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The prominent features of higher order nonlinear ion-acoustic waves involving quantum corrections in an unmagnetized quantum dusty plasma are revisited with the theoretical framework of Hossain et al. [1]. The fluid model is demonstrated here by its constituent inertial ions, Fermi electrons with quantum effect, and immovable dust grain with negative charge. We have used the ideology of Gardner equation. The well-known RPM method is employed to derive the equation. Indeed, the basic features of quantum dust ion-acoustic Gardner solitons (GSs) are pronounced here. GSs are shown to exist for the value of dust to ion ratio around 2/3 which is valid for space plasma [2], and are different from those of K-dV (Korteweg-de Vries) solitons, which do not exist for the value around 2/3. The implications of our results are suitable for cosmological and astrophysical environments.

Quantum plasmas have attracted a great deal of attention because of their potential applications in dense plasma particularly in different astrophysical and cosmological systems [3-5] (e.g. interstellar or molecular clouds, planetary rings, comets, interior of white dwarf stars, etc.), in nanostructures [

We consider a one-dimensional, collisionless, unmagnetized quantum dusty plasma system composed of inertial ions, massless Fermi electrons with quantum effect, and negatively charged immobile dust. Thus, at equilibrium we have. The nonlinear dynamics of these low-frequency (purely electrostatic) QDIA waves in such a plasma system is described by the normalized equations of the form

where is the ion (electron) number density normalized by its equilibrium value, is the ion fluid speed normalized by quantum ion-acoustic speed with being the ion rest mass, is the electron Fermi energy, is the Boltzmann constant, and T_{Fe} is the Fermi temperature of electron, is the electrostatic wave potential normalized by with e being the magnitude of the charge of an electron, ρ is the normalized surface charge density, and. The time variable is normalized by and the space variable is normalized by. In Equation (3) we have used the following Fermi pressure law for the electron species [40,41]

Also with and is the ratio between the plasmon energy and the electron Fermi energy where is the electron Fermi speed at temperature.

We first obtain the well known K-dV equation and see why Gardner equation is needed to find SW solution.

To obtain the QDIA K-dV equation, we introduce the stretched coordinates

where is the QDIA wave phase speed and is a smallness parameter measuring the weakness of the dispersion. We then expand, , , , and in power series of

and develop equations in various powers of. To the lowest order in, Equations (1)-(13) give

where. Equation (17) represents the linear dispersion relation for the QDIA waves. This clearly indicates that the QDIA wave phase speed increases with the increase of the dust charge density.

To the next higher order of, one can obtain another set of coupled equations for, , and, which -along with the first set of coupled linear equations for, , and -reduce to a nonlinear dynamical equation of the form

where

Equation (18) is known as K-dV equation. The stationary localized solution of Equation (18) is given by

where the amplitude and the width are given by and, respectively. is the mach number. As and, (22) clearly indicates that 1) small amplitude solitary waves with, i.e. positive soliton exists if, 2) small amplitude solitary waves with, i.e. negative soliton exists if, and 3) for i.e. the nonlinear term vanishes at μ = μ_{c} and is not valid near μ = μ_{c} which makes soliton amplitude large enough to break down its validity. To find soliton solution around, we now obtain gardner equation.

To study QDIA GSs by analyzing the ingoing solutions of Equations (1)-(5), we first introduce the stretched coordinates [

By using Equations (23) and (24) in Equations (1)-(6), and Equations (9)-(13) and to the lowest order in ε, we find the same values of, , , and as like as that of the K-dV. To the next higher order in, we obtain a set of equations, which, after using Equations (14)-(17), can be simplified as

It is obvious from Equation (28) that since. One can find that at its critical value (which is a solution of). So, for around its critical value, can be expressed as

where, is a small and dimensionless parameter, and can be taken as the expansion parameter, i.e., and for and for. So, can be expressed as

which, therefore, must be included in the third order Poissons equation. To the next higher order in, and after some mathematical calculations we obtain a set of equations

Now, combining Equations (31)-(33), we obtain a equation of the form

where

And is given in Equation (21). Equation (34) is known as Gardner equation. It is important to note that if we neglect term and put, the Gardner equation reduces to K-dV equation which has derived in Equation (18). However, in this K-dV equation the nonlinear term vanishes at, and is not valid near which makes soliton amplitude large enough to break down its validity. But the Gardner equation derived here is valid for near its critical value.

To analyze stationary GSs, we first introduce a transformation which allows us to write Equation (34), under the steady state condition, as

where the pseudo-potential is

It is obvious from Equation (38) that

The conditions of Equations (39) and (40) imply that SW solutions of (37) exist if

The latter can be solved as

where, and. Now, using Equations (38) and (43) in Equation (37) we have

where. The SW solution of Equations (37) or (44) is, therefore, directly given by

where are given in Equation (43) and SWs width is

Figures 1-4 show the variation of amplitude of positive (negative) GSs with μ for U_{0} = 0.5 and H = 0.3. These figures clearly indicate that both positive and negative GSs exist around the crical value,. It has been found that the amplitude (magnitude of the amplitude) of both positive and negative GSs decrease with the increase of μ. Figures 2-5 represent the variation of amplitude of positive (negative) GSs with U_{0} for μ = 0.66 (μ = 0.67) and H = 0.3. These figures indicate that the amplitude of both positive and negative GSs increase with the increase of U_{0}. We have found that the amplitude of positive and negative GSs does not vary with the quantum diffraction parameter, H but the width of the both positive and negative GSs vary with it. Figures 3-6 imply that the width of both positive and negative GSs decrease with the increase of H and increase with the increase of μ. We have also noticed that in our present system the GSs exist when the quantum effect of electron is neglected.

We have investigated QDIA GSs in quantum dusty plasma by deriving Gardner equation. The K-dV solitons are not valid for and, which vanish

at the nonlinear coefficients of the K-dV equation. However, the QDIA GSs investigated in our present work are valid for. The results, which have been obtained from this investigation, can be summarized as follows:

1) The quantum dusty plasma system under consideration supports finite amplitude GSs, whose basic features (polarity, amplitude, width, etc.) depend on the ion and dust number densities and quantum diffraction (tunneling) parameter, H.

2) GSs are shown to exist for, and are found to be different from K-dV solitons, which do not exist for.

3) It has been found that at, positive GSs exist, whereas at, negative GSs exist.

4) We have seen that the amplitude of positive and negative GSs decreases with μ, increases with, and does not depend on H.

5) We have also observed that the width of the GSs increases with μ but decreases with the increase of H.

It should be mentioned here that in our present investigation, we have neglected the quantum effect of ions since ions are heavier than electrons. However, QDIA solitary waves in quantum dusty plasma with or without the effects of obliqueness and external magnetic field are also problems of recent interest for many space and laboratory dusty plasma situations, but beyond the scope of our present investigation. In conclusion, we propose that a new experiment may be designed based on our results to observe such waves in both laboratory and space quantum dusty plasma system.

The Third World Academy of Science (TWAS) Research Grant for research equipment is gratefully acknowledged.