Physics Letters A 372 (2008) 4279–4282 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Planar and non-planar ion acoustic shock waves in electron–positron–ion plasmas Waqas Masood a,∗ , Nusrat Jehan b , Arshad M. Mirza b , P.H. Sakanaka c a Theoretical Plasma Physics Division, PINSTECH, P.O. Nilore, Islamabad, Pakistan b Theoretical Plasma Physics Group, Physics Department, Quaid-i-Azam University, Islamabad 45320, Pakistan c Instituto de Fisica “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP, Brazil article info abstract Article history: Received 4 December 2007 Received in revised form 15 February 2008 Accepted 31 March 2008 Available online 7 April 2008 Communicated by F. Porcelli PACS: 52.27.Ep 52.35.Fp 52.35.Tc Keywords: Shock waves e–p–i plasma KdVB Ion acoustic shock waves (IASW’s) are studied in an unmagnetized plasma consisting of electrons, positrons and adiabatically hot positive ions. This is done by deriving the Kortweg–deVries–Burger (KdVB) equation under the small amplitude perturbation expansion method. The dissipation is introduced by tak- ing into account the kinematic viscosity among the plasma constituents. It is found that the strength of ion acoustic shock wave is maximum for spherical, intermediate for cylindrical, and minimum for planar geometry. It is observed that the positron concentration, ratio of ion to electron temperature, and the plasma kinematic viscosity significantly modifies the shock structure. Finally, it is found that the tempo- ral evolution of the non-planar IASW’s is quite different by comparison with the planar geometry. The relevance of the present study with regard to the dense astrophysical environments is also pointed out.  2008 Elsevier B.V. All rights reserved. 1. Introduction It has been observed that the electron–positron plasmas behave differently as opposed to typical electron–ion (e–i) plasmas [1,2]. An interesting feature of electron–positron (e–p) plasma by com- parison with the usual electron–ion plasma is the same mass and equal magnitude of charge of the constituents of an e–p plasma. Electron–positron plasmas have been observed in active galactic nuclei [3], in pulsar magnetospheres [4], in the polar regions of neutron stars [5], as well as in the intense laser fields [6]. Elec- tron positron plasma is also believed to exist in the early uni- verse [7] as well as at the centre of our own galaxy [8]. Since in many astrophysical environments there exits a small number of ions along-with the electrons and positrons, therefore, it is im- portant to study linear and non-linear behavior of plasma waves in electron–positron–ion (e–p–i) plasmas. A lot of research has been carried out to study the e–p and e–p–i plasmas in the past few years [9–13]. For instance, Nejoh [9] investigated the effect of ion temperature on the large amplitude ion-acoustic waves in e–p–i plasma and observed that the ion temperature decreases the amplitude and increases the maximum Mach number of ion acoustic wave. Mushtaq et al. [11] studied the effect of positron concentration on the non-linear propagation of two dimensional * Corresponding author. Tel.: +92 322 4218922. E-mail address: waqasmas@gmail.com (W. Masood). magnetosonic waves and found that the solitary waves in e–p–i plasma behave quite differently than that of ordinary e–i plasma. In particular, they found that an increase in positron concentration increases the amplitude of slow magnetosonic solitary waves while the opposite was observed for fast magnetosonic solitons. Linear and non-linear propagation of ion acoustic waves in a non-planar geometry has also been investigated [14,15]. It is well known that in a non-linear dispersive media, shock- like solutions are formed. This happens due to the balance be- tween the non-linearity (causing wave steepening) and dissipa- tion (e.g., caused by viscosity, collisions, wave particle interaction, etc.). However, when a medium has both dispersive and dissipa- tive properties, the propagation of small amplitude perturbations can then be adequately described by Korteweg–deVries–Burgers (KdVB) equation. The dissipative Burger term in the non-linear KdVB equation arises by taking into account the kinematic vis- cosity among the plasma constituents [14–16]. When the wave braking due to non-linearity is balanced by the combined effect of dispersion and dissipation, a monotonic or oscillatory dispersive shock wave is generated in a plasma [16,17]. In this Letter, we study the non-linear propagation of ion- acoustic shock waves (IASW’s) in a plasma system comprising of electrons, positrons, and ions, respectively in a non-planar (cylin- drical/spherical) geometry. The Letter is organized in the following manner. In Section 2, we write down the basic set of equations for the system under consideration and also derive the KdVB equa- tion using small amplitude perturbation expansion method. In Sec- 0375-9601/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.03.058