Dynamics of fully nonlinear capillary–gravity solitary waves under normal electric fields

Two-dimensional capillary–gravity waves travelling under the effect of a vertical electric field are considered. The fluid is assumed to be a dielectric of infinite depth. It is bounded above by another fluid which is hydrodynamically passive and perfectly conducting. The problem is solved numerically by time-dependent conformal mapping methods. Fully nonlinear waves are presented, and their stability and dynamics are studied.

Configuration of the problem. The gravity acts in the negative y-direction. We denote the equation of the unknown free surface by y = ζ (x, t) of horizontal electric fields were studied. Fully nonlinear solutions were computed using a boundary integral equation method, and weakly nonlinear solutions were studied analytically by assuming the long-wave approximation limit. The effect of vertical electric fields was investigated in [9][10][11] where an asymptotic model equation for long waves was derived. Once again fully nonlinear solutions were computed by a boundary integral equation method and compared to the ones produced by the long-wave equation. To our knowledge, there have been, so far, no studies of time-dependent fully nonlinear water waves under the influence of electric fields.
The present work considers a two-dimensional dielectric fluid of infinite depth bounded above by a perfectly conducting gas such as plasma. The electric field is applied vertically throughout the space. This particular configuration converts a two-layer problem to the one of a one layer. Therefore, all the mathematical techniques which are used to solve the capillary-gravity problem can be inherited. Future work will involve cases where the upper layer also plays an active role in the dynamics.
The paper is structured as follows. In Sect. 2, we formulate the problem mathematically. Linear and weakly nonlinear asymptotic analyses are performed as well. In Sect. 3, a numerical method based on a time-dependent conformal map is introduced. A variety of numerical solutions are shown in Sect. 4. Finally conclusions are presented in Sect. 5.

Formulation
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 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 σ are both included in the formulation. The deformation of the free surface is denoted by ζ(x, t). A vertical electric field with voltage potential v is applied. We assume that v ∼ V 0 y as y → −∞, where V 0 is a constant. Since the fluid motion can be described by a velocity potential function φ(x, y, t), introducing dimensionless variables by choosing σ ρg 1 2 , σ ρg 3 1 4 , as the reference length, time and voltage potential, the governing equations can then be written as and where the subscripts denote partial derivatives and 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 Using condition (10) allows us to manipulate the electric field term in the dynamic boundary condition (8) to find The Hamiltonian of the system is defined by It reduces to the classical form of the Hamiltonian for capillary-gravity waves when β = 0. We denote the velocity potential on the free surface by ϕ(x, t) ≡ φ(x, ζ(x, t), t). The kinematic and dynamic boundary conditions can be written in the canonical variables ϕ and ζ as (see [12]) According to Saffman [13], a stability exchange of periodic waves due to superharmonic perturbations can only occur at 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 where is a small parameter, T = 2 t, and χ = (x − c g t).
Here c g is the group speed and c.c. denotes the complex conjugate. Details of the derivation can be found in [14][15][16]. Here we just present the results from the first three orders. At O( ), we retrieve the linear dispersion relation 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 ≤ β < 2. This minimum phase speed is denoted by c * . Wave-packet like solitary waves with decaying tails bifurcate from that point. We have immediately At O( 2 ), we obtain the expression for the group speed We proceed to the next order to get the cubic nonlinear Schrödinger equation (NLS) where λ = |k| 2ω and μ = |k| 3 k 4 + 12β|k| 3 + 1 2 k 2 − 3β|k| + 4 2ω(2k 2 − 1) .
The bifurcation to solitary waves takes place at k = 1, where the NLS reads The NLS is of focusing type for 0 < β < 2. It predicts the existence of bright solitons the envelope A of which has the explicit solution: where C is a constant. If β > 2, the electric field destabilises the system therefore no solitary waves can exist. When β approaches 2, the coefficients of A χχ and |A| 2 A in (22) both tend to infinity. The variable T is required to be of order O( √ 2 − β) to balance equation (22) which violates the initial assumptions when √ 2 − β = o(1). Consequently the amplitude and horizontal length need to be rescaled and for β near 2 the envelope becomes very broad and small (we note that ω → 0 when β → 2). It is expected that the solutions approach linear sinusoidal waves in the limit β → 2.

Numerical scheme
To find the unknown shape of the free surface, we use the time-dependent conformal mapping introduced in [17] which maps the free surface onto the horizontal axis in a new complex plane denoted by (ξ, η). The map can be formally defined as the solution of the following boundary value problem where Y (ξ ) = ζ(ξ, 0). The harmonic conjugate x(ξ, η) can be obtained via the Cauchy-Riemann equations for the analytic function z(ξ, η) = x(ξ, η) + iy(ξ, η). Similarly we can derive the harmonic conjugates of φ(ξ, η) and v(ξ, η), denoted by ψ(ξ, η) and w(ξ, η) respectively. In the mapped plane, we defined the surface variables as It can be shown that where H[.] is the Hilbert transform operator which is defined as where PV demotes the principal value of the integral. We note that V ξ = 0 as v is identically zero everywhere on the free surface. Next we follow [1] to derive the time-evolution equations where J = X 2 ξ + Y 2 ξ is the Jacobian of the conformal map. In the dynamical computations, we employ the fourth order Runge-Kutta method to advance in time. The mesh size in space and the time-step are usually chosen as dξ =0.05 and dt = 2.5 × 10 −4 respectively.
For travelling waves, all functions depend on x − ct. After similar calculations as those presented in [1], we have Then the resulting governing equation becomes The Hamiltonian (12) can be rewritten in dimensionless variables as where the terms appearing on the right hand side correspond to the kinetic energy, the potential energy due to gravity, surface tension and the electric field, respectively. For a given wave profile with phase speed c = c 1 and electric parameter β = β 1 , equation (34) has the same wave profile as its solution for a different value of β = β 2 if the corresponding phase speed c 2 satisfies For example, a depression wave with c = 1.4 for β = 0 has exactly the same profile as that of a depression wave with c = 0.6 for β = 1.6. In addition, we deduce from (35) that H is independent of β for a fixed wave profile due to (36). Travelling waves of our system are related directly to capillary-gravity waves through this relation, by setting β = 0. For example, the minimal speed of depression capillary-gravity solitary waves is about 0.9275 (see e.g [4]) where the wave profile has a point of contact with a trapped bubble. Applying (36) implies that the branch of depression waves can be continued down to c = 0 without trapped bubbles, for any 2 > β > 0.9275 2 0.86. For elevation waves, the complicated structure of the capillary-gravity bifurcation curve together with (36) implies that, with electric fields, isolated branches of waves will appear and disappear at certain values of β. This phenomenon will be discussed further below. Lastly, we note that the evolution equations (31),(32) do not have a simple transformation into their capillary-gravity counterparts.
We approximate the solitary waves by very long periodic waves and use the Fourier representation where the series is truncated after N terms. The coefficients a n and b n are the unknowns to be found. In particular for symmetric waves, we choose ξ = 0 to be the axis of symmetry, i.e. the coefficients b n are zero. The wavelength 2L and the number of modes N are chosen to be sufficiently large so that the solutions do not change when L and N are further increased. We discretise the domain [−L , L) in the mapped plane with a uniform mesh. The dynamic boundary condition (34) is satisfied on these grid points. The resultant system is solved by Newton's method for the unknown Fourier coefficients. In most computations, we use 4096 or 6144 collocation points. The iterations are stopped when the l ∞ -norm of residual errors are less than 10 −10 . This numerical approach has been successfully used in [1,4,18] for capillary-gravity waves and in [19][20][21] for flexural-gravity waves.

Travelling waves
We start by computing fully localised steady solutions for different values of β. When β = 0, our computations agree with the classical results for the capillary-gravity problem. The branch of depression waves is a simple monotonic curve whereas the branch of elevation waves has a complicated structure which snakes back and forth. The reader is referred to [22] for more details.
When β = 1, we obtain qualitatively similar results to those of capillary-gravity waves. The amplitude-speed and energy-speed bifurcation diagrams are presented in Figs. 2 and 3, respectively, where depression (lower branch) and elevation (upper branch) solitary waves bifurcate from infinitesimal amplitude periodic waves at the minimum phase speed defined in (18). We only show part of the branch for better visualisation. The typical wave profiles are shown in Sect. 4.2 as the initial data of the time-dependent dynamics. The cubic NLS (20) can be used to We present two numerical examples for values of β just smaller and larger than β † , namely β = 1.5 and β = 1.55, as shown in Fig. 4a, b. It can be seen clearly that the turning point approaches c = 0 line and touches it. When this happens, the branch of elevation waves split into two separate parts as shown in Fig. 4b. When the second turning point (marked by a pentagram in Figs. 2 and 3) touches the boundary c = 0 at another critical value β ‡ , the lower part of the branch for elevation waves disappears. The value of β ‡ can be estimated by using the same approach We computed the bifurcation diagrams with β = 1.97 (<β ‡ ) and β = 1.99 (>β ‡ ) as shown in Fig. 4c confirm (39). In particular when β = 1.99, simple bifurcation diagrams for both depression and elevation waves have been obtained (see Fig. 4d). The two branches emanate from the bifurcation point and continue to c = 0, where the corresponding wave profiles are plotted in Fig. 5c, d. In the energy-speed diagram shown in Fig. 2a, b, the branches are monotonic in the wave speed c and appear almost identical.
In conclusion, numerical evidence shows that the wave speeds are slowed down due to the effect of electric fields as predicted by (36). In addition, the branch of elevation waves split into two separate parts, when β > β † , and  Fig. 2; however, the problem does not end there. The branch continues to snake back and forth where more turning points are expected. There exist other separate solution branches when β > β † . As β approaches 2 with many branches reaching c = 0, there only remain two branches which are monotonic in c. In the next section, we investigate the stability of the computed solitary wave solutions with time-dependent computation.

Stability analysis
We now investigate numerically the stability of the solitary waves by introducing a small disturbance and solving the initial value problem. A frame of reference moving with the initial wave speed is always chosen, and we evolve the resultant initial condition in time. We recall that in the absence of the electric fields, the depression solitary waves are stable, whereas the elevation waves are unstable near the bifurcation point but become stable after the first turn of the branch of elevation waves at c 1.24 (see [1,3,4]).
We now describe our results with the inclusion of the electric fields. The numerical calculations presented here are carried out with β = 1. The results are qualitatively similar to those of the capillary-gravity problem. A variety of perturbations with ±5% of the amplitude of the depression waves show no sign of instabilities. One example is presented in Fig. 6. A depression solitary wave ensues with smaller amplitude and therefore travelling faster than the original waves. This explains why the wave moves rightwards as time increases since the wave speed of the smaller wave is greater than the speed of the reference frame. As there is no stationary point or turning point on the curve of H (c) for depression waves, no stability exchange occurs on this branch, i.e. the depression branch is stable.
Similarly, we apply a disturbance with ±1% of the amplitude to elevation waves from the elevation branch on the segment between the bifurcation point and the diamond symbol in Fig. 2. The wave eventually evolves into a depression solitary wave as presented in Fig. 7. On the next segment of elevation waves between the diamond and the pentagram, solitary waves are stable as confirmed by the numerical results in Fig. 8. Computations were carried out for other values of β also and yielded similar results to those presented above.
Asymmetric solitary waves also exist in the presence of electric fields. Two examples are shown on the top left and top right in Fig. 9. By virtue of (36), the bifurcation structures of the solution branches for these two asymmetric solitary waves are guaranteed to be qualitatively similar to those presented in [4,18]. In this work, we focus on the stability of these asymmetric waves. By adding an initial disturbance to these two asymmetric solitary waves, it is observed that both waves are unstable and evolve into two depression waves as confirmed in the figure. Similar numerical experiments in the absence of electric fields were carried out in [4].

Collision
In this section, we perform several numerical studies on wave interactions of stable solitary waves when β = 1.  ; it is seen that both waves survive after the interaction and continue to travel with a slightly faster speed due to energy dispersion in the form of small ripples. These numerical solutions match the case of capillary-gravity waves studied in [1].
To study overtaking collisions, we place a solitary wave with a slower speed ahead of a faster solitary wave. A frame of reference is chosen to move with the slower wave. When the difference of two wave speeds is small, both waves survive and maintain their form. We have performed three numerical experiments as shown in Figs. 12, 13 and 14. In Fig. 12, a depression solitary wave overtakes another. Both waves survive, and there are small ripples generated after the collision. It is interesting to note from the results in Fig. 13 that the double-trough solitary wave emerges from the collision as a breather: a travelling localised wave with a periodic oscillation (note the troughs oscillate in amplitude at t = 3000, 4000). This emergence of breathers has been noted previously in [24]. In Fig. 14, the energy has been transferred from one wave to another during the interaction. As a result of this, the depression wave travels faster after the collision.   Fig. 16 where a depression and an elevation solitary wave are chosen initially. In Fig. 17, an elevation wave with speed c = 0.9465 overtakes a depression wave with speed c = 0.87. After the collision, two depression waves and many small ripples emerge, i.e. the elevation wave evolves into a depression one. The reason for this is that the depression wave is much less energetic than the elevation wave travelling at same wave speed. When energy dispersion is large, the elevation wave cannot maintain its form but transforms into a less energetic wave instead of dispersing out completely. If we switch the roles of the two waves studied above, i.e. a depression wave overtaking an elevation wave, the outcomes are qualitatively similar. Such observations were reported for flexural-gravity wave in [19]. From the numerical experiments of collisions, we have observed that larger waves get larger (and slower) and smaller waves get smaller (and faster). This is due to the inelastic nature of the collisions where energy exchange always takes place.

Excitation
The problem of wave excitation is an important research area that attracts a lot of interest. The minimum of the phase speed is a critical value where the resonance between a moving load, for example, and the surface waves yields interesting phenomena. In this regime, linear theory becomes invalid, and nonlinearity is crucial in describing the physical behaviour of the system. Such types of problem were studied in [14,15,19,20,25,26] for flexural-gravity waves and [4] for capillary-gravity waves. In this section, we study this problem in the presence of gravity, surface tension and electric fields. We fix β = 1 and introduce a single moving disturbance P with a speed U = 0.99 in the transcritical range (slightly below the minimal phase speed); the disturbance is given by where the amplitude A is chosen to be 0.02. The forcing is initially placed at x = −250 and later switched off at t = 100. Snapshots taken at t = 25, 100, 400 are presented in Fig. 18a. Immediately after removing the pressure disturbance at t = 100, a depression solitary wave forms on the surface. The long time evolution (up to t = 400) shows that this depression wave continues to propagate without losing its main structure. Hence, the results confirm the stability of the fully localised response. In the next set of numerical experiments, we reduce the strength of electric field progressively by setting and at t = 500, the electric field is switched off completely by keeping β = 0. During this period, the depression solitary wave maintains its shape and becomes less steep due to energy radiation (see Fig. 18b, c). As the value of β decreases to zero, the wave speed rises to approximately 1.385 beyond t = 500. We continued the numerical computations up to t = 2000 and found that the depression wave keeps propagating. The stable elevation wave can be excited by means of two moving disturbances with an appropriate distance between them as in [19] for flexural-gravity waves and in [4] for capillary-gravity waves, for example. We introduce two moving loads where A = B = 0.03, U = 0.99 and d = 10. Both disturbances are switched on at t = 0 and switched off at t = 80. It can be seen clearly from Fig. 19 that an elevation wave is generated at t = 80 and does not disperse out after the removal of the moving loads. This wave also survives while we reduce the value of β progressively to zero for 400 ≤ t ≤ 500, and maintains its shape afterwards. The stability of this fully localised solution is confirmed by these results.
With the presence of electric fields, excitations can be realised with single or multiple moving pressure disturbances at a relatively low speed, compared to the case of capillary-gravity waves. By further turning down the electric field strength, the excited depression and elevation waves both maintain their wave structure, i.e. capillary-gravity waves emerge. We propose this as an alternative possible way to excite capillary-gravity solitary waves.

Conclusion
The problem of capillary-gravity flows under normal electric fields was considered. By means of a conformal mapping, we found steady and time-dependent solutions for various values of the dimensionless electric field strength β. In particular, when β < β † ≈ 1.5265, the properties of waves are qualitatively similar to those for the capillary-gravity waves. The branch of elevation waves split into two separate parts when β > β † , and the lower part disappears when β > β ‡ ≈ 1.9809. The stability problem was studied systematically using a timedependent numerical scheme for β = 1. Several numerical experiments of head-on and overtaking collisions were performed. The stable depression and elevation waves were excited, respectively, using a single load and multiple loads moving with a speed which is slightly less than the minimum phase speed. Overall, the electrified waves have similar properties to those of classical capillary-gravity waves. The main advantage with the presence of normal electric fields is that the waves can be slowed down while keeping the shape. This could open new scenarios for experimentation.