The effect of quantized anisotropic magnetic pressure, arising due to a strong magnetic field, is studied on the growth rate of Jeans instability of quantum electron–ion and classical dusty plasma. Two dispersion relations are formulated for the propagation of the gravitational waves that are perpendicular and parallel to the magnetic field, respectively. It is shown that for perpendicular propagation, the quantized magnetic pressure is stabilizing the Jeans instability, whereas for the parallel propagation the plasma become more unstable with the increase in the magnetic field or we can say that Jeans instabilities becomes anisotropic in this case. The corresponding Jeans wave number is calculated in the absence of tunneling effect which shows that the magnetic field anisotropy of the Galaxy is connected with Jeans effect. Further, it is shown that the Madelung term leads to the inhomogeneity of the plasma medium. Numerical results are presented to show the effect of the anisotropic magnetic pressure on the Jeans instability.

Magnetic field quantization is an important issue for degenerate environments such as neutron stars, radio pulsars and magnetars etc., due to the fact that these stars have a magnetic field higher than the quantum critical field strength of the order of $4.4\times 10^{13}~\text{G}$ , accordingly, the cyclotron energy may be equal to or even much more than the Fermi energy of degenerate particles. We shall formulate here the exotic physics of strongly magnetized neutron stars, known as pulsars, specifically focusing on the outcomes of the quantized magnetic pressure. In this scenario, while following the modified quantum hydrodynamic model, we shall investigate both linear and nonlinear fast magnetosonic waves in a strongly magnetized, weakly ionized degenerate plasma consisting of neutrons and an electron–ion plasma in the atmosphere of a pulsar. Here, linear analysis depicts that sufficiently long, fast magnetosonic waves may exist in a weakly dispersive pulsar having finite phase speed at cutoff. To investigate one-dimensional nonlinear fast magnetosonic waves, a neutron density expression as a function of both the electron magnetic and neutron degenerate pressures, is derived with the aid of Riemann’s wave solution. Consequently, a modified Korteweg–de Vries equation is derived, having a rarefractive solitary wave solution. It is found that the basic properties such as amplitude, width and phase speed of the fast magnetoacoustic waves are significantly altered by the electron magnetic and the neutron degenerate pressures. The results of this theoretical investigation may be useful for understanding the formation and features of the solitary structures in astrophysical compact objects such as pulsars, magnetars and white dwarfs etc.

We examine the degenerate characteristics of the weakly ionized neutron stars (WINSs), while considering magnetic field quantization. A new set of quantum (QMHD) equations is obtained, which describes the homogeneous, weakly ionized, collisional degenerate neutron star. Then, using the obtained QMHD model, we have shown that in a WINS, electrons are responsible for the appearance of an additional term in momentum equations of neutrons, and hence, we formulate the linear dispersive properties of the medium. It is found that the nonlinear dispersive effects cause the broadening of the wave profile to balance the nonlinear steepening, so that the nonlinear stationary waves can be formed in the dispersive neutrino stars. In this context, it is demonstrated that neutron sound waves of WINS are described by two different modified Korteweg–de Vries equations. The present astrophysical plasma is highly recommended to investigate the propagation of proper waves in a strongly magnetized, ionized hydrogen atom.

By applying the Maxwell and quantum hydrodynamic equations, we have studied the parametric instabilities of stimulated Raman scattering (SRS) and stimulated Brillouin scattering (SBS) in an unmagnetized electron-ion quantum plasma. In this context, we have derived the nonlinear dispersion relations of the large-amplitude electromagnetic (EM) waves, the electrostatic electron plasma waves, and the ion-acoustic waves. The nonlinear evolution equations are then solved by utilizing the Fourier transform to obtain expressions for the three-wave decay and modulational instabilities with quantum corrections. It is found that the growth rate of the instabilities is a strong function of large-amplitude EM waves, and quantum effects due to Fermi pressure and quantum correlations stabilize both SRS and SBS instabilities. Expressions for the maximum growth rates attributed to SRS and SBS instabilities are also derived by ignoring the nonlinear correction shift on the frequency of EM waves. The relevance of nonlinear interaction of EM waves with a quantum dense astrophysical plasma is highlighted in the perspective of electron Fermi pressure and exchange-correlation effects, where the plasma density is high enough.

The dispersive properties of Jeans surface instability at a charged plane interface of weakly coupled, classical self-gravitating, magnetized electron-ion plasma and vacuum, are investigated. The general dispersion relation is formulated by employing  one fluid magnetohydrodynamic (MHD) model and Poisson’s equations for both electrostatic and gravitational potentials. It is shown that both the surface charge and  mass density fluctuations are coupled  together effectively, to enhance the growth rate of  surface Jeans instability, whereas an increase in magnetic field and surface tension tends to stabilize Jeans surface instability. The modified Jeans criterion of gravitational instability is obtained by formulating  a new definition of Jeans wavelength. These results may be useful to examine the gravitational collapse of self-gravitating electron-ion plasma, leading to star formation in astro-cosmic environments.