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 [6], in microelectronic device [7] as well as in the next-generation intense laser [8]. Many authors have proposed some theories including the quantum corrections to the quantum plasma echoes [9], the self-consistent dynamics of Fermi gases [10], quantum beam instabilities [11], wave interactions in quantum magnetoplasmas [12], classical and quantum kinetics of the Zakharov system [13], quantum corrections to the Zakharov equations [14], expansion of quantum electron gas into vacuum [15], quantum ion acoustic waves [16], quantum Landau damping [17], magnetohydrodynamics of quantum plasmas [18], etc. Quantum plasmas have extremely high plasma number densities and low temperatures. At extremely low temperatures, the thermal de Broglie wavelength becomes comparable to the interelectron distance and the electron temperature becomes comparable to the electron Fermi temperature and the electrons follow Fermi Dirac distribution law. In this condition, quantum mechanical effects are expected to play a significant role in the behavior of charged particles [19-21]. As electrons are lighter than ions, the quantum behavior of electron is reached faster than ions. The dust particles are quite common in various plasma systems. The inclusion of immobile charged dust in electron-ion plasmas leads to introduce a new mode. Shukla and Silin [22] have first theoretically shown the existence of low-frequency dust ion-acoustic (DIA) waves in a dusty plasma, which was latter observed in laboratory experiments [23,24]. The phase speed of the DIA waves is much smaller (larger) than electron (ion) thermal speed. The inertia is provided by the ion mass while the restoring force comes from the electron thermal pressure. These waves differ from usual ion-acoustic waves [25] due to the conservation of equilibrium charge density and the strong inequality, where is the particle number density of the species with for electrons, for ions and for dust, is the number of electrons residing onto the dust grain surface, and is the magnitude of an electronic charge. Therefore, a dusty plasma can not support the usual ion-acoustic waves, but can do the DIA waves of Shukla and Silin [22]. The nonlinear waves associated with the DIA and QDIA waves particularly solitary waves (SWs) [26,27] and shock waves [28] have received a great deal of interest in understanding the basic properties of localized electrostatic perturbation in space [29,30] and laboratory dusty plasmas [31-34]. A number of investigations have been made on QDIA SWs [35] and shocks [28] by using K-dV equation. For plasmas with more than two species, it can arise cases where the K-dV equation is not valid near a critical value of a certain parameter (say). The nonlinear term vanishes at this critical value (at) [36] which makes soliton amplitude large enough to break down the validity of K-dV equation. The mmK-dV (mixed modified K-dV) equation, also known as Gardner equation, can give soliton solution around this critical value [37]. The technique of analyzing SWs is Gardner approach which leads to a standard Gardner equation. From the analysis of standard Gardner equation, SW of permanent profile is found, which is known as Gardner soliton (GS) [1,38,39]. In our present manuscript, we attempt to study the basic features of QDIA GSs by deriving modified Gardner equation, which is valid around, in a quantum dusty plasma containing inertial ions, Fermi electrons with quantum effect, and negatively charged immobile dust. The manuscript is organized as follows. The model equations are provided in Section 2. The Gardner equation is derived by using the reductive perturbation method in Section 3. The analytical solutions are presented in Section 4. A brief discussion is finally given in Section 5.

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 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 U₀ = 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 U₀ 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 U₀. 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.