Quantum effects in plasmas should be taken into account when the Fermi energy of the charged particles in the plasma exceeds its thermal energy [1–3]. Accordingly, quantum plasmas have been widely investigated in the laser solid interaction [4], and in astrophysical objects like neutron stars and white dwarfs star, in quantum x-ray free-electron lasers [5], in nuclear fusion reaction [6, 7], in warm dense matter [8, 9], and in streaming and wake effects [10].

In order to measure the interparticle interactions between electrons or ions, the quantum coupling parameter is preferred in the quantum plasma as ${{\rm{\Gamma }}}_{e,i}=4\pi {e}^{2}{n}_{e,i}^{1/3}/{E}_{\mathrm{Fe},\mathrm{Fi}}$ with ${E}_{\mathrm{Fe},\mathrm{Fi}}={{\hslash }}^{2}{(3{\pi }^{2}{n}_{e,i})}^{2/3}/2{m}_{e,i}$ being the Fermi energy, where e is the electron charge, n_e,i is the number density of electrons or ions, m_e,i is the rest mass of electron or ion, and ℏ is the reduced Plancks constant. The qantum coupling parameters of electrons and ions implies that a quantum plasma becomes even more ideal at higher densities [11–13]. The quantum diffraction effect stands for the fluctuation of microscopic particles, which was experimentally verified by x-ray Thomson scattering [14]. When the plasma density is high enough that the distance between charged particles is comparable or smaller than the De Broglie wavelength of the charged particles, the relativistic degeneracy pressure, which arises from the Pauli exclusion mechanism, must be taken into acount. The presence of upper limit of the mass of a white dwarf is one of the first important evidence which is supported by the pressure of degenerate plasma [15, 16]. The spin effects is also important in quantum plasmas, especially in astrophysical invironments such as magnetars and pulsars [17], which was studied by spin magnetohydrodynamic approach by using a non-relativistic Pauli equation for spin-1/2 particles [18]. The investigation of the magnetorotational instability should be also taken into acount with the effect of spin magnetization in a differentially rotating degenerate quantum plasma [19]. The combined effects of spin and polarization on the laser C plasma interactions is important for the quantum electrodynamics plasmas [20], where spin-polarized quantum radiation reaction strongly affects the dynamics of the plasma.

Dusty particles with nano-sized or micron-sized solid particles suspended in a plasma environment will deeply modify the collective excitation in the dusty plasma. Due to the large mass of the dusty particles, it should be taken that the Coulomb coupling strength for dusty plasma can be defined as ${{\rm{\Gamma }}}_{d}=({Z}_{d}^{2}{e}^{2}/{k}_{{\rm{B}}}{T}_{d}L)\exp (-L/{\lambda }_{d})$ with the dusty Debye length ${\lambda }_{d}={({k}_{{\rm{B}}}{T}_{d}/4\pi {n}_{d}{Z}_{d}^{2}{e}^{2})}^{1/2}$ and the mean interparticle distance L. The coupling strength will increase with the increase of the dusty plasma density. The dusty plasma can be described by the fluid model [21, 22] or the visco-elastic fluid model [23, 24] according to the magnitude of the Coulomb coupling strength. The strongly coupling effects can cause ring structural transition [25] and the formation of a void [26]. A pair of counter-rotating symmetric vortices was observed in the wake in a flowing dusty plasma, where the strongly coupled dusty plasmas flow with controllable velocity [27]. For dusty plasma in a perpendicular magnetic field, the shear viscosity can be modified substantially, which shows that the viscosity of the dusty plasma increases at low temperatures if the magnetic field is increased, while at high temperatures the viscosity of the dusty plasma decreases [28]. The strong correlation effects contribute to stabilizing the growth rate of linear Rayleigh–Taylor instability [29] and Jeans instability [30] in the strongly coupled dusty plasma.

Accordingly, for the ultradense dusty plasma, the quantum coupling effects of electrons and ions and the Coulomb coupling effects will dominate the dynamics of the strongly coupled dusty plasma [31, 32]. We will consider the combined quantum effects of quantum diffraction, quantum statistic pressure, as well as electron exchange and correlation effects due to the spin of electrons and ions on the nonlinear collective excitation of dust-ion-acoustic waves (DIAWs) in the strongly coupled dusty plasma. As will be shown, the strongly coupling effects will lead to the attenuation of DIAWs. The quantum effects will also affect the dispersion and nonlinearity of the DIAWs.

We consider the nonlinear DIAW propagation in a strongly coupled dust quantum plasma with arbitrary charged cold dust fluid, and electron and ion fluids. The electrons and ions are modeled by quantum hydrodynamic equation, where the quantum diffraction, quantum statistic pressure, as well as electron exchange and correlation effects due to electron spin are all considered in our model. The quantum hydrodynamic equations of the electrons and ions are given as [33–35] where n_j(j = e, i, d) (normalized by n_j0) is the electron, ion, or dust number density, respectively. The n_j0) is the equilibrium number density. u_j(j = e, i, d) (normalized by Fermi speed V_Fe) is the electron, ion, or dust fluid velocity, respectively. The time and space variables are normalized by ${\lambda }_{\mathrm{DFe}}\,={({k}_{{\rm{B}}}{T}_{\mathrm{Fe}}{\varepsilon }_{0}/{e}^{2}{n}_{e0})}^{1/2}$ and inverse electron plasma frequency ${\omega }_{\mathrm{pe}}^{-1}={({e}^{2}{n}_{e0}/{m}_{e}{\varepsilon }_{0})}^{-1/2}$. Here, T_Fe is the electron Fermi-temperature, k_B is the Boltzmann constant, m_e,i is electron (ion) rest mass. In equations (2) and (3) the third term stands for the electrostatic field force from the charge separation between electrons, ions, and dust particles. The fourth term in equations (2) and (3) is related to the electron and ion exchange-correlation force due to electron spin, of which the coefficients are Γ_xce = 0.985e²/εr_e0k_BT_Fe, and Γ_xci = 0.985e²M/εr_i0k_BT_Fe, where the ${r}_{e0,i0}={n}_{e0,i0}^{-1/3}$ are the Wigner–Seitz radius. The non-dimensional exchange-correlation potential V_xce,i are given as [36]where M = m_e/m_i and the Bohr radius is a_Be,i =εℏ²/m_e,ie²r_0e,i, which is normalized by the Wigner–Seitz radius. ε is the dielectric constant in vacuum. The fifth and sixth term of equations (2) and (3) is the degenerate pressure and quantum reoil force, respectively. Here we consider non-relativistic degenerate plasma and then the pressure can be given as ${P}_{e,i}={K}_{e,i}{n}_{e,i}^{5/3}$ and K_e,i =(3/5)(π/3)^1/3πℏ²/m_e,i [37]. ℏ is the Planck constant. The parameters in the coefficient ${V}_{\mathrm{Fe},i}^{2}/{C}_{\mathrm{se},i}^{2}$ are the Fermi speed ${V}_{\mathrm{Fe},i}={(1/3)}^{1/6}{({\hslash }/{m}_{e,i}^{* })({\pi }^{2}{n}_{e0,i0})}^{1/3}$ and quantum acoustic velocity ${C}_{\mathrm{se},i}={({k}_{{\rm{B}}}{T}_{\mathrm{Fe}}/{m}_{e,i}^{* })}^{1/2}$. The quantum recoil force is related to the Bohm potential, which stands for the electrons or ion tunneling effects. The quantum diffraction parameters are given as H_e = (ℏω_pe/2k_BT_Fe) and H_i =(ℏω_pe/2k_BT_Fe)M [38], which stands for the fluctuation of microscopic particles in the quantum mechanical effects.

For strongly coupled dust particles, a generalized hydrodynamic model is considered aswhere the coefficients are A_γ = γ_dm_eT_d/m_dT_Fe and A_β = m_eZ_d/m_d. Here m_d is the mass of the dust particles, and Z_d is the charged number of dust particles, and γ_d is the adiabatic index of the dust fluid, and T_d is the temperature of the dusty plasma, and ${\eta }^{* }={m}_{e}{\omega }_{\mathrm{pe}}/\left({n}_{d0}{m}_{d}{k}_{{\rm{B}}}{T}_{\mathrm{Fe}}\right)\left(\xi +\tfrac{4}{3}\eta \right)$ is the viscosity effects. ξ and η stand for shear and bulk viscosity effects, respectively. The non-dimensional ${\overline{\tau }}_{m}={\eta }^{* }({T}_{e}/{T}_{d}){\left[1-{\mu }_{d}+\tfrac{5}{14}u({\rm{\Gamma }})\right]}^{-1}$ refers to the visco-elastic relaxation time [39], in which ${\mu }_{d}=1\,+\tfrac{1}{3}u({\rm{\Gamma }})+\tfrac{{\rm{\Gamma }}}{9}\tfrac{\partial u({\rm{\Gamma }})}{\partial {\rm{\Gamma }}}$ is the compressibility and u(Γ) is a measure of the excess internal energy of the system. The system is enclosed by the following Poisson's equations:Here, α = n_i0/n_e0 and φ is the electrostatic potential normalized by k_BT_Fe/e. The equilibrium quasi-neutrality condition n_e0 + Z_dn_d0 = n_i0 is considered.

In order to study the nonlinear set of equations (1)–(6) we introduce the stretched variables ξ and τ with $\xi ={\varepsilon }^{\tfrac{1}{2}}(x-\lambda t)$, $\tau ={\varepsilon }^{\tfrac{3}{2}}t$, where ϵ is small parameter standing for the weak nonlinearity and λ is the phase velocity of the DIAWs [40]. We then expand the physical variables n_j, u_j, and φ in terms of the expansion parameter ϵ aswhere f = n_j, u_j, φ, f⁽⁰⁾ = 1 for n_j, and f⁽⁰⁾ = 0 for f = u_j, φ. The expansion (7) shows that the physical quantities have a small deviation from the equilibrium state f₀, whereas ϵ characterizes the strength of the nonlinearity. The equilibrium state of the physical quantities can be written as unperturbed quantities. Accordingly, the unperturbed number density is normalized by their own equilibrium quantities as ${n}_{e}^{(0)}={n}_{i}^{(0)}={n}_{d}^{(0)}=1$. While the unperturbed velocity of the quantum fluids and the electrostatic potential are ${u}_{e}^{(0)}={u}_{i}^{(0)}={u}_{d}^{(0)}={\phi }^{(0)}=0$. Then, we will use equation (7) in equations (1)–(6) to expand the physical parameters in various orders in ϵ. We will take the ordering of ${\overline{\tau }}_{m}$ and η^* on the expansion. We consider the limits as ${\overline{\tau }}_{m}\,={\omega }_{\mathrm{pe}}{\tau }_{m}\gg 1,{\eta }^{* }\sim O(1)$. In this limit, we use stretched variables and equation (7) in equations (1)–(6) and obtain the following equations in the first two lowest order in ϵ from each of the equations (1)–(6) as From the lowest order expansions, we obtain the relationship between the first-order quantities aswhere the parameters standing for quantum oscillation dispersion are determined as ${D}_{e}={\lambda }^{2}-({V}_{\mathrm{Fe}}^{2}/{C}_{\mathrm{se}}^{2}+{C}_{\mathrm{xce}}^{2}/{C}_{\mathrm{se}}^{2})$, ${D}_{i}\,=\,{\lambda }^{2}-({V}_{\mathrm{Fi}}^{2}/{C}_{\mathrm{si}}^{2}+{C}_{\mathrm{xci}}^{2}/{C}_{\mathrm{si}}^{2})$, and ${D}_{d}\,=\,-{\lambda }^{2}{\overline{\tau }}_{m}+{A}_{\gamma }{\overline{\tau }}_{m}+{\eta }^{* }$. From the first-order quantities, we can obtain the dispersion relation asEliminating ${n}_{j}^{(2)},{u}_{j}^{(2)}$, and φ⁽²⁾ from equations (9), (11), (13), (15), (17) and making use of equations (8), (10), (12), (14), (16), the following modified KdV (mKdV) equation is obtained:where coefficients A, B, C, and D of equation (18) can be written as follows:where the other coefficients are given in the appendix.

As an exact analytic solution of equation (20) is absent, we are devoted to seek for an approximate time-dependent soliton solution of equation (20), where we consider the effect of D is small limit, i.e., A ∼ B ≫ D. Here the coefficient D stands for the importance of strong coupling. As will be shown in the approximate solution, the coefficient D will cause the dissipation of the solitary waves. The coefficients A, and B represent nonlinear and dispersion effects, respectively. From the analytical expression of A, and B, the nonlinear and dispersion effects come from the quantum effects of electrons and ions. Accordingly one can expect that the approximate solution will be reasonable when quantum effects are much stronger than strong coupling effects. Then we will consider the limit X_e − αX_i ≫ (1 − α)(X_d1 + Z_d1) to obtain the approximate solution, where the quantities X_e and X_i stand for the quantum statistic and electron exchange and correlation effects and the quantities X_d1, and Z_d1 represent the strong coupling effects. In this limit, we can rewrite equation (20) after integrating with respect to ξ aswhere we have used the boundary conditions: φ⁽¹⁾→0, ∂φ⁽¹⁾/∂ξ → 0 as ξ → ± ∞ . Now, in absence of D, a travelling wave (stationary soliton) solution of equation (20) can be obtained aswhere $\Psi$ = 3U/A is the amplitude, $W={\left(12B/{\rm{\Psi }}A\right)}^{1/2}\,\equiv \sqrt{4B/U}$ is the width and U = $\Psi$A/3 is the constant phase speed of the DIAWs.

Next, to find the solitary wave solution of equation (22) with the effect of a small amount of D, we first integrate equation (22) with respect to ξ to obtainThis shows that equation (22) conserves the total number of particles. Furthermore, multiplying equation (22) by Φ and integrating over ξ yieldsin which the equality sign holds only when Φ = 0 for all ξ. Equation (25) states that an initial perturbation of the form (23) for whichwill decay to zero. Thus, the wave amplitude $\Psi$ is not a constant, but decreases slowly with time. Here, we perform a perturbation analysis of equation (22) assuming that D is a small parameter with 1 ∼ A ∼ B ≫ D (in magnitudes). So, we introduce a new space coordinate χ in a frame moving with the solitary wave and normalized to its width as χ = ξ − U(τ)τ The amplitude $\Psi$ is assumed to vary slowly with time and $\Psi$ = $\Psi$(D, τ). We also assume that Φ ≡ Φ(χ, τ). To determine the time-dependent speed U(τ), we use momentum conservation law in the presence of viscosity as [41]where the momentum of KdV system $I=(1/2){\int }_{-\infty }^{+\infty }{\phi }^{2}{\rm{d}}{X}$. Substituting equation (18) into equation (19) we obtain the analytical expression for time-dependent speed aswhere the parameters ${\tau }_{0}=5A\sqrt{B}/{U}_{0}^{3/2}D$ and U₀ is the velocity of the DIAWs when τ = 0. From the expression of the speed U(τ) one can see that the amplitude of DIA solitary waves will decrease with time going on.

In conclusion, the DIAWs solitary in the kinetic regime are investigated in strongly coupled quantum dusty plasmas. We used the quantum hydrodynamic model for electrons and ions and the generalized hydrodynamic model for the strongly coupled dust particles. According to the quantum coupling parameter, the electrons and ions can be considered by quantum plasma, where the combined quantum effects of quantum diffraction, quantum exchange-correlation were concluded in our model. For dusty particles with nano-sized or micron-sized solid particles, we introduce the Coulomb coupling parameters to determine the generalized hydrodynamic model. In order to consider the spin effects in the ultradense quantum plasma, we introduce the electron and ion exchange-correlation force in the quantum hydrodynamic model. By using the reductive perturbation method, we derive a mKdV equation for the DIAWs in the presence of strong coupling effects. We firstly obtain the travelling wave (stationary soliton) solution of the mKdV equation. Consequently, we perform a perturbation analysis of the mKdV equation and assume that the shear and bulk viscosity effects are very weak. In the weak visco-elastic effect limit, we obtained an analytically approximate time-dependent solution of the mKdV equation for the dust-ion-acoustic solitary waves, which shows that the amplitude will increase as time goes on.