3D-Modulational Stability of Envelope Soliton in a Quantum Electron–Ion Plasma—A Generalised Nonlinear Schrödinger Equation

Abstract

In physical reality, the phenomena of plasma physics is actually a three-dimensional one. On the other hand, a vast majority of theoretical studies only analyze a one-dimensional prototype of the situation. So, in this communication, we tried to treat the quantum electron–ion plasma in a full 3D setup and the modulational stability of envelope soliton was studied in a quantum electron–ion plasma in three dimensions. The Krylov–Bogoliubov–Mitropolsky method was applied to the three-dimensional plasma governing equations. A generalized form of the nonlinear Schrödinger (NLS) equation was obtained, whose dispersive term had a tensorial character, which resulted in the anisotropic behavior of the wave propagation even in absence of a magnetic field. The stability condition was deduced ab initio and the stability zones were plotted as a function of plasma parameters. The modulational stability of such a three-dimensional NLS equation was then studied as a function of plasma parameters. It is interesting to note that the nonlinear excitation of soliton took place again here due to the balance of nonlinearity and dispersion. The zones of contour plots are given in detail.

1. Introduction

The analysis of nonlinear wave propagation in plasma is one of the most important topics of present-day research. Though the physical situation is three-dimensional, for mathematical simplicity, many a times the study is done in one dimension only. However, one should have an idea of the actual theoretical prediction in 3D. With this idea in mind, we treated a quantum electron–ion plasma in three dimensions; we found that we arrived at an new type of generalized nonlinear Schrödinger equation (NLSE). The study of quantum plasma was initiated mainly by the elegant works of Haas [1], Manfredi [2], Shukla [3,4] and Brodin [5]. Up until now, many different situations have been analyzed by various researchers, but all are mainly in one space–time dimension. The basis of the formulation of quantum plasma lies in the unique phase-space 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. However, with its successful application in plasma physics, interest has been invoked and many new observations have been conducted.

In this context, one may note that many different situations of quantum plasma have been investigated in relation to the study of a nonlinear Schrödinger (NLS) equation and the existence of an envelope soliton; however, all are mainly in two dimensions. Even the NLS equation has been derived in many other situations of physics and mathematics, viz., nonlinear optics, pulse propagation, etc. [7,8,9,10,11,12,13,14,15,16], but all were in two dimensions. In particular, the case of degenerate quantum plasma has been considered by Siddki et al. [17]. Some more general situation has been analyzed by B. Eliasson et al. [18,19]. In this respect, one may mention that a thermodynamically open and dissipative system has been considered by Abourabia et al. [20] using the standard reductive perturbation theory, but the whole analysis was performed in one space dimension. The case of quantum pair-ion plasma has been studied by Chaudhuri et al. [21,22] and that of Alfven soliton in a Fermionic plasma has been discussed by Keane et al. [23]. On the other hand, the situation of plasma in an external electric field has been considered by Chowdhury and Pakira [24]. The case of electromagnetic envelope soliton has been taken up by Nusrat Jehan et al. [25]. The situation with a magnetic field has been dealt with by Aktar et al. [26]. However, all these analysis considered a one-dimensional plasma both in the classical and quantum cases.

With these ideas in mind, we considered a three-dimensional quantum plasma (electron–ion). A three dimensional nonlinear Schrödinger equation was derived by taking recourse to the Krylov–Bogoliubov–Mitropolsky method [27,28]. 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 has never been considered before. We obtained the corresponding envelope soliton solution, as well as the condition of modulational stability.

It may further be added that the properties of the nonlinear Schrödinger equation in three dimensions are still not fully studied. Over and above, it would be very much useful for the present-day renewed attempt for the fusion process. Nonlinear excitations such as solitons, breathers and shock waves all are to be computed, because their properties may be widely different from those of a one-dimensional system. That is why, as a first step, we studied the modulational stability in such a situation and calculated the different regions of stability with respect to the plasma parameters. The anisotropic character of the dispersive term is very significant in the sense that it was obtained even in the absence of a magnetic field.

2. Methods and Materials

Our plasma consisted of positively charged ions and electrons which were governed by their respective continuity and momentum equations, along with Poisson’s equation. Due to the small mass of electrons, we assumed that only electrons were quantum mechanical. So, the governing equations can be written as

$$\frac{\partial n_i}{\partial t}+\overrightarrow{\nabla }·\left(n_i\overrightarrow{u_i}\right)=0$$

(1)

$$\frac{\partial n_e}{\partial t}+\overrightarrow{\nabla }·\left(n_e\overrightarrow{u_e}\right)=0$$

(2)

$$\frac{\partial u_i}{\partial t}+(\overrightarrow{u_i}·\overrightarrow{\nabla })\overrightarrow{u_i}=\frac{e}{m_i}\overrightarrow{E}$$

(3)

$$\begin{array}{cc}\hfill & \frac{\partial u_e}{\partial t}+(\overrightarrow{u_e}·\overrightarrow{\nabla })\overrightarrow{u_e}=\hfill \\ \hfill & -\frac{e}{m_e}\overrightarrow{E}-\frac{1}{m_e{n}_e}\overrightarrow{\nabla }{P}_e+\frac{{\hslash }^2}{2m_e}\left\{\overrightarrow{\nabla }\left(\frac{{\nabla }^2\left(\sqrt{n_e}\right)}{\sqrt{n_e}}\right)\right\}\hfill \end{array}$$

(4)

$$\overrightarrow{\nabla }·\overrightarrow{E}=\frac{e}{{\epsilon }_0}\left(n_i-n_e\right)$$

(5)

where $n_i$ and $n_e$ are the densities of ions and electrons, whereas $u_i$ and $u_e$ are the velocities and $\overrightarrow{E}$ is the electric field. These dynamical variables were normalized in the usual manner by a scaling in the following way: $n_i\to n_i/n_{i0}$, $n_e\to n_e/n_{e0}$, $u_e\to u_e/C_s$, $u_i\to u_i/C_s$, along with $x\to x/\lambda De$, $t\to {\omega }_{pe}t$ where ${\lambda }_{De}=\sqrt{\frac{k_B{T}_e{\epsilon }_0}{n_{e0}{e}^2}}$, ${\omega }_{pe}=\sqrt{\frac{n_{e0}{e}^2}{{\epsilon }_0{m}_i}}$. Therefore, the normalized equations were written as

$$\frac{\partial n_i}{\partial t}+\overrightarrow{\nabla }·\left(n_i\overrightarrow{u_i}\right)=0$$

(6)

$$\frac{\partial n_e}{\partial t}+\overrightarrow{\nabla }·\left(n_e\overrightarrow{u_e}\right)=0$$

(7)

$$\frac{\partial u_i}{\partial t}+(\overrightarrow{u_i}·\overrightarrow{\nabla })\overrightarrow{u_i}=\overrightarrow{E}$$

(8)

$$\frac{\partial u_e}{\partial t}+(\overrightarrow{u_e}·\overrightarrow{\nabla })\overrightarrow{u_e}=-\overrightarrow{E}-\frac{{\omega }_{pe}^2}{n_e}\overrightarrow{\nabla }{n}_e+H^2\left\{\overrightarrow{\nabla }\left(\frac{{\nabla }^2\left(\sqrt{n_e}\right)}{\sqrt{n_e}}\right)\right\}$$

(9)

$$\overrightarrow{\nabla }·\overrightarrow{E}={\mu }_i{n}_i-n_e$$

(10)

where ${\mu }_i=\frac{n_{i0}}{n_{e0}}$ and $H^2=\frac{{\hslash }^2{\omega }_{pe}^2}{2k_B^2{T}_e^2}$.

To apply the Krylov–Bogoliubov–Mitropolsky (KBM) method [27] for nonlinear wave modulation, we expanded all the dependent variables by their equilibrium values

$$\left.\begin{array}{c}\hfill n_e=1+ϵn_e^{\left(1\right)}+{ϵ}^2{n}_e^{\left(2\right)}+{ϵ}^3{n}_e^{\left(3\right)}+\cdots \\ \hfill n_i=1+ϵn_i^{\left(1\right)}+{ϵ}^2{n}_i^{\left(2\right)}+{ϵ}^3{n}_i^{\left(3\right)}+\cdots \\ \hfill {\overrightarrow{u}}_i=ϵ{\overrightarrow{u}}_{\left(i\right)}^{\left(1\right)}+{ϵ}^2{\overrightarrow{u}}_i^{\left(2\right)}+{ϵ}^3{\overrightarrow{u}}_i^{\left(3\right)}+\cdots \\ \hfill {\overrightarrow{u}}_e=ϵ{\overrightarrow{u}}_e^{\left(1\right)}+{ϵ}^2{\overrightarrow{u}}_e^{\left(2\right)}+{ϵ}^3{\overrightarrow{u}}_e^{\left(3\right)}+\cdots \\ \hfill \overrightarrow{E}=ϵ{\overrightarrow{E}}^{\left(1\right)}+{ϵ}^2{\overrightarrow{E}}^{\left(2\right)}+{ϵ}^3{\overrightarrow{E}}^{\left(3\right)}+\cdots \end{array}\right\}$$

(11)

In order to consider nonlinear excitations, we assumed that all the perturbed quantities in all order depended upon x, y, z and t through the complex amplitude $\left(\overrightarrow{a}(x,y,z,t),{\overrightarrow{a}}^{*}(x,y,z,t)\right)$ and the phase factor $\psi$, where $\psi =\overrightarrow{k}·\overrightarrow{r}-\omega t$, $\overrightarrow{r}=(x,y,z)$ and ‘$\overrightarrow{k}=(k_x,k_y,k_z)$’. As per the KBM prescription, the space–time derivatives of $\overrightarrow{a}$ were written as

$$\left.\begin{array}{c}\hfill \frac{\partial \overrightarrow{a}}{\partial t}=ϵ{\overrightarrow{A}}_1+{ϵ}^2{\overrightarrow{A}}_2+{ϵ}^3{\overrightarrow{A}}_3+\cdots \\ \hfill \frac{\partial a_{\alpha }}{\partial x_{\beta }}=ϵB_{\alpha \beta }^{\left(1\right)}+{ϵ}^2{B}_{\alpha \beta }^{\left(2\right)}+{ϵ}^3{B}_{\alpha \beta }^{\left(3\right)}\cdots \end{array}\right\}$$

(12)

where $\alpha =x,y,z$ and $\beta =x,y,z$. The perturbed quantities of the electric field were written as

$$\overrightarrow{E}=ϵ{\overrightarrow{E}}^{\left(1\right)}(\overrightarrow{a},{\overrightarrow{a}}^{*},\psi )+{ϵ}^2{\overrightarrow{E}}^{\left(2\right)}(\overrightarrow{a},{\overrightarrow{a}}^{*},\psi )+{ϵ}^3{\overrightarrow{E}}^{\left(3\right)}(\overrightarrow{a},{\overrightarrow{a}}^{*},\psi )+\cdots$$

(13)

From the zeroth order of $ϵ$, we obtained the equilibrium condition to be ${\mu }_i=1$, i.e., $n_{i0}=n_{e0}$. Substituting Equations (11)–(13) in Equations (6)–(10) and equating like powers of $ϵ$, we obtained

$$n_e^{\left(1\right)}=\left(\frac{\overrightarrow{k}·{\overrightarrow{u}}_e}{\omega }\right)$$

(14)

$$n_i^{\left(1\right)}=\left(\frac{\overrightarrow{k}·{\overrightarrow{u}}_i}{\omega }\right)$$

(15)

along with

$$\omega \frac{\partial u_{ex}^{\left(1\right)}}{\partial t}=-E_x^{\left(1\right)}-{\omega }_{pe}^2{k}_x\frac{\partial n_e^{\left(1\right)}}{\partial \psi }+\frac{H^2}{2}{k}_x^3\frac{{\partial }^3{n}_e^{\left(1\right)}}{\partial {\psi }^3}$$

(16)

$$\omega \frac{\partial u_{ey}^{\left(1\right)}}{\partial t}=-E_y^{\left(1\right)}-{\omega }_{pe}^2{k}_y\frac{\partial n_e^{\left(1\right)}}{\partial \psi }+\frac{H^2}{2}{k}_y^3\frac{{\partial }^3{n}_e^{\left(1\right)}}{\partial {\psi }^3}$$

(17)

$$\omega \frac{\partial u_{ez}^{\left(1\right)}}{\partial t}=-E_z^{\left(1\right)}-{\omega }_{pe}^2{k}_z\frac{\partial n_e^{\left(1\right)}}{\partial \psi }+\frac{H^2}{2}{k}_z^3\frac{{\partial }^3{n}_e^{\left(1\right)}}{\partial {\psi }^3}$$

(18)

$$\omega \frac{\partial {\overrightarrow{u}}_i^{\left(1\right)}}{\partial \psi }={\overrightarrow{E}}^{\left(1\right)}$$

(19)

Using Equations (14)–(20), we obtained the following equation:

$$\left[D^2-\left\{\frac{{\mu }_i}{{\omega }^2}+\frac{1}{{\omega }^2}+\frac{{\mu }_i{\omega }_{pe}^2}{{\omega }^4}{k}^2\right\}+\frac{H^2{k}^4}{2}{D}^4\right]\left(\overrightarrow{k}·{\overrightarrow{E}}^{\left(1\right)}\right)=0$$

(20)

where D stands for the derivative operator $\frac{\partial }{\partial \psi }$. Substituting ${\overrightarrow{E}}^{\left(1\right)}=\overrightarrow{a}exp\left(j\psi \right)+{\overrightarrow{a}}^{*}exp(-j\psi )$, we obtained the linear dispersion relation as follows:

$$L{\omega }^4+M{\omega }^2+N=0$$

(21)

where $L=\left(1-\frac{H^2{k}^4}{2}\right)$, $M=(1+{\mu }_i)$, $N={\mu }_i{\omega }_{pe}^2{k}^2$.

The expressions for the first order quantities are

$$n_i^{\left(1\right)}=\frac{1}{{\omega }^2}\left[\left(\overrightarrow{k}·\overrightarrow{a}\right)exp\left(j\psi \right)+c.c\right]$$

(22)

$$n_e^{\left(1\right)}=\left({\mu }_i-\frac{1}{{\omega }^2}\right)\left[\left(\overrightarrow{k}·\overrightarrow{a}\right)exp\left(j\psi \right)+c.c\right]$$

(23)

$${\overrightarrow{u}}_i^{\left(1\right)}=-\frac{1}{\omega }(\overrightarrow{a}exp\left(j\psi \right)+c.c)$$

(24)

$${\overrightarrow{u}}_e^{\left(1\right)}=\left[-\frac{1}{\omega }-\frac{{\omega }_{pe}^2}{\omega }\left({\mu }_i-\frac{1}{{\omega }^2}\right)k^2-\frac{H^2{k}^4}{2}\left({\mu }_i-\frac{1}{{\omega }^2}\right)\right](\overrightarrow{a}exp\left(j\psi \right)+c.c)$$

(25)

Here, j($=\sqrt{-1})$ is used instead of i just to avoid confusion, because i is used, in this paper, as a suffix for denoting the parameters for ion.

Similarly, by equating the coefficients of ${ϵ}^2$, we obtained

$$\begin{array}{cc}& \omega \frac{\partial u_{el}^{\left(2\right)}}{\partial \psi }=A_m^{\left(1\right)}\frac{\partial u_{el}^{\left(1\right)}}{\partial a_m}+{\omega }_{pe}^2\left(B_{lm}^{\left(1\right)}\frac{\partial n_e^{\left(1\right)}}{\partial a_m}\right)-\frac{H^2}{2}{k}_l^3\frac{{\partial }^3{n}_e^{\left(2\right)}}{\partial {\psi }^3}+\\ & \frac{H^2}{2}{n}_e^{\left(1\right)}{k}_l^3\frac{{\partial }^3{n}_e^{\left(1\right)}}{\partial {\psi }^3}+{\omega }_{pe}^2{k}_l\frac{\partial n_e^{\left(2\right)}}{\partial \psi }-{\omega }_{pe}^2{k}_l{n}_e^{\left(1\right)}\frac{\partial n_e^{\left(1\right)}}{\partial \psi }+\\ & k_l{u}_{el}^{\left(1\right)}\frac{\partial u_{el}^{\left(1\right)}}{\partial \psi }+E_l^{\left(2\right)}\end{array}$$

(26)

with sum over repeated index and ($l=x,y,z$; $m=x,y,z$). Similarly,

$$\omega \frac{\partial u_{il}^{\left(2\right)}}{\partial \psi }=A_m\frac{\partial u_{il}^{\left(1\right)}}{\partial a_l}+k_l{u}_{il}^{\left(l\right)}\frac{\partial u_{il}^{\left(1\right)}}{\partial \psi }-E_l^{\left(2\right)}$$

(27)

$$\omega \frac{\partial n_e^{\left(2\right)}}{\partial \psi }=A_l^{\left(1\right)}\frac{\partial n_e^{\left(1\right)}}{\partial a_l}+B_{xl}^{\left(1\right)}\frac{\partial u_{ex}^{\left(1\right)}}{\partial a_l}+k_l\frac{\partial u_{el}^{\left(2\right)}}{\partial \psi }+B_{yl}\frac{\partial u_{ey}^{\left(1\right)}}{\partial a_l}+k_l\frac{\partial }{\partial \psi }\left(n_e^1{u}_{el}^1\right)+B_{zl}\frac{\partial u_{ez}}{\partial a_l}$$

(28)

$$\omega \frac{\partial n_i^{\left(2\right)}}{\partial \psi }=A_l^{\left(1\right)}\frac{\partial n_i^{\left(1\right)}}{\partial a_l}+B_{xl}^{\left(1\right)}\frac{\partial u_{ix}^{\left(1\right)}}{\partial a_l}+k_l\frac{\partial u_{il}^{\left(2\right)}}{\partial \psi }+B_{yl}\frac{\partial u_{iy}^{\left(1\right)}}{\partial a_l}+k_l\frac{\partial }{\partial \psi }\left(n_i^1{u}_{il}^1\right)+B_{zl}\frac{\partial u_{iz}^{\left(1\right)}}{\partial a_l}$$

(29)

Lastly, from Poisson’s equation, we obtained

$$k_l\frac{\partial E_l^{\left(2\right)}}{\partial \psi }+B_{lx}^{\left(1\right)}\frac{\partial E_x^{\left(1\right)}}{\partial a_l}+B_{ly}^{\left(1\right)}\frac{\partial E_y^{\left(1\right)}}{\partial a_l}+B_{lz}^{\left(1\right)}\frac{\partial E_z^{\left(1\right)}}{\partial a_l}={\mu }_i{n}_i^{\left(2\right)}-n_e^{\left(2\right)}$$

(30)

Using Equations (23)–(30), we obtained

$$(D^3+D)(\overrightarrow{k}·{\overrightarrow{E}}^{\left(2\right)})=\frac{\chi }{\overrightarrow{{\alpha }^{\prime }}}\left[\overrightarrow{A_1}+\overrightarrow{v_g}·{\overleftrightarrow{B}}_1\right]exp\left(i\psi \right)+\frac{1}{\overrightarrow{{\alpha }^{\prime }}}\mathrm{Terms}\mathrm{proportional}\mathrm{to}exp\left(2i\psi \right)$$

(31)

The above equation, in order to be secularity-free, requires $\left[{\overrightarrow{A}}_1+{\overrightarrow{v}}_g·{\overleftrightarrow{B}}_1\right]=0$, where ${\overrightarrow{v}}_g$ is the group velocity and is given by

$${\overrightarrow{v}}_g=\frac{\overrightarrow{{\alpha }^{\prime }}}{\chi }$$

(32)

where $\overrightarrow{{\alpha }^{\prime }}=\left(2{\omega }_{pe}^2{\omega }^2+2{\omega }_{pe}^2{\mu }_i+2H^2{k}^2{\omega }^2+2H^2{k}^2{\mu }_i\right)\overrightarrow{k}$, $\chi =2{\omega }^2+{\mu }_i-2\omega {\omega }_{pe}{k}^2-H^2{k}^4\omega -1$.

Next, proceeding to the perturbation terms, which are the third order in $ϵ$, we obtained

$$-\sigma \left\{1+\frac{{\partial }^2}{\partial {\psi }^2}\right\}\left(\overrightarrow{k}·{\overrightarrow{E}}^{\left(3\right)}\right)=L(1,2)$$

(33)

where

$$\begin{array}{c}\hfill L(1,2)=\frac{{\mu }_i}{{\omega }^2}({\omega }^2-{\omega }_{pe}^2{k}^2)J_1(1,2)-\\ \hfill ({\omega }^2-{\omega }_{pe}^2{k}^2)F(1,2)-\frac{H^2{k}^4}{2}{I}_1(1,2)+\\ \hfill \frac{{\mu }_i}{{\omega }^2}\frac{{\partial }^2{J}_1(1,2)}{\partial {\psi }^2}+G(1,2)\end{array}$$

(34)

with $\sigma =\left\{1+\frac{{\mu }_i}{{\omega }^2}({\omega }^2-{\omega }_{pe}^2{k}^2)\right\}$. In the same way as before, in order to have Equation (34) to be secularity-free, we set

$$\chi \left(A_2+{\overrightarrow{v}}_g·{\overleftrightarrow{B}}_2\right)+j{\Lambda }_{\alpha \beta }\left(B_{\alpha \beta }\frac{\partial B_{\alpha \beta }}{\partial a_l}\right)+j{Q}_0{|\overrightarrow{k}·\overrightarrow{a}|}^2+j{R}_0(\overrightarrow{k}·\overrightarrow{a})=0$$

(35)

Thus, by suitable scale transformation and taking $\overrightarrow{k}·\overrightarrow{a}=\varphi$, we obtained the standard nonlinear Schrödinger equation, given as

$$j\frac{\partial \varphi }{\partial \tau }+\sum P_{\alpha \beta }\frac{{\partial }^2\varphi }{\partial {\xi }_{\alpha }\partial {\xi }_{\beta }}+Q{\left|\varphi \right|}^2\varphi +R\varphi =0$$

(36)

where it should be noted that $P_{\alpha \beta }=-\frac{{\Lambda }_{\alpha \beta }}{\chi }$, $Q=-Q_0/\chi$ and $R=-R_0/\chi$. The expressions for ${\Lambda }_{\alpha \beta }$, $Q_0$ and $R_0$ are given in Appendix A.

Equation (36) is the generalized form of the nonlinear Schrödinger equation in three dimensions. The expressions for $F(1,2)$, $J(1,2)$ and $I(1,2)$ are all given in Appendix A. One should note that, in this equation, the usual Laplacian term is replaced by a second-order operator, where the coefficients of each second-order space derivative are different. In this equation, the last term, $Ra$, known as the linear interaction term, causes simply a phase shift. Using a simple substitution $\overrightarrow{a}\to \overrightarrow{a}exp\left[jR\tau \right]$, we obtained the standard nonlinear Schrödinger equation in 3D as

$$j\frac{\partial \varphi }{\partial \tau }+\sum P_{\alpha \beta }\frac{{\partial }^2\varphi }{\partial {\xi }_{\alpha }\partial {\xi }_{\beta }}+Q{\left|\varphi \right|}^2\varphi =0$$

(37)

It should be noted that, in the above equations (for example, Equations (36) or (37)), as well as in other places in this article, we use $j=\sqrt{-1}$ instead of i to avoid confusion with the notation of ions.

We know that, when there is a nonlinear evolution of a wave, it may so happen that the nonlinear effect is balanced by the dispersion effect and a stable wave structure is formed. In fact, the application of the external electric field on the plasma causes a local depression in density, called a ‘caviton’. Then, plasma waves trapped in this cavity form an isolated structure, called an ‘envelope soliton’, which is one of the solutions of Equation (37). A graphical representation of such a soliton is depicted in Figure 1 and its corresponding contour plot is given in Figure 2.

Dispersion and Stability

To proceed further, we started with the dispersion relation (21). Here, we plotted $\omega$ as a function of the wave vector $k\left(=\sqrt{k_x^2+k_y^2+k_z^2}\right)$ in Figure 3.

One may observe that the trend is quite similar to that observed in one dimension. Here, we show three different curves pertaining to three separate values of H. The next important quantity is the group velocity ${\overrightarrow{v}}_g$, as given in Equation (32). Here, in Figure 4a–c, we also depict the variation in $v_{gx}$ with respect to $k_x$, $k_y$ and $k_z$.

One may note that, whereas $v_{gx}$ has a growing behavior with respect to $k_x$, it has a decreasing nature when seen with respect to $k_y$ and $k_z$. This event is also 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 (Figure 5). The negative and positive regions are explicitly exhibited in the contour plot. A similar behavior can be seen for $P_{yy}$ in a subplot (Figure 6). Since $P_{\alpha \beta }$ is a symmetric tensor, we obtained a value of $P_{zz}$ similar to that of $P_{xx}$ and $P_{yy}$. On the other hand, due to the symmetric behavior of $P_{\alpha \beta }$, the variation in the off-diagonal elements of $P_{\alpha \beta }$ are similar (i.e., $P_{\alpha \beta }=P_{\beta \alpha }$). Thus, in this communication, we show the variation in $P_{xy}$ and $P_{xz}$ only, depicted in Figure 7 and Figure 8, respectively. The variation in the nonlinear coefficient Q is depicted in Figure 9 with respect to $k_x$. Since Q is a scalar, its variation with $k_y$ and $k_z$ is similar to the variation with respect to $k_x$.

To explore further, we tried to investigate the modulational instability of the system (37).

For that, we set

$$\varphi =({\mathsf{\Phi }}_0+\delta \mathsf{\Phi }\left(\zeta \right))exp[-i\Delta \tau ]$$

(38)

where ${\mathsf{\Phi }}_0$ is the carrier wave amplitude, K and $\mathsf{\Omega }$ are the modulational wave number and frequency, respectively, and $\zeta =\overrightarrow{K}·\overrightarrow{\xi }-\mathsf{\Omega }\tau$.

Using this expression of ‘$\mathsf{\Phi }$’ in Equation (38), from the zeroth-order term of $\delta \mathsf{\Phi }$, we obtained $\Delta =-Q|{\mathsf{\Phi }}_0{|}^2$.

Moreover, from the first-order term of $\delta \mathsf{\Phi }$, we obtained

$$\begin{array}{cc}\hfill & j\frac{\partial }{\partial \tau }\left(\delta \mathsf{\Phi }\right)+\Delta \left(\delta a\right)+P\frac{{\partial }^2}{\partial {\xi }_{\alpha }\partial {\xi }_{\beta }}\left(\delta \mathsf{\Phi }\right)+Q{\left|a_0\right|}^2(\delta \mathsf{\Phi }+\delta {\mathsf{\Phi }}^{*})\hfill \\ \hfill & +Q|{\mathsf{\Phi }}_0{|}^2\left(\delta \mathsf{\Phi }\right)=0\hfill \end{array}$$

(39)

where $\delta {\mathsf{\Phi }}^{*}$ is the complex conjugate of $\delta \mathsf{\Phi }$. Setting $\delta \mathsf{\Phi }=\mathcal{U}+i\mathcal{V}$ and using the expression of $\Delta$ obtained from the zeroth-order term of $\delta \mathsf{\Phi }$, one obtains, by separating the real and imaginary parts,

$$\left.\begin{array}{c}\hfill \frac{\partial \mathcal{U}}{\partial \tau }+P_{\alpha \beta }\frac{{\partial }^2\mathcal{V}}{\partial {\xi }_{\alpha }{\xi }_{\beta }}=0\\ \hfill -\frac{\partial \mathcal{V}}{\partial \tau }+P_{\alpha \beta }\frac{{\partial }^2\mathcal{U}}{\partial {\xi }_{\alpha }{\xi }_{\beta }}+2Q\mathcal{U}{\left|{\mathsf{\Phi }}_0\right|}^2=0\end{array}\right\}$$

(40)

Next, assuming plane wave solutions of $\mathcal{U}$ and $\mathcal{V}$, i.e.,

$$\left.\begin{array}{c}\hfill \mathcal{U}=\mathcal{S}exp(j[\overrightarrow{K}·\overrightarrow{\xi }-\mathsf{\Omega }\tau ])\\ \hfill \mathcal{V}=\mathcal{J}exp(j[\overrightarrow{K}·\overrightarrow{\xi }-\mathsf{\Omega }\tau ])\end{array}\right\}$$

(41)

These linearized equations led to

$$\left|\begin{array}{cc}j\mathsf{\Omega }& K_{\alpha }{K}_{\beta }{P}_{\alpha \beta }\\ \\ -K_{\alpha }{K}_{\beta }{P}_{\alpha \beta }+2Q{\left|{\mathsf{\Phi }}_0\right|}^2& j\mathsf{\Omega }\end{array}\right|=0$$

which reduced to

$${\mathsf{\Omega }}^2+2\alpha Q{\left|{\mathsf{\Phi }}_0\right|}^2-{\alpha }^2=0$$

(42)

where $\alpha =K_x{K}_x{P}_{xx}+K_x{K}_y{P}_{xy}+K_x{K}_z{p}_{xz}+K_y{K}_x{P}_{yx}+K_y{K}_y{P}_{yy}+K_y{K}_z{P}_{yz}+K_z{K}_x{P}_{zx}+K_z{K}_y{P}_{zy}+K_z{K}_z{P}_{zz}$, solution of (41), yields $\mathsf{\Omega }=\sqrt{{\alpha }^2-\alpha \left(2Q\right|{\mathsf{\Phi }}_0{|}^2)}$. Therefore, that condition of stability turned out to be the following: if $\left(\frac{\alpha }{2Q|{\mathsf{\Phi }}_0{|}^2}<1\right)$, then the wave is unstable and, if $\left(\frac{\alpha }{2Q|{\mathsf{\Phi }}_0{|}^2}>1\right)$, then the wave is stable. To ascertain this condition numerically, we plotted the contour plot of $\left(\frac{\alpha }{Q}\right)$ for various values of H and $K_x,K_y\mathrm{and}{K}_z$ in subplots (Figure 10a–c). We can see that, in the subplot (Figure 10a,c), for $H\gtrsim 0.1$ and for all values of modulational wave number $K_l$, the ratio $\alpha /Q$ becomes less than 1 and the wave instability sets in. On the other hand, in Figure 10b, the behavior of $\alpha /Q$ is slightly different from that of (a,c), but, here, the instability also sets in for $H⩾0.2$. The graphical representation in Figure 10 explicitly identifies the zones of stability and instability in the space of plasma parameters. From these figures, one can easily identify the experimental values needed for the requisite parameters. Thus, we can 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.

3. Discussion and Conclusions

In the above analysis, we studied the evolution of the envelope soliton using the Krylov–Bogoliubov–Mitropolsky method. In this case, we considered a quantum hydrodynamic (QHD) model. The nonlinear Schrödinger equation deduced in this case is three-dimensional, where the group velocity is a 3D-vector quantity and the dispersive coefficient $P_{\alpha \beta }$ is a symmetric tensor. Both the dispersion relation and the group velocity of the system have a clear variation, not only with the wave vector ($k_l$) but also with the quantum diffraction parameter (H). However, the most interesting part of this observation is the anisotropic behavior of the propagation wave. We saw that the variational nature of the group velocity ($v_g$) along x is different from that of y- and z-directions. Generally, this sort of anisotropy occurs in plasma when there is a presence of a magnetic field, but, in this case, even in absence of a magnetic field, the anisotropy occurred.

Next, we analyzed the modulational stability of the system. The phenomenon of modulational instability is one of the basic topics to be studied in the domain of nonlinear plasma physics. Initially, it was an idea put forward by Benjawin and Feir [29] in the case of water waves and by Bespalov and Talanov [30] for electromagnetic wave application in many branches of physics, such as nonlinear optics, condensed matter physics, etc. When the monochromatic wave (or its quasi counter part), as it propagates through the nonlinear dispersive medium, gives rise to two side-bands, the whole phenomenon is, in fact, related to the role these side-bands play in the growth and decay of the propagating side-bands. With the help of the nonlinear dispersion relation (41), we found the stability criteria. In this case, the anisotropic effect was also clear. We saw that, when the ratio $(\alpha /Q)>1$, the carrier wave was modulationally ‘stable’, while, for $(\alpha /Q)<1$, the carrier wave became modulationally ‘unstable’. The values of the plasma parameters used for the numerical analysis are as follows: $n_{i0}=n_{e0}\sim (1-4)\times {10}^{14}$ (meter)${}^{-3}$ and ${\omega }_{pe}\sim {10}^7$ Hz.