Nowadays, studies in quantum plasmas have become important due to their potential applications to quantum wells,^[1] to spintronics,^[2] plasmonics,^[3–4] to microelectronics,^[5] to nonlinear optics,^[6] to astro-physics,^[7] and to solid density target experiments.^[8–9] The quantum plasmas were first studied by Pines^[10] in regimes where we have a high density and a low temperature as compared to classical plasmas. At such high densities, the plasma behaves like a degenerate fluid and quantum mechanical effects play a pivot role in the dynamics. Generally, the quantum effects associated with the strong density correlation play a significant role in the plasma system when the de Broglie wavelength of the charged particles is larger than the Debye wavelength and is near to the Fermi wavelength. Under these conditions, the quantum hydrodynamic (QHD) model is a suitable method to describe the charged particle systems.^[11–12] The QHD model consists of a set of equations that describe the transport of momentum and energy of the charged particles and includes quantum statistical pressure via the Fermi pressure and the quantum tunneling effect via the Bohm potential.^[13] Numerous collective effects have been studied by a number of authors in the field of quantum plasmas.^[14–19] Misra et al.^[20] have studied the nonlinear propagation of two-dimensional quantum ion acoustic waves (QIAWs) in an electron-ion (EI) quantum plasma.

Most of the aforementioned works focus on the electron-ion plasma instead of the electron-positron-ion (EPI) plasma. Recently, there has been much attention in EPI plasmas for its potential application point of view to investigate new collective modes and instabilities. The EP plasma exists in active galactic nuclei,^[21] in Van Allen radiation belts and near the polar cap of fast rotating neutron stars,^[22–23] in semiconductor plasmas,^[24] intense laser fields.^[25] The EP plasmas that include an additional ion species, exhibiting a collective behavior and holding quasi-neutrality condition, will constitute an EPI plasma. In contrast to usual EI plasmas, it has been well established fact that the nonlinear propagation of acoustic modes behaves quite differently in EPI plasmas.^[26–27] Nejoh^[28] investigated the effect of ion temperature on the large amplitude IA waves in EPI plasma. Mushtaq and Shah^[29] studied the effect of positron concentration on the nonlinear propagation of magnetosonic waves and found that the solitary waves in EPI plasma behaved quite differently than that of ordinary EI plasma. Out of several mechanisms, such as the effects of turbulence, collisions between charged and neutrals particles, Landau damping, wave particle interaction, etc., one possible dissipative mechanism, which is peculiar to plasmas, may be due to the kinematic viscosity, the effects of which were previously examined by many authors.^[30–33] Rouhani et al.^[34] studied the characteristic of IA shock waves in a dissipative quantum pair plasma with dust particulates. Hanif et al.^[35] investigated the propagation characteristics of IA shock waves in an unmagnetized dense quantum plasma. Hossen and Mamun^[36] investigated theoretically the basic features of the quantum IA solitary and shock structures in a strongly coupled cryogenic quantum plasma.

However, most of the above metioned investigations are limited to one-dimensional (1D) planar geometry, which may not be a realistic situation in laboratory devices, since the waves observed in laboratory devices are certainly not bounded in one dimension. The objective of the present investigation is to study the nonlinear propagation of solitary and shock waves in dissipative EPI quantum plasma in a two-dimensional planar geometry. The kinematic viscosities amongst the constituent plasma particles give rise to the dissipative term in the nonlinear evolution equation. The paper is organized in the following way. In Sec. 2, the basic set of dynamic equations of our theoretical model for QIAWs are presented by using QHD model. In Sec. 3, the derivation of KPB equation is given to study the small amplitude solitary and shock structures. Section 4 deals with the discussion and solution of KPB equation, while Sec. 5 is kept for conclusion.

We consider a three component unmagnetized quantum plasma which is an admixture of degenerate electrons, degenerate positrons and singly charged non degenerate ions. The phase velocity of the wave is assumed to be v_Fi ≪ ω/k ≪ v_Fe, v_Fp (v_Fi, v_Fe, and v_Fp are the ion, electron, and positron Fermi velocities, respectively). We, therefore, ignore the quantum pressure and Bohm potential contributions of ions. Also, we have assumed that the positron annihilation time is larger than the inverse of the characteristic frequency of the IAWs. The basic equations governing the nonlinear dynamics of the IAWs in quantum plasma are given in dimensionless variables as follows^[33,37]

In Eqs. (1)–(6), n_e(i) is the electron (ion) number density normalized by their equilibrium value n_e0(i0), (u, v) are the ion velocity components normalized by the ion acoustic speed c_s = (k_BT_Fe/m_i)^1/2, with k_B denoting the Boltzmann constant, m_i is the ion mass, T_Fe = ℏ²(3π² n₀)^2/3/2 k_B m_e is the electron Fermi temperature and ℏ is the scaled Planck’s constant. Also, the nondimensional quantum parameter H = ℏω_pe/k_BT_Fe is denoting the ratio between the electron plasmon energy and the electron Fermi energy, where ω_pj = (n₀e²/ε₀ m_j)^1/2 is the plasma frequency for the j-th particle. Moreover, ϕ is the electrostatic potential normalized by k_B T_Fe/e. The space and time variables are respectively normalized by c_s/ω_pi and the inverse of ω_pi. , in which μ is the ion kinematic viscosity. At equilibrium the charge neutrality condition reads μ = n_e0/n_i0 = 1 + p, p = n_p0/n_i0, and δ = T_Fp/T_Fe = (1 − 1/μ)^2/3. We assume that the ions are cold, and electrons and positrons obey the following pressure law:^[38]
where with j = e, p is the particle Fermi thermal speed.

To investigate the nonlinear propagation of the IAWs in quantum plasma, we shall employ the reductive perturbation method.^[37,39] According to this method, we choose the stretching space time coordinates as
where ϵ is a small parameter measuring the weakness of perturbations of the wave amplitude (0 < ϵ < 1) and V is the normalized (by c_s) wave phase speed. We can expand the dynamical variables as



where we have considered the perturbations for the transverse velocity component v as higher-order effects, i.e., weaker than those of the longitudinal component u. This is due to the fact that since the wave propagation is assumed to propagate along the x-axis or ξ direction in the moving frame of reference, the effects of dispersion due to separation of electron and ion charges and the quantum tunneling associated with the Bohm potential on QIAWs will appear only in the ξ direction, i.e., along the direction of the ion velocity component u. Also, we assume that the effect of the viscosity is small and the constant η is the same for both the velocity components u and v, so we may set η = ϵ^1/2η₀, η₀ is O(1). Now substituting the expansions into the basic equations and collecting the terms in different powers of ϵ, we obtain in the lowest order , , , u⁽¹⁾ = (1/V) ϕ⁽¹⁾, v⁽¹⁾ = (1/V) ϕ⁽¹⁾, and .

From the next order of ϵ, we obtain the following equations

Finally, eliminating the second order quantities from Eqs. (13)–(18) and substituting the expressions for the first-order quantities we obtain the KPB equation as
where ϕ = ϕ⁽¹⁾, and

In conclusion, we have investigated the existence of compressive and rarefactive IASWs in a dense quantum plasma consisting of the electrons, positrons, and ions. It is shown that the evolution of two-dimensional nonlinear waves is governed by the KPB equation. Both the dissipative (due to kinematic viscosity) and dispersive (due to Bohm potential) effects are taken into consideration for the formation of QIA shock and soliton structures. The small amplitude compressive and rarefactive QIA shock and solitonic structures are obtained analytically and numerically from the KPB equation. It is observed that the kinematic viscosity, positron concentration, and Bohm potential affect the propagation characteristics of nonlinear QIA waves. Numerical simulation reveals that the transition from oscillatory to monotonic shocks occurs when the value of kinematic viscosity parameter increases. It is found that the energy of solitons is decreased when the positron concentrations are increased. The present investigation may be beneficial to understand the propagation of the nonlinear IA solitary waves and shocks in degenerate plasmas such as those in compact astrophysical objects and in laboratory plasmas.