Modulational Stability of Envelope Soliton in a Quantum Electron-Ion Plasma in Three Dimension-A generalised Nonlinear Schrödinger Equation in 3D

Modulational stability of envelope soliton is studied in a quantum dusty plasma in three dimension. The Krylov-Bogoliubov-Mitropolsky method is applied to the three dimension plasma governing equations. A generalised form of Nonlinear Schrödinger equation is obtained whose dispersive term has a tensorial character. Stability condition is deduced abintio and the stability zones are plotted as a function of plasma parameters.


I. INTRODUCTION
The study of quantum plasma was initiated mainly by the elegant works of Haas [1], Manfredi [2], Shukla [3,4] and Brodin [5]. Up-till now many different situations have been analysed by various researchers, but all are mainly in one space and one time dimension. The basis of the formulation of quantum plasma lies in the unique phasespace quantization advocated by E.P. Wigner [6] long ago. For a long time this methodology was ignored due to the conceptual difficulty of phase space in quantum physics. But with its successful application in plasma physics, interest have been invoked and many new observations are done.
In this context one may note that many different situations of quantum plasma have been investigated in relation to the study of Nonlinear Schrödinger(NLS) equation and existence of envelope soliton but all are mainly in two dimensions. In particular the case of degenerate quantum plasma was considered by Siddki et.al [7]. Some more general situation was analysed was by B. Eliasson et.al [8,9]. In this respect one may mention that a thermodynamically open and dissipative system was considered by Abourabia et.al [10] using the standard reductive perturbation theory, but the whole analysis was in one space dimension. The case of quantum pair-ion plasma has been studied Chaudhuri et.al [11,12] and that of Alfven soliton in a Fermionic plasma was discussed by Keane et. al [13]. On the other-hand the situation of plasma in an external electric field was considered by Chowdhury and Pakira [14]. The case of electromagnetic envelope soliton was taken up by Nusrat Jehan et. al [15]. The situation with a magnetic field was dealt by Aktar et.al [16]. But all these analysis considered a one dimensional plasma both in the classical and quantum case.
With three ideas in mind we have considered a three dimensional quantum plasma(electron-ion). A three dimensional Nonlinear Schrödinger equation is derived by taking recourse to the Krylov-Bogoliubov-Mitropolsky method [17,18]. The equation so derived is more general than the standard Nonlinear Schrödinger equation in the sense that its dispersive term has a tensorial character. To the authors knowledge this type of equation was never considered before. We have obtained the corresponding envelope soliton solution and the condition of modulational stability is obtained.

II. FORMULATION
Our plasma consist of positive charged ion and electron which are governed by their respective continuity and momentum equation respectively, along with the Poisson's equation. Due to small mass of electron we assume that only electrons are quantum mechanical. So, the governing equations can be written as where 'n i ','n e ' are the densities of ions and electrons whereas 'u i ', 'u e ' are the velocities, ' E' is the electric field. These dynamical variables are normalized in the usual manner by a scaling in the following way, n i → n i /n i0 , n e → n e /n e0 , u e → u e /C s , ǫ0mi , so that the normalized equations are written as; ∂n e ∂t + ∇ · (n e u e ) = 0 (7) where, To apply the Krylov-Bogoliubov-Mitropolsky(KBM) method [17,18] for nonlinear wave modulation we expand all the dependent variables about their equilibrium values, n e = 1 + ǫn (1) e + ǫ 2 n (2) e + ǫ 3 n (3) e + · · · n i = 1 + ǫn In order to consider nonlinear excitations we assume that all the perturbed quantities in all order depend upon 'x', 'y' 'z' and 't' through the complex amplitude ( a(x, y, z, t), a * (x, y, z, t)) and the phase factor 'ψ' where, ψ = k · r − ωt, r = (x, y, z), ' k = (k x , k y , k z )'. As per the KBM prescription the space-time derivatives of ' a' is written as; where α = x, y, z and β = x, y, z. The perturbed quantities of the electric field are written as E = ǫ E (1) ( a, a * , ψ)+ǫ 2 E (2) ( a, a * , ψ)+ǫ 3 E (3) ( a, a * , ψ)+· · · (13) From the zeroth order of ǫ we get the equilibrium condition to be µ i = 1 i.e. n i0 = n e0 . Substituting equations 11 to 13 in equation 6 to 10 and equating like powers of ǫ we get; ω ∂u Using 14 to 20, we get obtain the following equation; where, 'D' stands for the derivative operator ' ∂ ∂ψ '. Substituting, E (1) = a exp(iψ) + a * exp(−iψ) we get the linear dispersion relation as follows, The expressions for the first order quantities are; Similarly equating coefficients of ǫ 2 we get; ω ∂u with sum over repeated index and (l = x, y, z ; m = x, y, z). Similarly; ω ∂u Lastly from the Poisson's equation we get, Using the equations (23) to (30) we get; The above equation in order to be secularity free, requires is the group velocity and is given by Next proceeding to the perturbation terms which are third order in ǫ, we get; where, In the same way as before, in order to have equation(34) to be secularity free, we set ∂B αβ ∂a l +iQ 0 | k· a| 2 +iR 0 ( k· a) = 0 (35) Thus by suitable scale transformation and taking k · a = φ we get the standard Nonlinear Schrödinger equation given as; It should be noted that in the above equations (for example equations( 36) or ( 37)) and also in other places in this article we have used j = √ −1 instead of 'i' to avoid confusion with notation of ions.
We know that, when there is a nonlinear evolution of a wave, it may so happen that te nonlinear effect is balanced by the dispersion effect and a stable wave structure is formed. Actually the application of the external electric field on the plasma causes a local depression in density called a 'caviton'. Plasma waves trapped in this cavity then form an isolated structure called an 'envelope soliton' which is one of the solution of the equation( 37). A graphical representation of such a soliton is depicted by figure( 1) and its corresponding contour plot is given in figure( 2).

III. DISPERSION AND STABILITY
To proceed further we start with the dispersion relation (21). Here we have plotted 'ω' as a function of the wave vector k(= k 2 x + k 2 y + k 2 z ) in figure( 3). One may observe that the trend is quite similar to that observed in one dimension. Here we have shown three different curves pertaining to three separate values of 'H'. The next important quantity is the group velocity ' v g ' as given in equation (32). Here also in the figure 4(a,b,c) we have depicted the variation of v gx with respect to k x , k y and k z . One may note that where as v gx has a growing behaviour with respect to k x , it has a decreasing nature when seen with respect to k y and k z . This event has also been repeated for v gy and v gz . Our next important feature is the tensorial components of the dispersive terms. Let us start with P xx and consider its variation with k x ,k y and k z by subplot ( 5). The negative and positive regions are explicitly exhibited in the contour plot. A similar behaviour is seen for P yy in subplot ( 6). Since P αβ is a symmetric tensor, hence we obtained a of P zz similar to that of P xx and P yy . On the other-hand due to the symmetric behaviour of P αβ the variation of the off-diagonal elements of P αβ are similar (i.e. P αβ = P βα ). Thus in this communication we have shown the variation of P xy and P xz only, depicted in the figures ( 7) and ( 8) respectively. The variation of the nonlinear coefficient Q is depicted by the figure( 9) with respect to k x . Since Q is a scalar, so its variation with k y and k z were similar as the variation with respect to k x .
To explore further we try to investigate the modulational instability of the system (37).
For that we set Φ 0 being the carrier wave amplitude, K and Ω the modulational wave number and frequency respectively and ζ = K · ξ − Ωτ Using this expression of 'Φ' in equation (38), from the zeroth order term of δΦ we get; ∆ = −Q|Φ 0 | 2 And from the first order term of δΦ we get; j ∂ ∂τ (δΦ) + ∆(δa) + P ∂ 2 ∂ξ α ∂ξ β (δΦ) + Q|a 0 | 2 (δΦ + δΦ * ) where, δΦ * is the complex conjugate of δΦ. Setting, δΦ = U + iV and using the expression of '∆' obtained from the zeroth order term of δΦ, one gets by separating the real  and imaginary parts; Next assuming plane wave solutions of U and V i.e., These linearised equations leads to which reduces to, where, α = K x K x P xx + K x K y P xy + K x K z p xz + K y K x P yx + K y K y P yy + K y K z P yz + K z K x P zx + K z K y P zy + K z K z P zz , solution of (41) yields; Ω = α 2 − α(2Q|Φ 0 | 2 ). So that condition of stability turns out to be :: if α 2Q|Φ0| 2 < 1 then the wave is unstable and if α 2Q|Φ0| 2 > 1 then the wave is stable. To ascertain this condition numerically we have plotted the contour plot of α Q for various values of H and K x , K y and K z in subplot( 10a,b,c).
We see that in the subplot( 10a,c), for the H 0.1 and for all values of modulational wave number K l the ratio 'α/Q' becomes less than 1 and the wave instability sets in. While in figure( 10b) though the behaviour of α/Q is bit different from that of (a,c) but here also the instability set in for H 0.2. Thus we see that the quantum diffraction parameter has a good impact on the stability of the wave.
At this point we may note that the modulational stability condition is slightly changed in this 3D case.

ACKNOWLEDGEMENT
One of the author Shatadru Chaudhuri (SC) is grateful to the Department of Physics, Jadavpur University for a fellowship under the scheme RUSA. SC would also like to thank Mr. Jyotirmoy Goswami and Mr. Jit Sarkar for thier support and inspiration.