Nonlinear stability, excitation and soliton solutions in electrified dispersive systems

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Abstract

A weakly nonlinear theory of wave propagation in two superposed dielectric fluids in the presence of a horizontal electric field is investigated in (2+1)-dimensions. The equation governing the evolution of the amplitude of the progressive waves is obtained in the form of a two-dimensional nonlinear Schrödinger equation. A three-wave resonant interaction for nonlinear excitations created from electrohydrodynamic capillary-gravity waves is observed to be possible in a dispersive medium with a self-focusing cubic nonlinearity. Under suitable conditions, the nonlinear envelope equations for the resonant interaction are derived by using multiple scales and inverse scattering methods, and an explicit three-wave soliton solution is discussed. Both the dynamic properties and the modulational instability of finite amplitude electrohydrodynamic wave are studied for the cubic nonlinear Schrödinger equation by means of linearized stability analysis and the nonlinear interaction coefficient. We show that the trajectories in phase space exhibit different behavior with the increase of nonlinear perturbations, and we determine the electric field and wavenumber ranges at which the original point is elliptic or hyperbolic, respectively. It is found also that the presence of the electric field in the equation modifies the nature of wave stability and soliton structures, and that the amplitude and width of the soliton are decreased and increased, respectively, when the electric field value increases.

Introduction

Resonant wave interaction can be described as a nonlinear process in which energy is transferred between different natural modes of an oscillatory system by resonance. Consider a nonlinear system that is oscillating by one or more of its natural modes. As the system is nonlinear, the motion is not simply a summation of the linear modes, but consists of the linear harmonics plus their nonlinear coupling [15]. Under resonance conditions, the nonlinear coupling between some modes may lead to excitation of a neutral mode or modes. The behavior of this excited mode(s) depends on the properties of the original modes and the system. An interesting situation occurs when the created mode(s) grows rapidly in time, being of primary importance in studies of hydrodynamic stability [19]. The phenomenon of resonant wave interaction was first studied by Philips [30] and subsequently by Longuett-Higgins [24]. Textbooks of Drazin and Reid [7], Craik [5], and Komen et al. [21] as well as articles by Philips [31], and Hammack and Handerson [16] give excellent reviews of the subject. In wave interaction problems, the resonance conditions are expressed in terms of certain relations between the wavenumbers and the frequencies of the waves involved, and these resonance conditions may differ from one class of waves to another. McGoldrick [25], [26], [27] studied interaction between capillary-gravity waves in a series of papers. He studied the problem when the following conditions of resonance hold between three capillary-gravity waves 1, 2, and 3.k1=k2+k3,ω1=ω2+ω3,where ki and ωi are the vector wavenumber and frequency of the i-th wave, respectively. McGoldrick derived the following evolution equations for the wave amplitudes.da1dt=iαω1a2a3,da2dt=iαω1a1a3,da3dt=iαω1a1a2,where α is a constant, and ai is the amplitude of the i-th wave. The superscript denotes the complex conjugate.

The subject of electrohydrodynamics, on the other hand, has drawn considerable interest over the past few decades, and it has a wide range of importance in various physical situations (see for example, refs. [8], [14], [28], and references therein). Problems of nonlinear electrohydrodynamic stability have been considered by many authors in recent years. El-Sayed and Callebaut [9], [10], [11], in a series of papers, studied the slow modulation of the interfacial capillary-gravity waves of two superposed dielectric fluids of uniform depth under the influence of a general applied electric field (tangential or normal) to the interface between the two fluids, in the presence (or absence) of surface charges at their interfaces. They [12] have also obtained the envelope solutions of the steady state in (2+1)-dimensions, in terms of the Jacobian elliptic functions. Callebaut and El-Sayed [2] have investigated the nonlinear electrohydrodynamic stability of solitary wave packets of capillary-gravity in (2+1)-dimensions using the tanh method. They [3] had also used a variation of the tanh method to obtain alternative kinds of solitary wave solutions of the extended mKdV–KdV–Burgers equation. More recently, El-Sayed [13] has investigated the nonlinear electrohydrodynamic wave propagation of two superposed dielectric fluids in the presence of a horizontal electric field using the multiple time scales method in (2+1)-dimensions. In all the above mentioned elecyrohydrodynamic studies, the three-wave resonance interaction has not been investigated yet.

In this paper, we observe that, for two superposed electrified fluids in (2+1)-dimensions, a three-wave resonance can occur in a self-focusing dispersive medium with only a cubic nonlinearity. The excited waves considered here are generated from capillary-gravity waves, and the interaction between them is shown to have quadratic character. Under suitable phase-matching conditions, the envelope equations for the three-wave resonance are derived by using multiple scales and inverse scattering methods, and explicit three-wave soliton solution is provided and discussed. The dynamic properties and modulational instability of the cubic nonlinear Schrödinger equation are also discussed. We show that the inclusion of the electric field to the analysis will modify the nature of wave instability and soliton structures.

Section snippets

Nonlinear Schrödinger equation in (2+1)-dimensional system

We consider the finite amplitude three-dimensional wave propagation on the interface z=0 which separates two semi-infinite dielectric inviscid incompressible fluids. A fluid with density ρ(1), and dielectric constant ɛ(1) occupies the region z<0, whereas the region z>0 is occupied by a fluid of density ρ(2), and dielectric constant ɛ(2). The fluids are influenced by a constant electric field E0 in the x-direction. We nondimensionalize the various quantities using the characteristic length L=T/ρ(

Wave resonant interaction

Using the transformation A=2/Q1Ψexp(iτ), we can reduce Eq. (11) to the normalized form2iΨτ+2ΨX2+2ΨY22(|Ψ|21)Ψ=0.Letting Ψ=Qexp(iR) with Q and R being two real functions, Eq. (12) is recast into the formQτ+Q22RX2+2RY2+QXRX+QYRY=0,QRτ122QX2+2QY2+Q2RY2+RY2+Q3Q=0.

We are interested in a possible three-wave resonance of excitation waves. For an efficient three-wave resonance, the phase-matching conditionsω1+ω2=ω3andK1+K2=K3,should be required. It is easy to show

Behavior and instability of cubic nonlinear Schrödinger equation

In this section, we investigate the modulational instability of the cubic nonlinear Schrödinger equation, for simplicity [32]. This equation can be obtained from the previous Eq. (2) with (l=0), and it readsiAτ+p2Ax12+q|A|2A=0,wherep=12ωvg212ωk(1+ρ)(1ρ)+kE02(ɛ(2)ɛ(1))3(ɛ(1)+ɛ(2))+1kvg,q=kω(1+ρ)ω2(1ρ)+k2E02(ɛ(2)ɛ(1))3(ɛ(1)+ɛ(2))2Λ+k244(1ρ)+k2,in which Λ and ω2 are defined by Eqs. (7) and (8), respectively with K=k (or l=0), and vg=dω/dk is the group velocity in the x-direction.

Using

Conclusions

In conclusion, based on a self-focusing nonlinear Schrödinger equation resulting from the nonlinear study for two superposed electrified fluids system in (2+1)-dimensions, we have investigated a wave resonant interaction in an electrohydrodynamic dispersive medium with cubic nonlinearity. We have observed that a three-wave resonance is indeed possible for the excitation waves created from the capillary-gravity waves. By adequately choosing the wave vectors and frequencies of the three exciting

Acknowledgements

It is a pleasure to thank Prof. Dr. Wily Malfliet (University of Antwerp, Belgium) for his useful comments and his critical reading of the original manuscript. I would like also to thank the three referees for their interest in this work and their valuable suggestions that improved the original manuscript.

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