].$ (3.3)

It’s easy to see that

$\displaystyle\partial_{\eta}F(\lambda,E_{0},0)[\eta]=-2\overline{\psi}_{p}^{2}\eta+2g\eta-2\sigma\eta^{\prime\prime}-2\epsilon_{0}\overline{V}_{p}^{2}\eta$
(3.4)

and

$\displaystyle\partial_{\psi}F(\lambda,E_{0},0)[w]=2\overline{\psi}_{p}w_{p},\qquad\partial_{V}F(\lambda,E_{0},0)[h]=-2\epsilon_{0}\overline{V}_{p}h_{p}.$
(3.5)

On the other hand, it follows from (2.10) that

$\overline{V}=E_{0}y=E_{0}(p(1-\eta)+\eta)$

and

$\overline{\psi}=\frac{\gamma y^{2}}{2}+\lambda
y=\frac{\gamma}{2}\left(p(1+\eta)+\eta\right)^{2}+\lambda\left(p(1+\eta)+\eta\right)$

Thus, (3.4) and (3.5) would become that

$\displaystyle\partial_{\eta}F(\lambda,E_{0},0)[\eta]=-2\lambda^{2}\eta+2g\eta-2\sigma\eta^{\prime\prime}-2\epsilon_{0}E_{0}^{2}\eta$
(3.6)

and

$\displaystyle\partial_{\psi}F(\lambda,E_{0},0)[w]=2\lambda
w_{p},\qquad\partial_{V}F(\lambda,E_{0},0)[h]=-2\epsilon_{0}E_{0}h_{p}.$ (3.7)

Based on (3.2)-(3.3) and (3.6)-(3.7), it remains for us to determine
$w_{p}(q,0)$ and $h_{p}(q,0)$. By observation, differentiating the equations
of (2.15) with respect to $\eta$, then
$w=\partial_{\eta}\psi(\lambda,0)[\eta]$ is the solution of the following
problem

$\left\\{\begin{array}[]{llll}{\Delta
w=2\overline{\psi}_{pp}\eta+(1+p)\overline{\psi}_{p}\eta^{\prime\prime}=2\gamma\eta+(1+p)(\gamma
p+\lambda)\eta^{\prime\prime}}&~{}{p\in(-1,0)},\\\
{w=0}&~{}\text{on}~{}~{}{p=0},\\\ {w=0}&~{}\text{on}~{}~{}{p=-1}.\\\
\end{array}\right.$ (3.8)

Similarly, differentiating the equations of (2.16) with respect to $\eta$,
then we have that $h=\partial_{\eta}V(E_{0},0)[\eta]$ is the solution of the
following problem

$\left\\{\begin{array}[]{llll}{\Delta
h=-2\overline{V}_{pp}\eta+(1-p)\overline{V}_{p}\eta^{\prime\prime}=(1-p)E_{0}\eta^{\prime\prime}}&~{}{p\in(0,1)},\\\
{h=0}&~{}\text{on}~{}~{}{p=0},\\\ {h=0}&~{}\text{on}~{}~{}{p=1}.\\\
\end{array}\right.$ (3.9)

Since $\eta\in X$ and (3.8)-(3.9), we consider now the Fourier series
expansions of $\eta,w$ and $h$ by

$\displaystyle\eta(q)=\sum_{k=1}^{\infty}\eta_{k}\cos(kq),~{}w=\sum_{k=1}^{\infty}\eta_{k}w_{k}(p)\cos(kq),~{}h=\sum_{k=1}^{\infty}\eta_{k}h_{k}(p)\cos(kq).$

Then the Fourier coefficients $w_{k}(p)$ of the solution $w$ to (3.8) can be
written as

$\displaystyle w_{k}(p)=\int^{0}_{-1}G_{1}(p,r)\left(2\gamma-(1+r)(\gamma
r+\lambda)k^{2}\right)\,dr$ (3.10)

where

$\displaystyle
G_{1}(p,r)=\frac{1}{k\sinh(k)}\left\\{\begin{array}[]{llll}{\sinh(k(1+p))\sinh(kr)}&{\text{
for }~{}p\leq r},\\\ {}\\\ {\sinh(kp)\sinh(k(1+r))}&{\text{ for }~{}p\geq
r}\end{array}\right.$ (3.14)

is Green’s function of the operator $\frac{\partial}{\partial p^{2}}-k^{2}$ on
the interval $[-1,0]$ with Dirichlet boundary conditions (see Subsection 5.1).
Similarly, the Fourier coefficients $h_{k}(p)$ of the solution $h$ to (3.9)
can be written as

$\displaystyle h_{k}(p)=\int^{1}_{0}G_{2}(p,r)E_{0}(r-1)k^{2}\,dr$ (3.15)

where

$\displaystyle
G_{2}(p,r)=-\frac{1}{k\sinh(k)}\left\\{\begin{array}[]{llll}{\sinh(kp)\sinh(k(1-r))}&{\text{
for }~{}p\leq r},\\\ {}\\\ {\sinh(k(1-p))\sinh(kr)}&{\text{ for }~{}p\geq
r}\end{array}\right.$ (3.19)

is Green’s function of the operator $\frac{\partial}{\partial p^{2}}-k^{2}$ on
the interval $[0,1]$ with Dirichlet boundary conditions (see Subsection 5.1).

Thus, it follows from (3.10) and (3.15) that

$\displaystyle w_{k,p}(0)$
$\displaystyle=\frac{1}{\sinh(k)}\int_{-1}^{0}(2\gamma-(1+p)(\gamma
p+\lambda)k^{2})\sinh(k(1+p))dp$ (3.20)
$\displaystyle=\frac{(\gamma+\lambda-\lambda k)e^{k}-(\gamma+\lambda+\lambda
k)e^{-k}}{2\sinh(k)}$

and

$\displaystyle h_{k,p}(0)$
$\displaystyle=-\frac{1}{\sinh(k)}\int_{0}^{1}E_{0}(p-1)k^{2}\sinh(k(1-p))dp$
(3.21) $\displaystyle=\frac{(k-1)e^{k}+(k+1)e^{-k}}{2\sinh(k)}E_{0}.$

Summarising, we find that the Fréchet derivative of
$\partial_{\eta}F(\lambda,E_{0},0)$ is the Fourier multiplier, that is to say

$\displaystyle\partial_{\eta}F(\lambda,E_{0},0)\sum_{k=1}^{\infty}\eta_{k}\cos(kq)=\sum_{k=1}^{\infty}D_{k}(\lambda,E_{0})\eta_{k}\cos(kq)$
(3.22)

where

$\displaystyle D_{k}(\lambda,E_{0}):$ $\displaystyle=-2\lambda^{2}+2g+2\sigma
k^{2}-2\epsilon_{0}E_{0}^{2}-\epsilon_{0}E_{0}^{2}\frac{(k-1)e^{k}+(k+1)e^{-k}}{\sinh(k)}$
$\displaystyle+\lambda\frac{(\gamma+\lambda-\lambda
k)e^{k}-(\gamma+\lambda+\lambda k)e^{-k}}{\sinh(k)}$
$\displaystyle=-\frac{2k}{\tanh(k)}\left(\lambda^{2}-\frac{\gamma}{k}\tanh(k)\lambda+\epsilon_{0}E_{0}^{2}-\frac{g+\sigma
k^{2}}{k}\tanh(k)\right).$

Indeed, this formula is called dispersion relation. Once the dispersion
relation vanishes, which means the linearized operator
$\partial_{\eta}F(\lambda,E_{0},0)$ is degenerate and the nontrivial solutions
may occur. It is obvious that

$\displaystyle\lambda^{*}_{k,\pm}=\frac{\gamma}{2k}\tanh(k)\pm\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}$ (3.23)

are solutions to $D_{k}(\lambda,E_{0})=0$ if there holds that

$\displaystyle\frac{(g+\sigma
k^{2})\tanh(k)}{k}>\epsilon_{0}E_{0}^{2},~{}~{}\text{for
any}~{}~{}k\in\mathbb{N}^{+}.$ (3.24)

###### Remark 1

In fact, the wave number $k$ and the electric field $E_{0}$ satisfying

$\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}>\epsilon_{0}E_{0}^{2}$

are enough to conclude the occurrence of electrohydrodynamic waves, which
means physically that the electric field is not too strong. However, a
slightly stronger assumption (3.24) would ensure that

$\lambda^{*}_{k,+}>0,\qquad\lambda^{*}_{k,-}<0,$

which play a key role in the ripples. That is to say that we will let
$\lambda^{*}_{k,+}=\lambda^{*}_{l,+}$ (or
$\lambda^{*}_{k,+}=\lambda^{*}_{l,+}$) for any $k\neq l$ in Section 4, which
leads to $E_{0}^{2}=E_{k,l}$ where $E_{k,l}$ is defined by (4.5) and the case
of bifurcation from two-dimensional kernels would occur.

###  3.2 The existence of local primary bifurcation

In this subsection, we will give our first main result on the existence of
electrohydrodynamic capillary-gravity waves by using Theorem 5.1 in Appendix.

Based on the argument in Subsection 3.1, it is obvious that

$\displaystyle F(\lambda,E_{0},0)=0$ (3.25)

for any $\lambda\in\mathbb{R}$, which implies that (H1) in Theorem 5.1 holds.
It follows from (3.22) and (3.23) that the bounded linear operator
$\partial_{\eta}F(\lambda,E_{0},0):\mathbb{R}^{2}\times X\rightarrow Y$ is
invertible whenever $\lambda\neq\lambda^{*}_{k,\pm}$ for any integer $k\geq
1$. By the implicit function theorem, these points are not bifurcation points.

However, we claim that (H2) in Theorem 5.1 holds for each
$\lambda^{*}\in\\{\lambda^{*}_{k,\pm}:k\in\mathbb{N}^{+}\\}$. From (3.22) and
(3.23), it is obvious that
$\mathcal{N}(\partial_{\eta}F(\lambda^{*},E_{0},0))$ is one-dimensional and
generated by $\eta^{*}=\cos(kq)\in X$. While
$\mathcal{R}(\partial_{\eta}F(\lambda^{*},E_{0},0))$ is the closed subspace of
$Y$ formed by the elements $f\in Y$ satisfying that

$\int^{\pi}_{-\pi}f(q)\cos(kq)dq=0.$

Then it follows from (3.22)-(3.24) that

$\partial_{\lambda\eta}F(\lambda^{*},E_{0},0)[1,\eta^{*}]=-\frac{2k}{\tanh(k)}\left(2\lambda^{*}-\frac{\gamma}{k}\tanh(k)\right)\eta^{*}\notin\mathcal{R}(\partial_{\eta}F(\lambda^{*},E_{0},0)),$
(3.26)

which shows that (H2) holds for every
$\lambda^{*}\in\\{\lambda^{*}_{k,\pm}:k\in\mathbb{N}^{+}\\}$. With (3.25) and
(3.26) in hands, then we can obtain the following result by applying the
Theorem 5.1 in Appendix.

###### Theorem 3.1

(Primary branches of bifurcation) Assume that the electric field $E_{0}$ such
that (3.24) holds and $E_{0}^{2}\neq E_{k,l}$ for any positive integer $k,l$
and the vorticity function $\gamma\in\mathbb{R}$, let $\lambda_{k,\pm}^{*}$ be
shown as in (3.23). For any
$\lambda\in\mathbb{R}\backslash\left\\{\lambda_{k,\pm}^{*}:k\in\mathbb{N}^{+}\right\\}$,
the solutions of (3.1) are trivial. For $\lambda\in\lambda_{k,\pm}^{*}$ and
each choice of sign $\pm$, there exists in the space $\mathbb{R}\times X$ a
continuous curve
$\mathcal{K}_{k,\pm}=\left\\{\left(\lambda_{k,\pm}(s),\eta(s)\right):s\in\mathbb{R}\right\\}$
of solutions of (3.1) satisfying

(i)
$\left(\lambda_{k,\pm}(0),\eta(0)\right)=\left(\lambda_{k,\pm}^{*},0\right)$;

(ii) $\eta=\eta(s)$ in $X$ with $\eta(s)=s\eta^{*}+o(s)$ for
$|s|<\varepsilon$, where $\eta^{*}=\cos(kq)$ and $\varepsilon>0$ sufficiently
small;

(iii) there exist a neighbourhood $\mathcal{U}_{k,\pm}$ of
$\left(\lambda_{k,\pm}^{*},0\right)$ in $\mathbb{R}\times X$ and
$\varepsilon>0$ sufficiently small such that

$\left\\{\left(\lambda_{k,\pm},\eta\right)\in\mathcal{U}_{k,\pm}:\eta\not\equiv
0\,\,\text{and}\,\,f(\lambda,\eta)=0\right\\}=\left\\{\left(\lambda_{k,\pm}(s),\eta(s)\right):0<|s|<\varepsilon\right\\}.$

###  3.3 The local stability of the primary bifurcation

Inspired by ConstantinS1 ; DaiLZ , we now establish the local stability of the
electrohydrodynamic waves lying on the primary branch $\mathcal{K}_{k,\pm}$
obtained in Theorem 3.1. We would like to mention that the concept of formal
stability has been given in Appendix.

###### Theorem 3.2

(The stability of the local primary bifurcation curve)
(i) For the fixed primary bifurcation branch $(\lambda_{k,\pm}(s),\eta(s))$
emanating from $\lambda_{k,\pm}^{*}$, we have that

$\lambda_{k,\pm}^{\prime}(0)=0$

and there holds that

$\lambda_{k,+}^{\prime\prime}(0)<0,~{}~{}~{}\lambda_{k,-}^{\prime\prime}(0)>0$

when the constant vorticity $\gamma\in[-\varepsilon,\varepsilon]$ for
$\varepsilon$ small enough.
(ii) As the constant vorticity $\gamma\in[-\varepsilon,\varepsilon]$, near the
bifurcation point $\lambda_{k,+}^{*}$, the laminar solution is formally stable
for $\lambda>\lambda_{k,+}^{*}$ but unstable for $\lambda<\lambda_{k,+}^{*}$.
However, near the bifurcation point $\lambda_{k,-}^{*}$, the laminar solution
is unstable for $\lambda>\lambda_{k,-}^{*}$ but formally stable for
$\lambda<\lambda_{k,-}^{*}$.
(iii) As the constant vorticity $\gamma\in[-\varepsilon,\varepsilon]$, the
nontrivial solution curves are formally stable near the bifurcation points
$\lambda_{k,\pm}^{*}$ in local. (see Fig.3)

Figure 3: The local stability near bifurcation points $\lambda_{k,\pm}^{*}$.

###### Proof

Let us first use the Theorem 5.2 to prove the bifurcation direction (i). For
any $\eta\in X$, define

$\displaystyle\widetilde{l}(\eta)=\frac{1}{\pi}\int_{-\pi}^{\pi}\eta\cos(kq)dq.$

Then $\widetilde{l}$ is a linear functional on $X$ such that
$\widetilde{l}\left(\cos(kq)\right)=1$. It is clearly
$\mathcal{R}(F_{\eta}\left(\lambda_{k,\pm}^{*},0\right))=\left\\{\eta:\widetilde{l}(\eta)=0\right\\}$.
Due to $\eta^{*}=\cos(kq)$, it follows (3.26) that

$\left\langle\widetilde{l},F_{\lambda\eta}\left(\lambda_{k,\pm}^{*},0\right)[1,\eta^{*}]\right\rangle=\mp\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}.$ (3.27)

Let $T_{k}:=\frac{\tanh(k)}{k}$ for $k\in\mathbb{N}^{+}$, it follows from
(5.18) that

$\displaystyle F_{\eta\eta}(\lambda_{k,\pm}^{*},0)[\eta^{*},\eta^{*}]~{}$
(3.28)
$\displaystyle=2(\lambda_{k,\pm}^{*2}-\epsilon_{0}E_{0}^{2})k^{2}\sin^{2}(kq)+6(\lambda_{k,\pm}^{*2}-\epsilon_{0}E_{0}^{2})\cos^{2}(kq)$
$\displaystyle-2\epsilon_{0}\frac{(1-T_{k})^{2}}{T_{k}^{2}}E_{0}^{2}\cos^{2}(kq)+2\frac{\left((\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}\right)^{2}}{T_{k}^{2}}\cos^{2}(kq)$
$\displaystyle-8\epsilon_{0}E_{0}^{2}\frac{1-T_{k}}{T_{k}}\cos^{2}(kq)-8\lambda_{\pm}^{*}\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}\cos^{2}(kq)$

which leads to that

$\displaystyle\left\langle\widetilde{l},F_{\eta\eta}(\lambda_{k,\pm}^{*},0)[\eta^{*},\eta^{*}]\right\rangle=0,$
(3.29)

where we use the following facts

$\int^{\pi}_{-\pi}\cos^{3}(kq)dq=0,\quad\int^{\pi}_{-\pi}\sin^{2}(kq)\cos(kq)dq=0.$

Combining (3.27) with (3.29) and applying Theorem 5.2, we obtain that

$\lambda_{k,\pm}^{\prime}(0)=-\frac{\left\langle\widetilde{l},F_{\eta\eta}(\lambda_{k,\pm}^{*},0)[\eta^{*},\eta^{*}]\right\rangle}{2\left\langle\widetilde{l},F_{\lambda\eta}\left(\lambda_{k,\pm}^{*},0\right)[1,\eta^{*}]\right\rangle}=0,$

which indicates that the primary bifurcation is pitchfork bifurcation.

It follows from (3.20)-(3.21) that

$\displaystyle w_{k,p}(0)=\frac{(\gamma+\lambda)T_{k}-\lambda}{T_{k}},\quad
h_{k,p}(0)=\left(\frac{1-T_{k}}{T_{k}}\right)E_{0}.$

Combining this with (5.20), we can easily deduce that

$\displaystyle
F_{\eta\eta\eta}(\lambda_{k,\pm}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]~{}$
$\displaystyle=-12k^{2}\left(\lambda_{k,\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)\sin^{2}(kq)\cos(kq)-24\left(\lambda_{k,\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)\cos^{3}(kq)$
$\displaystyle-18k^{4}\sigma\sin^{2}(kq)\cos(kq)+36\lambda_{k,\pm}^{*}\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}\cos^{3}(kq)$
$\displaystyle+12k^{2}\lambda_{k,\pm}^{*}\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}\sin^{2}(kq)\cos(kq)$
$\displaystyle-12k^{2}\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)\sin^{2}(kq)\cos(kq)-36\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)\cos^{3}(kq)$
$\displaystyle-4\left(\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}\right)^{2}\cos^{3}(kq)-4\epsilon_{0}\left(\frac{1-T_{k}}{T_{k}}\right)^{2}E_{0}^{*2}\cos^{3}(kq).$

Based on (3.3), we obtain that

$\displaystyle\left\langle\widetilde{l},F_{\eta\eta\eta}(\lambda_{k,\pm}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]\right\rangle$
(3.31)
$\displaystyle=-3k^{2}\left(\lambda_{k,\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-18\left(\lambda_{k,\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-\frac{9}{2}k^{4}\sigma$
$\displaystyle+3k^{2}\lambda_{k,\pm}^{*}\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}+27\lambda_{k,\pm}^{*}\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}$
$\displaystyle-3k^{2}\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)-27\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)$
$\displaystyle-3\left(\frac{(\gamma+\lambda_{k,\pm}^{*})T_{k}-\lambda_{k,\pm}^{*}}{T_{k}}\right)^{2}-3\epsilon_{0}\left(\frac{1-T_{k}}{T_{k}}\right)^{2}E_{0}^{*2}.$

Based on (3.31), as $\gamma=0$, we find that

$\displaystyle\left\langle\widetilde{l},F_{\eta\eta\eta}(\lambda_{k,+}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]\right\rangle$
(3.32)
$\displaystyle=-3k^{2}\left(\lambda_{k,+}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-18\left(\lambda_{k,+}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-\frac{9}{2}k^{4}\sigma$
$\displaystyle+3k^{2}\lambda_{k,+}^{*}\frac{\lambda_{k,+}^{*}(T_{k}-1)}{T_{k}}+27\lambda_{k,+}^{*}\frac{\lambda_{k,+}^{*}(T_{k}-1)}{T_{k}}$
$\displaystyle-3k^{2}\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)-27\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)$
$\displaystyle-3\left(\frac{\lambda_{k,+}^{*}T_{k}-\lambda_{k,+}^{*}}{T_{k}}\right)^{2}-3\epsilon_{0}\left(\frac{1-T_{k}}{T_{k}}\right)^{2}E_{0}^{*2}<0$

and

$\displaystyle\left\langle\widetilde{l},F_{\eta\eta\eta}(\lambda_{k,-}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]\right\rangle$
(3.33)
$\displaystyle=-3k^{2}\left(\lambda_{k,-}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-18\left(\lambda_{k,-}^{*2}+\epsilon_{0}E_{0}^{*2}\right)-\frac{9}{2}k^{4}\sigma$
$\displaystyle+3k^{2}\lambda_{k,-}^{*}\frac{\lambda_{k,-}^{*}(T_{k}-1)}{T_{k}}+27\lambda_{k,-}^{*}\frac{\lambda_{k,-}^{*}(T_{k}-1)}{T_{k}}$
$\displaystyle-3k^{2}\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)-27\epsilon_{0}E_{0}^{*2}\left(\frac{1-T_{k}}{T_{k}}\right)$
$\displaystyle-3\left(\frac{\lambda_{k,-}^{*}T_{k}-\lambda_{k,-}^{*}}{T_{k}}\right)^{2}-3\epsilon_{0}\left(\frac{1-T_{k}}{T_{k}}\right)^{2}E_{0}^{*2}<0,$

where we use the fact $T_{k}-1<0$ for any integer $k>1$. By continuity, as
$\gamma\in[-\varepsilon,+\varepsilon]$ for $\varepsilon$ small enough,
applying Theorem 5.2 again, we obtain that

$\displaystyle\lambda_{k,+}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{\left\langle\widetilde{l},F_{\eta\eta\eta}(\lambda_{k,+}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]\right\rangle}{-3\left\langle\widetilde{l},F_{\lambda\eta}\left(\lambda_{k,+}^{*},0\right)[1,\eta^{*}]\right\rangle}<0$
(3.34)

and

$\displaystyle\lambda_{k,-}^{\prime\prime}(0)$ $\displaystyle=$
$\displaystyle\frac{\left\langle\widetilde{l},F_{\eta\eta\eta}(\lambda_{k,-}^{*},0)[\eta^{*},\eta^{*},\eta^{*}]\right\rangle}{-3\left\langle\widetilde{l},F_{\lambda\eta}\left(\lambda_{k,-}^{*},0\right)[1,\eta^{*}]\right\rangle}>0$
(3.35)

which indicates that the bifurcation at $\lambda_{k,+}^{*}$ is subcritical,
however the bifurcation at $\lambda_{k,-}^{*}$ is supercritical.

In order to prove (ii) and (iii), we need to apply the exchange of stability
Theorem 5.3. Based on the arguments above, we have known that $0$ is a simple
eigenvalue of $T:=F_{\eta}\left(\lambda_{k,\pm}^{*},0\right)$ and the
transversal condition is satisfied. Let the operator $K$ be the identical
operator. Since $0$ is a simple eigenvalue of $T$, then $0$ is also a
$K$-simple eigenvalue of $T$ (see the Definition 1 in Appendix). It is obvious
that all of assumptions of Theorem 5.3 hold.

Applying Theorem 5.3 by letting $\psi^{*}=\cos(kx)$, we have that there are
eigenvalues $\mu(s)$, $\beta(\lambda_{k,\pm})\in\mathbb{R}$ and their
eigenfunctions $u(s)$, $\psi(\lambda_{k,\pm})\in X$, such that

$\displaystyle F_{\eta}(\lambda_{k,\pm}(s),w(s))u(s)=\mu_{\pm}(s)u(s),$
$\displaystyle
F_{\eta}(\lambda_{k,\pm},0)\psi(\lambda_{k,\pm})=\beta(\lambda_{k,\pm})\psi(\lambda_{k,\pm})$

with

$\displaystyle\mu(0)=\beta\left(\lambda^{*}_{k,\pm}\right)=0,~{}~{}u(0)=\psi\left(\lambda^{*}_{k,\pm}\right)=\cos(kx)$

and

$\displaystyle\beta^{\prime}\left(\lambda^{*}_{k,\pm}\right)\neq
0,~{}~{}\lim_{s\rightarrow 0,\beta(s)\neq
0}\frac{s\lambda_{k,\pm}^{\prime}(s)}{\mu_{\pm}}=-\frac{1}{\beta^{\prime}\left(\lambda^{*}_{k,\pm}\right)}.$

It follows from (3.22) that

$\displaystyle\left[F_{\eta}(\lambda,0)\right]\cos(kx)$ $\displaystyle=$
$\displaystyle-\frac{2k}{\tanh(k)}\left(\lambda^{2}-\frac{\gamma}{k}\tanh(k)\lambda+\epsilon_{0}E_{0}^{2}-\frac{g+\sigma
k^{2}}{k}\tanh(k)\right)\cos(kx).$

So we have that

$\displaystyle\beta^{\prime}\left(\lambda^{*}_{k,\pm}\right)=\mp\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}.$

When $k>1$, we deduce that

$\displaystyle\beta^{\prime}\left(\lambda^{*}_{k,+}\right)=-\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}<0$

and

$\displaystyle\beta^{\prime}\left(\lambda^{*}_{k,-}\right)=\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}>0.$

Since $\beta\left(\lambda^{*}_{k,\pm}\right)=0$, then we can deduce that
$\beta\left(\lambda\right)<0$ if $\lambda>\lambda^{*}_{k,+}$ and
$\beta\left(\lambda\right)>0$ if $\lambda<\lambda^{*}_{k,+}$ for
$\left|\lambda-\lambda^{*}_{k,+}\right|$ sufficiently small. Similarly,
$\beta\left(\lambda\right)<0$ if $\lambda>\lambda^{*}_{k,-}$ and
$\beta\left(\lambda\right)>0$ if $\lambda<\lambda^{*}_{k,-}$ for
$\left|\lambda-\lambda^{*}_{k,-}\right|$ sufficiently small. This implies near
the bifurcation point $\lambda^{*}_{k,+}$ that the laminar solution is
unstable for $\lambda<\lambda^{*}_{k,+}$ but formally stable for
$\lambda>\lambda^{*}_{k,+}$. Similarly, near the bifurcation point
$\lambda^{*}_{k,-}$, the laminar solution is formally stable for
$\lambda<\lambda^{*}_{k,-}$ but unstable for $\lambda>\lambda^{*}_{k,-}$.

On the other hand, by using the Theorem 5.3 again, we have that

$\displaystyle\lim_{s\rightarrow 0,\beta(s)\neq
0}\frac{s\lambda_{k,\pm}^{\prime}(s)}{\mu_{\pm}(s)}=\frac{1}{\pm\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}}.$ (3.36)

It follows from (3.36) that

$\displaystyle\lim_{s\rightarrow 0,\beta(s)\neq
0}\frac{s\lambda_{k,+}^{\prime}(s)}{\mu_{+}(s)}=\frac{1}{\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}},$ (3.37)
$\displaystyle\lim_{s\rightarrow 0,\beta(s)\neq
0}\frac{s\lambda_{k,-}^{\prime}(s)}{\mu_{-}(s)}=\frac{1}{-\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}}.$ (3.38)

It is known that $\lambda_{k,\pm}^{\prime}(0)=0$. By using the Taylor
expansion of $\lambda_{k,\pm}^{\prime}(s)$ at $s=0$, we have that

$\displaystyle\lambda_{k,\pm}^{\prime}(s)=s\lambda_{k,\pm}^{\prime\prime}(0)+o\left(s\right).$

It follows from (3.37) and (3.38) that

$\displaystyle\lim_{s\rightarrow
0}\frac{s^{2}\lambda_{k,+}^{\prime\prime}(0)+o\left(s^{2}\right)}{\mu_{+}(s)}=\frac{1}{\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}}>0$

and

$\displaystyle\lim_{s\rightarrow
0}\frac{s^{2}\lambda_{k,-}^{\prime\prime}(0)+o\left(s^{2}\right)}{\mu_{-}(s)}=\frac{1}{-\frac{4k}{\tanh(k)}\sqrt{\frac{\gamma^{2}\tanh^{2}(k)}{4k^{2}}+\frac{(g+\sigma
k^{2})\tanh(k)}{k}-\epsilon_{0}E_{0}^{2}}}<0.$

Furthermore, by using (3.34) and (3.35), we obtain that

$\displaystyle\mu_{+}(s)<0,\qquad\mu_{-}(s)<0.$

for $\left|s\right|$ ($\neq 0$) small enough. This implies that the nontrivial
solutions are formally stable near the bifurcation points
$\lambda_{k,\pm}^{*}$.

##  4 The existence of electrohydrodynamic ripples

In section 3, the local bifurcation problem of (3.1) was studied in the case
when the kernel is one-dimensional. There we only choose $\lambda$ as the
bifurcation parameter by fixing the electric field $E_{0}$. In this section,
we will regard $E_{0}$ as another parameter and find that if $E_{0}^{2}$ is
sufficiently close to the constant $E_{k,l}$ and the resonance phenomenon
occurs by using the secondary bifurcation Theorem 5.4 stated in the Appendix,
that is to say there exist at least one secondary bifurcation branch emerging
from some primary branch $\mathcal{K}_{k,\pm}$ at a point far from the set of
$\\{\eta=0\\}$. These secondary bifurcation solutions are indeed corresponding
to the electrohydrodynamic ripples.

To this end, let’s first introduce a new functional analytic setting by
choosing a Hilbert space because closed subspaces of Hilbert space would
possess in general a closed complement. Then we redefine

$X:=\\{w:=\sum_{n=1}^{\infty}a_{n}\cos(nq):a_{n}\in\mathbb{R},~{}~{}\sum_{n=1}^{\infty}a_{n}^{2}n^{6}<\infty\\}$

and

$Y:=\\{v:=\sum_{n=0}^{\infty}a_{n}\cos(nq):a_{n}\in\mathbb{R},~{}~{}\sum_{n=1}^{\infty}a_{n}^{2}n^{2}<\infty\\},$

which are identified as subspaces of $H^{3}(\mathbb{R}/(2\pi\mathbb{Z}))$ and
$H^{1}(\mathbb{R}/(2\pi\mathbb{Z}))$, respectively. It is worth noting that
$H^{3}(\mathbb{R}/(2\pi\mathbb{Z}))\hookrightarrow
C^{2,\alpha}(\mathbb{R}/(2\pi\mathbb{Z}))$, which ensures that (3.1) holds in
the classic sense. In addition, here we take $E_{0}$ as another parameter by
considering

$\displaystyle F(\lambda,E_{0},\eta):$
$\displaystyle=\psi_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1+\eta)^{2}}\psi_{p}^{2}-\frac{2\eta^{\prime}}{1+\eta}\psi_{q}\psi_{p}+2g\eta-2\sigma\frac{\eta^{\prime\prime}}{\left(1+\eta^{\prime
2}\right)^{\frac{3}{2}}}$ (4.1)
$\displaystyle-\epsilon_{0}\left(V_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1-\eta)^{2}}V_{p}^{2}-\frac{2\eta^{\prime}}{1-\eta}V_{q}V_{p}\right)\bigg{|}_{p=0}-Q=0$

again, where $F:\mathbb{R}^{2}\times X\rightarrow Y$ and
$V=V(E_{0},\eta),\psi=\psi(\lambda,\eta)$. Define the set
$\mathcal{O}:=\\{\eta\in X:|w|<1\\}$ and it is obvious that the operator $F$
defined by (4.1) is real-analytic from
$\mathbb{R}^{2}\times\mathcal{O}\subset\mathbb{R}^{2}\times X\rightarrow Y$.
Based on the arguments in the previous section, we have that

###### Lemma 1

Let $T_{k}:=\frac{\tanh(k)}{k}$ for $k\in\mathbb{N}^{+}$ and assume that
(3.24) holds and

$\sigma\gamma^{2}\left(l^{2}-k^{2}\right)T_{k}T_{l}(T_{k}-T_{l})>\left((g+\sigma
k^{2})T_{k}-(g+\sigma l^{2})T_{l}\right)^{2},$ (4.2)

then the Frechet derivative $F_{\eta}(\lambda,E_{0},0):X\rightarrow Y$ is a
Fourier multiplier. Furthermore, there holds that

$F_{\eta}(\lambda,E_{0},0)\sum_{k=1}^{\infty}\eta_{k}\cos(kx)=\sum_{k=1}^{\infty}D_{k}(\lambda,E_{0})\eta_{k}\cos(kx),$
(4.3)

where $D_{k}(\lambda,E_{0})$ is defined in (3.22) and the operator
$F_{\eta}(\lambda,E_{0},0):X\rightarrow Y$ is a Fredholm operator of index
zero for any $(\lambda,E_{0})\in\mathbb{R}^{2}$. More precisely, defining

$\lambda_{k,\pm}^{*}:=\frac{\gamma}{2}T_{k}\pm\sqrt{\frac{\gamma^{2}}{4}T_{k}^{2}+(g+\sigma
k^{2})T_{k}-\epsilon_{0}E_{0}^{2}}$ (4.4)

and

$E_{k,l}:=\frac{\sigma\gamma^{2}\left(l^{2}-k^{2}\right)T_{k}T_{l}(T_{k}-T_{l})-\left((g+\sigma
k^{2})T_{k}-(g+\sigma
l^{2})T_{l}\right)^{2}}{(T_{k}-T_{l})^{2}\gamma^{2}\epsilon_{0}}$ (4.5)

for any $k,l\in\mathbb{N}^{+}$, we have that

(1) $\lambda_{k,+}^{*}>0$, $\lambda_{k,-}^{*}<0$ and $E_{k,l}>0$;

(2) if $\lambda\notin\\{\lambda_{k,\pm}^{*}:k\in\mathbb{N}^{+}\\}$, then
$F_{\eta}(\lambda,E_{0},0)$ is an invertible operator;

(3) if $\lambda=\lambda_{k,\pm}^{*}$ and $E_{0}^{2}=E_{k,l}$ for some positive
integers $k\neq l$, then $\lambda_{k,+}^{*}=\lambda_{l,+}^{*}$ (or
$\lambda_{k,-}^{*}=\lambda_{l,-}^{*}$). Moreover, $0$ is an eigenvalue of the
operator $F_{\eta}(\lambda,E_{0},0)$ and the corresponding eigenspace is two-
dimensional.

###### Proof

We note that the sign of $\lambda_{k,\pm}^{*}$ follows from (3.24) and the
sign of the constant $E_{k,l}$ can be determined directly from (4.2).

In order to prove _(2)_ , it follows from the arguments in previous section
that zero is an eigenvalue of $F_{\eta}(\lambda,E_{0},0)$ if and only if
$D_{k}(\lambda,E_{0})=0$ for any positive integer $k$. It is easy to check
that $\lambda=\lambda_{k,\pm}^{*}$ are solutions to $D_{k}(\lambda,E_{0})=0$,
whereby $\lambda_{k,\pm}^{*}$ are given by (4.4). Thus, if
$\lambda\notin\\{\lambda_{k,\pm}^{*}:k\in\mathbb{N}^{+}\\}$, then
$F_{\eta}(\lambda,E_{0},0)$ is an invertible operator.

Now let’s prove _(3)_ by letting

$\lambda_{k,+}^{*}=\lambda_{l,+}^{*}~{}~{}\text{or}~{}~{}\lambda_{k,-}^{*}=\lambda_{l,-}^{*}$

for some $k\neq l$, we obtain by using algebraic manipulations that

$E_{0}^{2}=E_{k,l}.$

This means, as $E_{0}^{2}=E_{k,l}$, that the kernel space of
$F_{\eta}(\lambda_{k,\pm}^{*},E_{0},0)$ is generated by
$\\{\cos(kx),\cos(lx)\\}\in X$. In fact, we are left to show that there cannot
be another integer $m\notin\\{k,l\\}$ such that
$\lambda_{k,+}^{*}=\lambda_{l,+}^{*}=\lambda_{m,+}^{*}$ or
$\lambda_{k,-}^{*}=\lambda_{l,-}^{*}=\lambda_{m,-}^{*}$ when
$E_{0}^{2}=E_{k,l}$. By contradiction arguments, if
$\lambda_{k,+}^{*}=\lambda_{l,+}^{*}=\lambda_{m,+}^{*}$ or
$\lambda_{k,-}^{*}=\lambda_{l,-}^{*}=\lambda_{m,-}^{*}$ holds. Then we have
that $E_{0}^{2}=E_{k,l}=E_{k,m}$, which leads to $T_{l}=T_{m}$ which is
contradicted due to $m\neq l$.

###### Remark 2

By the property of the sequence $(T_{k})_{k}$ of being decreasing, the
assumption (4.2) holds provided $\gamma^{2}$ is not small enough. Different
from the case in MartinM , the existence of ripples here is ensured not only
by letting the electric field $E_{0}^{2}$ be close to $E_{k,l}$ but also by
taking the suitable vorticity function $\gamma$.

In the following, we will show that if (3.24) and (4.2) hold and the electric
field $E_{0}^{2}$ is sufficiently close to the constant $E_{k,l}$ defined by
(4.5) with

$\frac{k}{l}\in\mathbb{N}^{+}\setminus\\{1\\},$

then there would be secondary bifurcation branches emerging from some of these
primary branches obtained in Theorem 3.1. Now we further make a decomposition
of Hilbert spaces $X$ and $Y$ by setting

$X_{1}:=\\{w\in X:w=\sum_{n=1}^{\infty}a_{n}\cos(knq)\\},$ $X_{2}:=\\{w\in
X:w=\sum_{\frac{n}{k}\notin\mathbb{N}}a_{n}\cos(nq)\\}$

and

$Y_{1}:=\\{v\in
Y:v=\sum_{n=1}^{\infty}a_{n}\cos(knq)\\},~{}~{}~{}Y_{2}:=\\{v\in
Y:v=\sum_{\frac{n}{k}\notin\mathbb{N}}a_{n}\cos(nq)\\}.$

Once the electric field $E_{0}^{*}$ is chosen such that $E_{0}^{*2}=E_{k,l}$
for $k,l\in\mathbb{N}^{+}$, then it follows from Lemma 1 that either
$\lambda_{k,+}^{*}=\lambda_{l,+}^{*}$ or
$\lambda_{k,-}^{*}=\lambda_{l,-}^{*}$. For convenience, we will denote the
bifurcation points by

$\lambda_{\pm}^{*}=\lambda_{k,\pm}^{*}=\lambda_{l,\pm}^{*}$

Then the operator $F_{\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)$ has a two-
dimensional kernel. Moreover,

$\mathcal{N}(F_{\eta}(\lambda_{\pm}^{*},E_{0}^{*},0))=span\\{x_{1},x_{2}\\},$

where $x_{1}:=\cos(kq)$ and $x_{2}:=\cos(lq)$. Now let us state the main
theorem of this section.

###### Theorem 4.1

(The existence of secondary bifurcation branches) Assume that (3.24) and (4.2)
hold and $|E_{0}-E_{0}^{*}|<\delta$ for small $\delta>0$. Then there exists a
smooth curve $\mathcal{S}_{E_{0}}$ that intersects $\mathcal{K}_{k,\pm}$ and
the secondary bifurcation curve $\mathcal{S}_{E_{0}}$ consists only of
solutions $(\lambda,\eta)$ of problem (4.1) with minimal period
$\frac{2\pi}{l}$.

Figure 4: The secondary bifurcation curve $\mathcal{S}_{E_{0}}$.

###### Proof

To finish the proof, we need to verify the assumptions (H1)-(H8) in Theorem
5.2. Let us first make a constraint for spaces by

$F:\mathbb{R}^{2}\times X_{1}\rightarrow Y_{1}.$

Then it follows from (3.25) that $F(\lambda,E_{0},0)=0$ for any
$(\lambda,E_{0})\in\mathbb{R}^{2}$. The Fredholm property of operator
$F_{\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)$ from $X_{1}$ to $Y_{1}$ can been
deduced similarly as in Subsection 3.2. In addition, it follows from (3.26)
that

$F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,x_{1}]=-\frac{2k}{\tanh(k)}\left(2\lambda_{\pm}^{*}-\frac{\gamma}{k}\tanh(k)\right)x_{1}\notin\mathcal{R}(F_{\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)).$
(4.6)

It is obvious that $x_{1}=\psi_{1}=\cos(kq)$ and $x_{2}=\psi_{2}=\cos(lq)$ in
our setting. The fact $x_{i}\in X_{i}$ and $\psi_{i}\in Y_{i}$ for $i=1,2$
together with the Lemma 1 ensures that (H1)-(H6) hold naturally. Similarly as
(4.6), we have that

$F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,x_{2}]=-\frac{2l}{\tanh(l)}\left(2\lambda_{\pm}^{*}-\frac{\gamma}{l}\tanh(l)\right)x_{2}\notin\mathcal{R}(F_{\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)),$
(4.7)

which verifies the general transversality condition (H7).

Finally, we are left to prove the nondegeneracy condition (H8) holds. Based on
(4.3), we have that

$\displaystyle
F_{E_{0}\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,x_{1}]=-\frac{4k}{\tanh(k)}\epsilon_{0}E_{0}^{*}x_{1},$
$\displaystyle
F_{E_{0}\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,x_{2}]=-\frac{4l}{\tanh(l)}\epsilon_{0}E_{0}^{*}x_{2}.$
(4.8)

With (4.6)-(4) in hands, we obtain that

$\displaystyle\left|\begin{array}[]{cc}\langle
F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(kq)]|\cos(kq)\rangle{}&~{}\langle
F_{E_{0}\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(kq)]|\cos(kq)\rangle\\\
{}\hfil&{}\hfil\\\ \langle
F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(lq)]|\cos(lq)\rangle{}&~{}\langle
F_{E_{0}\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(lq)]|\cos(lq)\rangle\\\
\end{array}\right|$ (4.12) (4.13)
$\displaystyle=\frac{8\gamma\epsilon_{0}E_{0}^{*}(T_{l}-T_{k})}{T_{k}T_{l}}\neq
0,$

where we use the facts of $k\neq l$ and $\\{T_{k}\\}_{k}$ is a decreasing
sequence. On the other hand, it follows from (3.20)-(3.21) that

$\displaystyle w_{k,p}(0)=\frac{(\gamma+\lambda)T_{k}-\lambda}{T_{k}},\quad
h_{k,p}(0)=\left(\frac{1-T_{k}}{T_{k}}\right)E_{0}.$ (4.14)

Combining (4.14) with (5.18) (see Subsection 5.2), we have that

$\displaystyle
F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\cos(kq),\cos(kq)]~{}$ (4.15)
$\displaystyle=2(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2})k^{2}\sin^{2}(kq)+6(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2})\cos^{2}(kq)$
$\displaystyle-2\epsilon_{0}\frac{(1-T_{k})^{2}}{T_{k}^{2}}E_{0}^{*2}\cos^{2}(kq)+2\frac{\left((\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}\right)^{2}}{T_{k}^{2}}\cos^{2}(kq)$
$\displaystyle-8\epsilon_{0}E_{0}^{*2}\frac{1-T_{k}}{T_{k}}\cos^{2}(kq)-8\lambda_{\pm}^{*}\frac{(\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}}{T_{k}}\cos^{2}(kq)$

and

$\displaystyle
F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\cos(kq),\cos(lq)]~{}$ (4.16)
$\displaystyle=2(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2})kl\sin(kq)\sin(lq)+6(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2})\cos(kq)\cos(lq)$
$\displaystyle-2\epsilon_{0}\frac{(1-T_{k})(1-T_{l})}{T_{k}T_{l}}E_{0}^{*2}\cos(kq)\cos(lq)$
$\displaystyle+2\frac{\left((\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}\right)\left((\gamma+\lambda_{\pm}^{*})T_{l}-\lambda_{\pm}^{*}\right)}{T_{k}T_{l}}\cos(kq)\cos(lq)$
$\displaystyle-8\epsilon_{0}E_{0}^{*2}\frac{1-T_{k}}{T_{k}}\cos(kq)\cos(lq)-8\lambda_{\pm}^{*}\frac{(\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}}{T_{k}}\cos(kq)\cos(lq).$

Therefore, it follows from (4.6)-(4.7) and (4.15)-(4.16) that

$\displaystyle\left|\begin{array}[]{cc}\langle
F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(kq)]|\cos(kq)\rangle{}&~{}\langle
F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\cos(kx),\cos(kq)]|\cos(kq)\rangle\\\
{}\hfil&{}\hfil\\\ \langle
F_{\lambda\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[1,\cos(lq)]|\cos(lq)\rangle{}&~{}\langle
F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\cos(kx),\cos(lq)]|\cos(lq)\rangle\\\
\end{array}\right|$ (4.20) (4.21)
$\displaystyle=-\frac{2k}{\tanh(k)}\left(2\lambda_{\pm}^{*}-\frac{\gamma}{k}\tanh(k)\right)\frac{M}{\pi}\int^{\pi}_{-\pi}\cos(kq)\cos^{2}(lq)dx\neq
0,$

where we use the following facts for $k\neq l$ that

$\int^{\pi}_{-\pi}\cos^{3}(kq)dq=0,\quad\int^{\pi}_{-\pi}\sin^{2}(kq)\cos(kq)dq=0,\quad\int^{\pi}_{-\pi}\cos(kq)\cos^{2}(lq)dq\neq
0$

and

$\displaystyle
M=6(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2})-2\epsilon_{0}\frac{(1-T_{k})(1-T_{l})}{T_{k}T_{l}}E_{0}^{*2}$
$\displaystyle+2\frac{\left((\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}\right)\left((\gamma+\lambda_{\pm}^{*})T_{l}-\lambda_{\pm}^{*}\right)}{T_{k}T_{l}}-8\epsilon_{0}E_{0}^{*2}\frac{1-T_{k}}{T_{k}}$
$\displaystyle-8\lambda_{\pm}^{*}\frac{(\gamma+\lambda_{\pm}^{*})T_{k}-\lambda_{\pm}^{*}}{T_{k}}.$
(4.22)

Indeed, we can check that $M\neq 0$ in (4) by using (4.4) and (4.5) for some
integers $k\neq l$. Up to now, the assumptions (H1)-(H8) are verified, thus we
finish the proof by using the Theorem 5.4 in Appendix.

###### Remark 3

The local stability of primary solution curves $\mathcal{K}_{k,\pm}$ has been
given in Subsection 3.3. However, as far as we know that the local stability
of secondary bifurcation curve $\mathcal{S}_{E_{0}}$ is still open. To this
end, maybe some new exchange of stability theorems are needed to establish,
which is a very interesting project in future.

##  5 Proofs and calculations

We first give the Green’s function $G_{2}(x,t)$ of the operator
$\frac{\partial}{\partial x^{2}}-k^{2}$ on the interval $[0,1]$ with Dirichlet
boundary conditions and the Green’s function $G_{1}(x,t)$ of the operator
$\frac{\partial}{\partial x^{2}}-k^{2}$ on the interval $[-1,0]$ with
Dirichlet boundary conditions, which are used in Section 3. In addition, we
show the second derivative of the abstract operator $F(\lambda,E_{0},\eta)$
with responding to $\eta$, which plays a key role in checking the
nondegeneracy condition in Section 4. Moreover, we also give the third
derivative of the abstract operator $F(\lambda,E_{0},\eta)$ with responding to
$\eta$, which is used to judge the direction of bifurcation.

###  5.1 The Green’s functions

To get the Green’s function $G_{2}(x,t)$, let us first consider the following
ODE

$\displaystyle\left\\{\begin{array}[]{llll}{\left(\frac{\partial}{\partial
x^{2}}-k^{2}\right)G_{1}(x,t)=\delta(x-t)},\\\ {G_{2}(0,t)=0,\quad
G_{2}(1,t)=0},\end{array}\right.$ (5.3)

where $\delta$ is the Dirac function. For $x<t$, by using the boundary
condition $G_{1}(0,t)=0$, we have that

$G_{2}(x,t)=A(t)\left(e^{kx}-e^{-kx}\right).$

For $x>t$, by using the boundary condition $G_{1}(1,t)=0$, we have that

$G_{2}(x,t)=B(t)\left(e^{kx}-e^{k(2-x)}\right).$

Then, for $x=t$, we have that

$A(t)\left(e^{kt}-e^{-kt}\right)=B(t)\left(e^{kt}-e^{k(2-t)}\right)$

and

$B(t)\frac{\partial\left(e^{kx}-e^{k(2-x)}\right)}{\partial
x}\bigg{|}_{t}-A(t)\frac{\partial\left(e^{kx}-e^{-kx}\right)}{\partial
x}\bigg{|}_{t}=1,$

which gives that

$A(t)=\frac{e^{kt}-e^{k(2-t)}}{2k(e^{2k}-1)},\quad
B(t)=\frac{e^{kt}-e^{-kt}}{2k(e^{2k}-1)}=\frac{\sinh(kt)}{k(e^{2k}-1)}.$

Taking $A(t)$ and $B(t)$ into the formulas above, we have that

$\displaystyle
G_{2}(x,t)=-\frac{1}{k\sinh(k)}\left\\{\begin{array}[]{llll}{\sinh(kx)\sinh(k(1-t))}&{\text{
for }~{}x\leq t},\\\ {}\\\ {\sinh(k(1-x))\sinh(kt)}&{\text{ for }~{}x\geq
t}.\end{array}\right.$ (5.7)

Similarly, by considering the ODE

$\displaystyle\left\\{\begin{array}[]{llll}{\left(\frac{\partial}{\partial
x^{2}}-k^{2}\right)G_{1}(x,t)=\delta(x-t)},\\\ {G_{1}(-1,t)=0,\quad
G_{1}(0,t)=0},\end{array}\right.$ (5.10)

we can obtain that

$\displaystyle
G_{1}(p,r)=\frac{1}{k\sinh(k)}\left\\{\begin{array}[]{llll}{\sinh(k(1+x))\sinh(kt)}&{\text{
for }~{}x\leq t},\\\ {}\\\ {\sinh(kx))\sinh(k(1+t))}&{\text{ for }~{}x\geq
t}.\end{array}\right.$ (5.14)

###  5.2 The second and third derivative

It is known by letting $d=1$ that

$\displaystyle F(\lambda,E_{0},\eta):$
$\displaystyle=\psi_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1+\eta)^{2}}\psi_{p}^{2}-\frac{2\eta^{\prime}}{1+\eta}\psi_{q}\psi_{p}+2g\eta-2\sigma\frac{\eta^{\prime\prime}}{\left(1+\eta^{\prime
2}\right)^{\frac{3}{2}}}$
$\displaystyle-\epsilon_{0}\left(V_{q}^{2}+\frac{\eta^{\prime
2}+1}{(1-\eta)^{2}}V_{p}^{2}-\frac{2\eta^{\prime}}{1-\eta}V_{q}V_{p}\right)\bigg{|}_{p=0}-Q=0,$

where $F:\mathbb{R}^{2}\times X\rightarrow Y$,
$V=V(E_{0},\eta),\psi=\psi(\lambda,\eta)$ and $F$ is analytic in the open set
$\mathcal{O}$. Therefore, we have that

$\displaystyle F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\eta,\eta]~{}$
(5.16) $\displaystyle=\left(F_{\eta\eta}[\eta,\eta]+F_{\eta
V}[\eta,\frac{\partial
V}{\partial\eta}[\eta]]+F_{\eta\psi}[\eta,\frac{\partial\psi}{\partial\eta}[\eta]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+\left(F_{V\eta}[\frac{\partial
V}{\partial\eta}[\eta],\eta]+F_{VV}[\frac{\partial
V}{\partial\eta}[\eta],\frac{\partial
V}{\partial\eta}[\eta]]+F_{V\psi}[\frac{\partial
V}{\partial\eta}[\eta],\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+\left(F_{\psi\eta}[\frac{\partial\psi}{\partial\eta}[\eta],\eta]+F_{\psi
V}[\frac{\partial\psi}{\partial\eta}[\eta],\frac{\partial
V}{\partial\eta}[\eta]]+F_{\psi\psi}[\frac{\partial\psi}{\partial\eta}[\eta],\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle=\left(F_{\eta\eta}[\eta,\eta]+F_{VV}[\frac{\partial
V}{\partial\eta}[\eta],\frac{\partial
V}{\partial\eta}[\eta]]+F_{\psi\psi}[\frac{\partial\psi}{\partial\eta}[\eta],\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+2\left(F_{\eta V}[\eta,\frac{\partial
V}{\partial\eta}[\eta]]+F_{\eta\psi}[\eta,\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle=2\left(\overline{\psi}_{p}^{2}\left(\eta^{\prime
2}+3\eta^{2}\right)-\epsilon_{0}\overline{V}_{p}^{2}\left(\eta^{\prime
2}+3\eta^{2}\right)-\epsilon_{0}\left(h_{q}^{2}+h_{p}^{2}\right)+w_{q}^{2}+w_{p}^{2}\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+4\left(-2\epsilon_{0}\overline{V}_{p}\eta
h_{p}+\epsilon_{0}\overline{V}_{p}\eta^{\prime}h_{q}-2\overline{\psi}_{p}\eta
w_{p}-\overline{\psi}_{p}\eta^{\prime}w_{q}\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$

It follows from (2.10) that

$\overline{V}=E_{0}y=E_{0}(p(1-\eta)+\eta)$

and

$\overline{\psi}=\frac{\gamma y^{2}}{2}+\lambda
y=\frac{\gamma}{2}\left(p(1+\eta)+\eta\right)^{2}+\lambda\left(p(1+\eta)+\eta\right).$

Thus, the formula (5.16) would become

$\displaystyle F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\eta,\eta]~{}$
(5.17)
$\displaystyle=2\left(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2}\right)\left(\eta^{\prime
2}+3\eta^{2}\right)-2\epsilon_{0}\left(h_{q}^{2}(q,0)+h_{p}^{2}(q,0)\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle+2w_{q}^{2}(q,0)+2w_{p}^{2}(q,0)-8\epsilon_{0}E_{0}^{*}\eta
h_{p}(q,0)+4\epsilon_{0}E_{0}^{*}\eta^{\prime}h_{q}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle-8\lambda_{\pm}^{*}\eta
w_{p}(q,0)-4\lambda_{\pm}^{*}\eta^{\prime}w_{q}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}.$

In addition, from (3.10) and (3.15), we have that

$\displaystyle w_{k}(0)=0,\quad h_{k}(0)=0,$

which implies that

$\displaystyle F_{\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\eta,\eta]~{}$
(5.18)
$\displaystyle=2\left(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2}\right)\left(\eta^{\prime
2}+3\eta^{2}\right)-2\epsilon_{0}\left(h_{q}^{2}(q,0)+h_{p}^{2}(q,0)\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle+2w_{q}^{2}(q,0)+2w_{p}^{2}(q,0)-8\epsilon_{0}E_{0}^{*}\eta
h_{p}(q,0)+4\epsilon_{0}E_{0}^{*}\eta^{\prime}h_{q}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle-8\lambda_{\pm}^{*}\eta
w_{p}(q,0)-4\lambda_{\pm}^{*}\eta^{\prime}w_{q}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle=2\left(\lambda_{\pm}^{*2}-\epsilon_{0}E_{0}^{*2}\right)\left(\eta^{\prime
2}+3\eta^{2}\right)-2\epsilon_{0}h_{p}^{2}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}$
$\displaystyle+2w_{p}^{2}(q,0)-8\epsilon_{0}E_{0}^{*}\eta
h_{p}(q,0)-8\lambda_{\pm}^{*}\eta
w_{p}(q,0)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0)}.$

In addition, it is obvious that

$\displaystyle
F_{\eta\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\eta,\eta,\eta]~{}$ (5.19)
$\displaystyle=\left(F_{\eta\eta\eta}[\eta,\eta,\eta]+F_{VVV}[\frac{\partial
V}{\partial\eta}[\eta],\frac{\partial V}{\partial\eta}[\eta],\frac{\partial
V}{\partial\eta}[\eta]]+F_{\psi\psi\psi}[\frac{\partial\psi}{\partial\eta}[\eta],\frac{\partial\psi}{\partial\eta}[\eta]],\frac{\partial\psi}{\partial\eta}[\eta]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+3\left(F_{\eta\eta V}[\eta,\eta,\frac{\partial
V}{\partial\eta}[\eta]]+F_{\eta\eta\psi}[\eta,\eta,\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+3\left(F_{\eta VV}[\eta,\frac{\partial
V}{\partial\eta}[\eta],\frac{\partial
V}{\partial\eta}[\eta]]+F_{\eta\psi\psi}[\eta,\frac{\partial\psi}{\partial\eta}[\eta],\frac{\partial\psi}{\partial\eta}[\eta]]\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$

Similarly, considering the $\overline{V}$, $\overline{\psi}$ and
$w_{k}(0)=0,h_{k}(0)=0$, then we have that

$\displaystyle
F_{\eta\eta\eta}(\lambda_{\pm}^{*},E_{0}^{*},0)[\eta,\eta,\eta]~{}$ (5.20)
$\displaystyle=\left(\overline{\psi}_{p}^{2}\left(-12\eta^{\prime
2}\eta-24\eta^{3}\right)-\epsilon_{0}\overline{V}_{p}^{2}\left(12\eta^{\prime
2}\eta+24\eta^{3}\right)+18\sigma\eta^{\prime\prime}\eta^{\prime
2}\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+12\left(\eta^{\prime
2}+3\eta^{2}\right)\overline{\psi}_{p}w_{p}-12\epsilon_{0}\left(\eta^{\prime
2}+3\eta^{2}\right)\overline{V}_{p}h_{p}-4\eta w_{p}^{2}-4\epsilon_{0}\eta
h_{p}^{2}\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle=\left(-\lambda_{\pm}^{*2}\left(12\eta^{\prime
2}\eta+24\eta^{3}\right)-\epsilon_{0}E_{0}^{*2}\left(12\eta^{\prime
2}\eta+24\eta^{3}\right)+18\sigma\eta^{\prime\prime}\eta^{\prime
2}\right)\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+12\left(\eta^{\prime
2}+3\eta^{2}\right)\lambda_{\pm}^{*}w_{p}-12\epsilon_{0}\left(\eta^{\prime
2}+3\eta^{2}\right)E_{0}^{*}h_{p}-4\eta w_{p}^{2}-4\epsilon_{0}\eta
h_{p}^{2}\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle=-12\left(\lambda_{\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)\eta^{\prime
2}\eta-24\left(\lambda_{\pm}^{*2}+\epsilon_{0}E_{0}^{*2}\right)\eta^{3}+18\sigma\eta^{\prime\prime}\eta^{\prime
2}\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle+12\lambda_{\pm}^{*}\eta^{\prime
2}w_{p}+36\lambda_{\pm}^{*}\eta^{2}w_{p}-12\epsilon_{0}E_{0}^{*}\eta^{\prime
2}h_{p}-36\epsilon_{0}E_{0}^{*}\eta^{2}h_{p}\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}$
$\displaystyle-4\eta w_{p}^{2}-4\epsilon_{0}\eta
h_{p}^{2}\bigg{|}_{(\lambda_{\pm}^{*},E_{0}^{*},0),p=0}.$

## Appendix: Quoted results

This appendix not only collects the remarkable Crandall-Rabinowitz local
bifurcation theorem CrandallR , which are designed to deal with the case of
linearized operator with one dimensional kernel, but also gives the secondary
bifurcation theorem due to Shearer Shearer . In addition, we give the
definition of stability and recall the stability exchange theorem due to
Crandall and Rabinowitz CrandallR1  and some useful formulas used to judge the
direction of local bifurcation.

###### Theorem 5.1

(Local bifurcation theorem CrandallR ) Let $X$ and $Y$ be Banach spaces and
$F:\mathbb{R}\times X\rightarrow Y$ be a $C^{k}$ function with $k\geq 2$. Let
$\mathcal{N}(T)$ and $\mathcal{R}(T)$ denote the null space and range of any
operator $T$. Suppose that

(H1) $F(\lambda,0)=0$ for all $\lambda\in\mathbb{R}$;

(H2) For some $\lambda_{*}\in\mathbb{R}$, $F_{x}\left(\lambda_{*},0\right)$ is
a Fredholm operator with
$\mathcal{N}\left(F_{x}\left(\lambda_{*},0\right)\right)$ and
$Y/\mathcal{R}\left(F_{x}\left(\lambda_{*},0\right)\right)$ are
$1$-dimensional and the null space generated by $x_{*}$, and the
transversality condition

$F_{\lambda,x}\left(\lambda_{*},0\right)[1,x_{*}]\not\in\mathcal{R}\left(F_{x}\left(\lambda_{*},0\right)\right)$

holds, where $\mathcal{N}\left(F_{x}\left(\lambda_{*},0\right)\right)$ and
$\mathcal{R}\left(F_{x}\left(\lambda_{*},0\right)\right)$ denote null space
and range space of $F_{x}\left(\lambda_{*},0\right)$, respectively.

Then $\lambda_{*}$ is a bifurcation point in the sense that there exists
$\varepsilon>0$ and a primary branch $\mathcal{P}$ of solutions

$\\{(\lambda,x)=(\Lambda(s),s\psi(s)):s\in\mathbb{R},|s|<\varepsilon\\}\subset\mathbb{R}\times
X$

with $F(\lambda,x)=0$, $\Lambda(0)=0$, $\psi(0)=x_{*}$ and the maps

$s\mapsto\Lambda(s)\in\mathbb{R},~{}~{}~{}~{}s\mapsto s\psi(s)\in X$

are of class $C^{k-1}$ on $(-\varepsilon,\varepsilon)$.

###### Theorem 5.2

(The formula of bifurcation direction Kielhofer ) If $F$ satisfies the
hypotheses of Theorem 5.1, then

$\lambda^{\prime}(0)=-\frac{\left\langle
l,F_{x}^{(2)}\left(\lambda_{*},0\right)x_{*}^{2}\right\rangle}{2\left\langle
l,F_{\lambda x}(\lambda_{*},0)x_{*}\right\rangle},$

where $l\in X^{*}$ satisfying
$\mathcal{N}(l)=\mathcal{R}\left(D_{x}F(\lambda_{*},0)\right)$, and $X^{*}$
being the dual space of $X$. Furthermore, for $n\geq 2$, if
$F_{x}^{(j)}(\lambda_{*},0)x_{*}^{j}=0$ for $j\in\\{1,\ldots,n\\}$, then
$\lambda^{(j)}(0)=0$, $\psi^{(j)}(0)=0$ for $j\in\\{1,\ldots,n-1\\}$ and

$\lambda^{(n)}(0)=-\frac{\left\langle
l,F_{x}^{(n+1)}\left(\lambda_{*},0\right)x_{*}^{n+1}\right\rangle}{(n+1)\left\langle
l,F_{\lambda x}(\lambda_{*},0)x_{*}\right\rangle},$

where $F_{x}^{(j)}(\lambda_{*},0)x_{*}^{j}$ means the value of the $j$-th
Fréchet derivative of $F(\lambda_{*},x)$ with respect to $x$ at
$(\lambda_{*},0)$ evaluated at the $j$-tuple each of whose entries is $x_{*}$.

Now let us recall the concept of stability and the definition of $K-$simple
eigenvalue introduced in CrandallR1 . The operator equation $F(\lambda,x)=0$
can be regarded as the equilibrium form of the evolution equation

$\displaystyle\frac{dx}{dt}=F(\lambda,x).$

Assume that $F\left(\lambda_{0},x_{0}\right)=0$. If all of eigenvalues of
$F_{x}\left(\lambda_{0},x_{0}\right)$ are negative, then the solution $x_{0}$
is called asymptotically linear stable solution of
$\frac{dx}{dt}=F(\lambda,x)$. If there exists a positive eigenvalue of
$F_{x}\left(\lambda_{0},x_{0}\right)$, $x_{0}$ is called unstable. If there
exists a zero eigenvalue of $F_{x}\left(\lambda_{0},x_{0}\right)$, $x_{0}$ is
called neutral stable.

###### Definition 1

Let $T,K$: $X\rightarrow Y$ be two bounded linear operators from a real Banach
space $X$ to another one $Y$. A complex number $\beta$ is called a $K-$simple
eigenvalue of $T$ if

$dim\mathcal{N}(T-\beta K)=1=codim\mathcal{R}(T-\beta K)$

and

$K\psi^{*}\notin\mathcal{R}(T-\beta
K)~{}~{}for~{}~{}0\neq\psi^{*}\in\mathcal{N}(T-\beta K).$

If $K$ is the identity operator, then $K-$simple eigenvalue is called simple.
Next, let us show the Crandall-Rabinowitz exchange of stability theorem
CrandallR1 .

###### Theorem 5.3

Let $X$ and $Y$ be real Banach spaces and let $K,T:X\rightarrow Y$ be two
bounded linear operators. Assume $F:\mathbb{R}\times X\rightarrow Y$ is
$C^{2}$ near $\left(\lambda_{*},0\right)\in\mathbb{R}\times X$ with
$F(\lambda,0)=0$ for $\left|\lambda_{*}-\lambda\right|$ sufficiently small.
Let $T=F_{x}\left(\lambda_{*},0\right)$. If $\beta=0$ is a $F_{\lambda
x}\left(\lambda_{*},0\right)-$simple eigenvalue of operator $T$ and a
$K-$simple eigenvalue of $T$, then there exists locally a curve
$(\lambda(s),x(s))\in\mathbb{R}\times X$ such that

$(\lambda(0),x(0))=\left(\lambda_{*},0\right)~{}~{}and~{}~{}F(\lambda(s),x(s))=0.$

Moreover, if $F(\lambda,x)=0$ with $x\neq 0$ and $(\lambda,x)$ near
$\left(\lambda_{*},0\right)$, then

$(\lambda,x)=(\lambda(s),x(s))~{}~{}for~{}~{}some~{}~{}s\neq 0.$

Furthermore, there are eigenvalues $\beta(s)$,
$\beta_{triv}(\lambda)\in\mathbb{R}$ with eigenvectors $\psi(s)$,
$\psi_{triv}(\lambda)\in X$, such that

$\displaystyle F_{x}\left(\lambda(s),x(s)\right)\psi(s)=\beta(s)K\psi(s),$
$\displaystyle
F_{x}\left(\lambda,0\right)\psi_{triv}(\lambda)=\beta_{triv}(\lambda)K\psi_{triv}(\lambda)$

with

$\displaystyle\beta(0)=\beta_{triv}\left(\lambda_{*}\right)=0,~{}~{}\psi(0)=\psi_{triv}\left(\lambda_{*}\right)=\psi^{*}.$

Each curve is $C^{1}$ if $F$ is $C^{2}$, then

$\displaystyle\frac{d\beta_{triv}\left(\lambda\right)}{d\lambda}|_{\lambda=\lambda_{*}}\neq
0,~{}~{}\lim_{s\rightarrow 0,\beta\left(s\right)\neq
0}\frac{s\lambda^{\prime}(s)}{\beta(s)}=-\frac{1}{\beta_{triv}^{\prime}\left(\lambda_{*}\right)}.$

It’s worth noting that
$\frac{d\beta_{triv}\left(\lambda\right)}{d\lambda}|_{\lambda=\lambda_{*}}\neq
0$ and $\beta_{triv}\left(\lambda_{*}\right)=0$, the trivial solution is
stable at one side of $\lambda_{*}$ and is unstable at another side of
$\lambda_{*}$. Based on the arguments above, the nontrivial solution $x(s)$ is
called to be formally stable if $\beta(s)<0$ and unstable if $\beta(s)>0$.
Finally, we quote below the secondary bifurcation theorem that forms the basis
of the existence of ripples in Section 4 (cf. Shearer  for a detailed
discussion). The version that we use comes from MartinM .

###### Theorem 5.4

(Secondary bifurcation theorem) _Let $X$ and $Y$ be real Banach spaces and
assume $F:\mathbb{R}^{2}\times X\rightarrow Y$ is $C^{2}$ and there exist
closed subspaces $X_{1}$ of $X$ and $Y_{1}$ of $Y$ such that
$F:\mathbb{R}^{2}\times X_{1}\rightarrow Y_{1}$ and_

(H1) $F(\lambda,\beta,0)=0$ for all $(\lambda,\beta)\in\mathbb{R}^{2}$;

(H2) For some $(\lambda_{*},\beta_{*})\in\mathbb{R}^{2}$,
$F_{x}\left(\lambda_{*},\beta_{*},0\right)$ is a Fredholm operator with
$\mathcal{N}\left(F_{x}\left(\lambda_{*},\beta_{*},0\right)\right)$ and
$Y/\mathcal{R}\left(F_{x}\left(\lambda_{*},\beta_{*},0\right)\right)$ are
$1$-dimensional and the null space generated by $x_{1}$, and the
transversality condition

$F_{\lambda,x}\left(\lambda_{*},\beta_{*},0\right)[1,x_{1}]\not\in\mathcal{R}\left(F_{x}\left(\lambda_{*},\beta_{*},0\right)\right)$

holds;

(H3) There exist closed spaces $X_{2},Y_{2}$ such that $X=X_{1}\oplus X_{2}$
and $Y=Y_{1}\oplus Y_{2}$;

(H4) $\mathcal{N}(F_{x}(\lambda_{*},\beta_{*},0))\cap X_{i}=span\\{x_{i}\\}$
for $i=1,2$;

(H5) There are $0\neq\psi_{i}\in Y$ with
$span\\{\psi_{i}\\}\oplus\left(Y_{i}\cap\mathcal{R}(F_{x}(\lambda_{*},\beta_{*},0))\right)=Y_{i}$
for $i=1,2$;

(H6) $F_{x}(\lambda_{*},\beta_{*},0)X_{2}\subset Y_{2}$;

(H7)
$F_{\lambda,x}(\lambda_{*},\beta_{*},0)[1,x_{i}]\not\in\mathcal{R}\left(F_{x}\left(\lambda_{*},\beta_{*},0\right)\right)$
for $i=1,2$;

(H8)

$\\\ \left|\begin{array}[]{cc}\langle F_{\lambda
x}(\lambda_{*},\beta_{*},0)[1,x_{1}]|\psi^{\prime}_{1}\rangle{}&~{}\langle
F_{\beta x}(\lambda_{*},\beta_{*},0)[1,x_{1}]|\psi^{\prime}_{1}\rangle\\\
{}\hfil&{}\hfil\\\ \langle F_{\lambda
x}(\lambda_{*},\beta_{*},0)[1,x_{2}]|\psi^{\prime}_{2}\rangle{}&~{}\langle
F_{\beta x}(\lambda_{*},\beta_{*},0)[1,x_{2}]|\psi^{\prime}_{2}\rangle\\\
\end{array}\right|\neq 0$

and

$\\\ \left|\begin{array}[]{cc}\langle F_{\lambda
x}(\lambda_{*},\beta_{*},0)[1,x_{1}]|\psi^{\prime}_{1}\rangle{}&~{}\langle
F_{xx}(\lambda_{*},\beta_{*},0)[x_{1},x_{1}]|\psi^{\prime}_{1}\rangle\\\
{}\hfil&{}\hfil\\\ \langle F_{\lambda
x}(\lambda_{*},\beta_{*},0)[1,x_{2}]|\psi^{\prime}_{2}\rangle{}&~{}\langle
F_{xx}(\lambda_{*},\beta_{*},0)[x_{1},x_{2}]|\psi^{\prime}_{2}\rangle\\\
\end{array}\right|\neq 0$

hold, where $\langle\psi_{i}|\psi^{\prime}_{j}\rangle=\delta_{ij}$ and
$\mathcal{N}(\psi^{\prime}_{i})=\mathcal{R}(F_{x}(\lambda_{*},\beta_{*},0))$
for $i=1,2$ with $\langle\cdot|\cdot\rangle$ denoting the duality pairing on
$Y\times Y^{\prime}$.

Then, there exists an interval $\mathcal{I}$ containing zero and smooth
functions $\chi,\kappa:\mathcal{I}\times\mathcal{I}\rightarrow\mathbb{R}$ such
that $\kappa(0,0)\neq 0$, and for $\beta=\beta_{*}+\theta$ with
$0\neq\theta\in\mathcal{I}$, the curve

$\mathcal{S}_{\beta}:=\\{(\lambda(\beta-\beta_{*},s),\beta,x(\beta-\beta_{*},s)):s\in\mathcal{I}\\}$

with

$\lambda(\theta,s):=\lambda_{*}+\theta\chi(\theta,s),$
$x(\theta,s):=\theta\kappa(\theta,s)x_{1}+\theta
sx_{2}+z(\lambda_{*}+\theta\chi(\theta,s),\beta,\theta\kappa(\theta,s)x_{1}+\theta
sx_{2})$

is a secondary branch of solutions intersecting the Primary branch
$\mathcal{P}$ of bifurcation at $(\lambda(\theta,0),x(\theta,0))$ whereby
$x(\theta,0)\neq 0$.

## Acknowledgments

Dai was supported by National Natural Science Foundation of China (No.
12371110). Xu was supported by the Postdoctoral Science Foundation of China
(2023M731381). Zhang was supported by National Natural Science Foundation of
China (No. 12301133), the Postdoctoral Science Foundation of China
(2023M741441) and Jiangsu Education Department (No. 23KJB110007).

## Data Availability Statements

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * (1) D.M. Ambrose, W.A. Strauss, J.D. Wright, Global bifurcation theory for periodic traveling interfacial gravity-capillary waves, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2016, 33:1081-1101.
  * (2) C.J. Amick, R.E.L. Turner, Small internal waves in two-fluid systems, Arch. Ration. Mech. Anal., 1989, 108:111-139.
  * (3) J.F. Chu, Q.X. Ding, J. Escher, Variational formulation of rotational steady water waves in two-layer flows, J. Math. Fluid Mech., 2021, 23:17pp.
  * (4) A. Constantin, Nonlinear water waves with applications to wave-current interactions and tsunamis, CBMS-NSF Conference Series in Applied Mathematics 81, SIAM, Philadelphia, 2011.
  * (5) A. Constantin, W. Strauss, Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math., 2004, 57:481-527.
  * (6) A. Constantin, W. Strauss, Stability properties of steady water waves with vorticity, Commun. Pure Appl. Math., 2007, 60:911-950.
  * (7) M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 1971, 8:321-340.
  * (8) M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearied stability, Arch. Ration. Mech. Anal., 1973, 52:161-180.
  * (9) G.W. Dai, F.Q. Li, Y. Zhang, Bifurcation structure and stability of steady gravity water waves with constant vorticity, J. Differential equations 332, 306-332 (2022).
  * (10) A. Doak, T. Gao, J.M. Vanden-Broeck, J.J.S. Kandola, Capillary-gravity waves on the interface of two dielectric fluid layers under normal electric fields, Quart. J. Mech. Appl. Math., 2020, 73: 231-250.
  * (11) A. Doak, T. Gao, J.M. Vanden-Broeck, Global bifurcation of capillary-gravity dark solitary waves on the surface of a conducting fluid under normal electric fields, Quart. J. Mech. Appl. Math., 2022, 75:215-234.
  * (12) M.L. Dubreil-Jacotin, Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie., J. Math. Pures Appl., 1934, 13:217-291.
  * (13) M. Ehrnström, J. Escher, E. Wahlén, Steady water waves with multiple critical layers, SIAM J. Math. Anal., 2011, 43:1436-1456.
  * (14) D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, Heidelberg, 2001.
  * (15) T. Gao, Z. Wang, Zhan; J.M. Vanden-Broeck, Nonlinear wave interactions on the surface of a conducting fluid under vertical electric fields, Phys. D, 2023, 446:12pp.
  * (16) H. Gleeson, P. Hammerton, D.T. Papageorgiou, J.M. Vanden-Broeck, A new application of the Korteweg-de Vries Benjamin-Ono equation in interfacial electrohydrodynamics, Phys. Fluids, 2007, 19:031703.
  * (17) D. Henry, B.V. Matioc, On the existence of steady periodic capillary-gravity stratified water waves, Ann. Sc. Norm. Super. Pisa CI. Sci., 2013, 12:955-974.
  * (18) J.M. Hunt, D. Dutykh, Free surface flows in electrohydrodynamics with a constant vorticity distribution, Water Waves, 2021, 3:297-317.
  * (19) M. Jones, J.F. Toland, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. Roy. Soc. London Ser. A, 1985, 399:391-417.
  * (20) H. Kielhöfer, Bifurcation theory. An introduction with applications to partial differential equations. Second edition. Applied Mathematical Sciences, Springer, New York, 2012, 156.
  * (21) Z. Lin, Y. Zhu, Z. Wang, Local bifurcation of electrohydrodynamic waves on a conducting fluid, Phys. Fluids, 2017, 29:032107.
  * (22) C.I. Martin, B.V. Matioc, Existence of Wilton ripples for water waves with constant vorticity and capillary effects, SIAM J. Appl. Math., 2013, 73:1582-1595.
  * (23) J.R. Melcher, G.I. Taylor, Electrohydrodynamics: A review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech., 1969, 1:111-146.
  * (24) D.T. Papageorgiou, Film flows in the presence of electric fields, Ann. Rev. Fluid Mech., 2019, 51:155-187.
  * (25) M. Shearer, Secondary bifurcation near a double eigenvalue, SIAM J. Math. Anal., 1980, 11:365-389.
  * (26) M. Smit Vega Garcia, E. Vărvǎrucǎ, G.S. Weiss, Singularities in axisymmetric free boundaries for electrohydrodynamic equations. Arch. Ration. Mech. Anal., 2016, 222:573-601.
  * (27) D. Sinambela, Large-amplitude solitary waves in two-layer density stratified water, SIAM J. Math. Anal., 2021, 53:4812-4864.
  * (28) E. Wahlén, Steady water waves with a critical layer, J. Differential Equations, 2009, 246:2468-2483.
  * (29) S. Walsh, O. Bühler, J. Shatah, Steady water waves in the presence of wind, SIAM J. Math. Anal., 2013, 45:2182-2227.
  * (30) Z. Wang, Modelling nonlinear electrohydrodynamic surface waves over three-dimensional conducting fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2017, 473:20160817.
  * (31) M.H. Wheeler, On stratified water waves with critical layers and Coriolis forces, Discrete Contin. Dyn. Syst., 2019, 39:4747-4770.
  * (32) J.M. Vanden-Broeck, Gravity-Capillary Free-Surface Flows, Cambridge University Press, Cambridge, 2010.