We consider the two-dimensional irrotational flow of an inviscid incompressible fluid of infinite depth, which is bounded above by a hydrodynamically passive region also of infinite extent vertically. The fluid is assumed to be a perfect dielectric with permittivity
\(\epsilon _0\). The passive region above the fluid is assumed to be perfectly conducting. The problem can be formulated by means of cartesian coordinates with the
y-axis directed vertically upwards and
\(y=0\) at the undisturbed level (Fig.
1). The gravity
g and the surface tension
\(\sigma \) are both included in the formulation. The deformation of the free surface is denoted by
\(\zeta (x,t)\). A vertical electric field with voltage potential
v is applied. We assume that
\(v \sim V_0y\) as
\(y\rightarrow -\infty \), where
\(V_0\) is a constant. Since the fluid motion can be described by a velocity potential function
\(\phi (x,y,t)\), introducing dimensionless variables by choosing
$$\begin{aligned} \left( \frac{\sigma }{\rho g}\right) ^{\frac{1}{2}},\quad \left( \frac{\sigma }{\rho g^3}\right) ^{\frac{1}{4}},\quad V_0 \end{aligned}$$
(1)
as the reference length, time and voltage potential, the governing equations can then be written as
$$\begin{aligned}&\nabla ^2\phi =0, \qquad \qquad \quad \text {for}\quad y<\zeta (x,t), \end{aligned}$$
(2)
$$\begin{aligned}&\nabla ^2 v=0, \qquad \qquad \quad \text {for}\quad y<\zeta (x,t),\end{aligned}$$
(3)
$$\begin{aligned}&\zeta _t=\phi _y-\phi _x\zeta _x, \quad \; \text {on}\quad y=\zeta (x,t),\end{aligned}$$
(4)
$$\begin{aligned}&v=0, \qquad \qquad \qquad \; \text {on}\quad y=\zeta (x,t),\end{aligned}$$
(5)
$$\begin{aligned}&v_y\sim 1, \qquad \qquad \qquad \text {as}\quad y\rightarrow -\infty ,\end{aligned}$$
(6)
$$\begin{aligned}&\phi _y\rightarrow 0, \qquad \qquad \quad \; \text {as}\quad y\rightarrow -\infty . \end{aligned}$$
(7)
and
$$\begin{aligned} \phi _t= & {} -\frac{1}{2}|\nabla \phi |^2-y + \frac{\beta }{1+\zeta _x^2} \Big [\frac{1}{2}\big (1-\zeta _x^2\big )\big (v_x^2-v_y^2\big )+2\zeta _x v_xv_y\Big ] +\frac{\zeta _{xx}}{\big (1+\zeta _x^2\big )^{3/2}},\quad \text {on}\quad y=\zeta (x,t), \end{aligned}$$
(8)
where the subscripts denote partial derivatives and
$$\begin{aligned} \beta =\epsilon _0 V_0^2\sqrt{\rho g/\sigma ^3} \end{aligned}$$
(9)
is a parameter which measures the ratio of electric forces to gravitational or surface tension forces [the latter two are in balance by our scaling (
1)]. The last three terms of (
8) are the forces due to gravity, the Maxwell stresses due to the electric field and surface tension. Equations (
4) and (
7) are the kinematic boundary conditions and the zero flow at minus infinity condition. The condition (
5) expresses the fact that the region above the fluid is a perfect conductor, and in turn implies
$$\begin{aligned} v_x+v_y\zeta _x=0,\quad \text {on}\quad y=\zeta (x,t). \end{aligned}$$
(10)
Using condition (
10) allows us to manipulate the electric field term in the dynamic boundary condition (
8) to find
$$\begin{aligned} \phi _t=-\frac{1}{2} |\nabla \phi |^2-y-\frac{\beta }{2}|\nabla v|^2+\frac{\zeta _{xx}}{\left( 1+\zeta _x^2\right) ^{3/2}}\quad \text {on}\quad y=\zeta (x,t). \end{aligned}$$
(11)
The Hamiltonian of the system is defined by
$$\begin{aligned} H =\frac{1}{2}\int _{\mathbb {R}}\int _{-\infty }^\zeta |\nabla \phi |^2 \mathrm{{d}}y \, \mathrm{{d}}x+\frac{1}{2}\int _{\mathbb {R}}\zeta ^2 \mathrm{{d}}x+\frac{\beta }{2}\int _{\mathbb {R}}\int _{-\infty }^\zeta |\nabla v|^2 \mathrm{{d}}y \, \mathrm{{d}}x+\int _{\mathbb {R}}\left( \sqrt{1+\zeta _x^2}-1\right) \,\mathrm{{d}}x. \end{aligned}$$
(12)
It reduces to the classical form of the Hamiltonian for capillary–gravity waves when
\(\beta =0\). We denote the velocity potential on the free surface by
\(\varphi (x,t) \equiv \phi (x,\zeta (x,t),t)\). The kinematic and dynamic boundary conditions can be written in the canonical variables
\(\varphi \) and
\(\zeta \) as (see [
12])
$$\begin{aligned} \zeta _t=\frac{\delta H}{\delta \varphi },\qquad \varphi _t=-\frac{\delta H}{\delta \zeta } . \end{aligned}$$
(13)
According to Saffman [
13], a stability exchange of periodic waves due to superharmonic perturbations can only occur at
$$\begin{aligned} \text {either}\qquad \frac{\partial H}{\partial c}=0,\qquad \text {or}\qquad \frac{\partial c}{\partial H}=0, \end{aligned}$$
(14)
where
c is the phase velocity. Saffman’s work was based on the Hamiltonian formulation (see [
12]) where only gravity is considered. It can be generalised to include surface tension and electric fields due to the Hamiltonian structure (
12), (
13). Since all the perturbations are superharmonic for solitary waves, this argument is particularly useful in our numerical studies of stability.
A normal form analysis can be carried out by substituting the following ansatz into the governing equations
$$\begin{aligned} \zeta&=\epsilon A(\chi ,T)e^{ikx-i\omega t}+\epsilon ^2 \zeta _1+\epsilon ^3 \zeta _2+\cdots +\text {c.c.}, \end{aligned}$$
(15)
$$\begin{aligned} \phi&=\epsilon B(\chi ,T)e^{ikx-i\omega t+|k|y}+\epsilon ^2 \phi _1+\epsilon ^3 \phi _2+\cdots +\text {c.c.}, \end{aligned}$$
(16)
where
\(\epsilon \) is a small parameter,
\(T=\epsilon ^2 t\), and
\(\chi =\epsilon (x-c_gt)\). Here
\(c_g\) is the group speed and c.c. denotes the complex conjugate. Details of the derivation can be found in [
14‐
16]. Here we just present the results from the first three orders. At
\(O(\epsilon )\), we retrieve the linear dispersion relation
$$\begin{aligned} \omega ^2=|k|(1+k^2)-\beta k^2\qquad \text {or}\qquad c^2=\frac{1}{|k|}+|k|-\beta , \end{aligned}$$
(17)
where we have assumed that waves are travelling in the positive
x-direction. The dispersion relation admits a minimum of
c at
\(k=1\) whenever
\(0\le \beta <2\). This minimum phase speed is denoted by
\(c^*\). Wave-packet like solitary waves with decaying tails bifurcate from that point. We have immediately
$$\begin{aligned} c^*=\sqrt{2-\beta }. \end{aligned}$$
(18)
At
\(O(\epsilon ^2)\), we obtain the expression for the group speed
$$\begin{aligned} c_g=\frac{1}{2\omega }(1+3k^2-2\beta |k|). \end{aligned}$$
(19)
We proceed to the next order to get the cubic nonlinear Schrödinger equation (NLS)
$$\begin{aligned} iA_T+\lambda A_{\chi \chi }+\mu |A|^2A=0, \end{aligned}$$
(20)
where
$$\begin{aligned} \lambda =\frac{|k|}{2\omega }\qquad \text {and} \qquad \mu =\frac{|k|^3\left( k^4+12\beta |k|^3+\frac{1}{2} k^2-3\beta |k| +4\right) }{2\omega (2k^2-1)}. \end{aligned}$$
(21)
The bifurcation to solitary waves takes place at
\(k=1\), where the NLS reads
$$\begin{aligned} iA_T+ \frac{1}{2\sqrt{2-\beta }} A_{\chi \chi }+\frac{11+18\beta }{4\sqrt{2-\beta }} |A|^2A=0. \end{aligned}$$
(22)
The NLS is of focusing type for
\(0<\beta <2\). It predicts the existence of bright solitons the envelope
A of which has the explicit solution:
$$\begin{aligned} A(\chi ,T)=\sqrt{\frac{2C}{\mu }}\mathrm{sech}\,\left( \sqrt{\frac{C}{\lambda }}\chi \right) \mathrm{e}^{iCT}, \end{aligned}$$
(23)
where
C is a constant. If
\(\beta >2\), the electric field destabilises the system therefore no solitary waves can exist. When
\(\beta \) approaches 2, the coefficients of
\(A_{\chi \chi }\) and
\(|A|^2 A\) in (
22) both tend to infinity. The variable
T is required to be of order
\(O( {\sqrt{2-\beta }})\) to balance equation (
22) which violates the initial assumptions when
\(\sqrt{2-\beta }=o(1)\). Consequently the amplitude and horizontal length need to be rescaled and for
\(\beta \) near 2 the envelope becomes very broad and small (we note that
\(\omega \rightarrow 0\) when
\(\beta \rightarrow 2\)). It is expected that the solutions approach linear sinusoidal waves in the limit
\(\beta \rightarrow 2\).