# Stability of two-dimensional ion-acoustic wave packets in quantum plasmas

Abstract

The nonlinear propagation of two-dimensional (2D) quantum ion-acoustic waves (QIAWs) is studied in a quantum electron–ion plasma. By using a 2D quantum hydrodynamic model and the method of multiple scales, a new set of coupled nonlinear partial differential equations is derived which governs the slow modulation of the 2D QIAW packets. The oblique modulational instability (MI) is then studied by means of a corresponding nonlinear Schrödinger equation derived from the coupled nonlinear partial differential equations. It is shown that the quantum parameter H (ratio of the plasmon energy density to Fermi energy) shifts the MI domains around the kθ -plane, where k is the carrier wave number and θ is the angle of modulation. In particular, the ion-acoustic wave (IAW), previously known to be stable under parallel modulation in classical plasmas, is shown to be unstable in quantum plasmas. The growth rate of the MI is found to be quenched by the obliqueness of modulation. The modulation of 2D QIAW packets along the wave vector k is shown to be described by a set of Davey–Stewartson-like equations. The latter can be studied for the 2D wave collapse in dense plasmas. The predicted results, which could be important to look for stable wave propagation in laboratory experiments as well as in dense astrophysical plasmas, thus generalize the theory of MI of IAW propagations both in classical and quantum electron–ion plasmas.

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- "Furthermore, low and arbitrary amplitude nonlinear structures have also been studied in dense e-p-i plasmas and/or e-i plasmas37383940. However, in most of the recent investigations, quantum effects are considered mostly involving electron-ion or dusty plasmas without external magnetic field41424344454647484950. Recently, Shukla et al. [51] have carried out a study on the electromagnetic solitary pulses in a magnetized e-p plasma. "

[Show abstract] [Hide abstract]**ABSTRACT:**Amplitude modulation of a compressional electromagnetic wave in a strongly magnetized electron-positron pair plasma is considered in the quantum magnetohydrodynamic regime. The important ingredients of this study are the inclusion of the external strong magnetic field, Fermi quantum degeneracy pressure, particle exchange potential, quantum diffraction effects via the Bohm potential, and dissipative effect due to collision of the charged carriers. A modified-nonlinear Sch¨odinger equation is developed for the compressional magnetic field of the electromagnetic wave by employing the standard reductive perturbation technique. The linear and nonlinear dispersions of the electromagnetic wave are discussed in detail. For some parameter ranges, relevant to dense astrophysical objects such as the outer layers of white dwarfs, neutron stars, andmagnetars, etc., it is found that the compressional electromagnetic wave is modulationally unstable and propagates as a dissipated electromagnetic wave. It is also found that the quantum effects due to the particle exchange potential and the Bohm potential are negligibly small in comparison to the effects of the Fermi quantum degeneracy pressure. The numerical results on the growth rate of the modulation instability is also presented.- [Show abstract] [Hide abstract]
**ABSTRACT:**We study the nonlinear propagation of electrostatic wave packets in a collisional plasma composed of strongly coupled ions and relativistically degenerate electrons. The equilibrium of ions is maintained by an effective temperature associated with their strong coupling, whereas that of electrons is provided by the relativistic degeneracy pressure. Using a multiple-scale technique, a (3 + 1)-dimensional coupled set of nonlinear Schrödinger-like equations with nonlocal nonlinearity is derived from a generalized viscoelastic hydrodynamic model. These coupled equations, which govern the dynamics of wave packets, are used to study the oblique modulational instability of a Stoke's wave train to a small plane-wave perturbation. We show that the wave packets, though stable to the parallel modulation, become unstable against oblique modulations. In contrast to the long-wavelength carrier modes, the wave packets with short wavelengths are shown to be stable in the weakly relativistic case, whereas they can be stable or unstable in the ultrarelativistic limit. Numerical simulation of the coupled equations reveals that a steady-state solution of the wave amplitude exists together with the formation of a localized structure in (2 + 1) dimensions. However, in the (3 + 1)-dimensional evolution, a Gaussian wave beam self-focuses after interaction and blows up in a finite time. The latter is, however, arrested when the dispersion predominates over the nonlinearities. This occurs when the Coulomb coupling strength is higher or a choice of obliqueness of modulation, or a wavelength of excitation is different. Possible application of our results to the interior as well as in an outer mantle of white dwarfs are discussed. - [Show abstract] [Hide abstract]
**ABSTRACT:**We present a theoretical study of the nonlinear propagation of electrostatic wave envelopes in a plasma which consists of strongly coupled ions and relativistically degenerate electrons. A generalized viscoelastic hydrodynamic model is considered, and the standard reductive perturbation technique is used to derive a (3+1)-dimensional coupled nonlocal nonlinear Schrödinger-like equations which describe the dynamics of electrostatic wave packets in strongly coupled degenerate plasmas. The modulational instability of a plane wave is studied, and it is shown that the wave packet is stable to the parallel modulation, whereas it is unstable against the oblique modulation. The stable and unstable regions for both the long-wavelength and short-wavelength modes are obtained in weakly relativistic as well as ultra-relativistic regimes. The coupled set of nonlinear equations is numerically solved, and is shown to exhibit a steady state solution of the wave amplitude as well as the formation of coherent structures in (2+1) dimensions. However, in (3+1)-dimensions, a Gaussian wave beam self-focuses leading to collapse in a finite time. Such a collapse can be arrested when the dispersion predominates over the nonlinearities.

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