Electron-acoustic solitons and double layers in a plasma system with Tsallis distributed hot electrons

The electron acoustic (EA) wave is a high frequency wave that occurs in plasma with two distinct populations of electrons, referred as “cool” and “hot” [1]. Electron acoustic wave may also exist in electron-ion plasmas with ions hotter than electrons [2]. It is basically an electrostatic wave in which the inertia is provided by the cool electrons and the restoring force comes from the hot electrons. The ions play the role of a neutralizing background, i.e., the ion dynamics does not influence the EA waves because the EA wave frequency is much higher than the ion plasma frequency.  The EA wave phase speed must be intermediate between the cold and hot electron thermal speeds such that the wave avoids damping by both the cold and the hot electron species. Electron acoustic waves often occur in laboratory plasmas [3-5] and space plasmas, e.g., in the Earth’s bow shock [6-8], in the auroral magnetosphere [9,10] and in geomagnetic tail [11]. In the case of Earth’s bow shock, heliospheric termination shock and planetary and neutron star magnetospheres, in addition to two electron populations, the presence of magnetic field-aligned electron beam is considered to be the  main energy source for the excitation of this mode. 

The electron and ion distributions play crucial role in characterizing the physics of nonlinear wave structures by increasing richness and variety of the wave motion in plasma. Also, they significantly influence the conditions required for the formation of solitons and double layers. Moreover, it is also known that electrons and ion distribution can be significantly modified in the presence of large amplitude waves. Over past few years, new statistical approach nonextensive statistics or Tsallis statistics has attracted much attention [12]. A suitable nonextensive generalization of the Boltzmann-Gibbs-Shannon (BGS) entropy for statistical equilibrium was first recognized by Renyi [13] and subsequently proposed by Tsallis [12] suitably extending the standard additivity of the entropies to the nonlinear, nonextensive case where one particular parameter, the entropic index q, characterizes the degree of nonextensivity of the considered system. The Maxwellian distribution in Boltzmann-Gibbs (BG) statistics is believed valid universally for the macroscopic ergodic equilibrium systems. But this concept fails to explain some phenomena which have complex behaviours such as long range interactions such as plasma and gravitational systems. Spacecraft measurements of plasma velocity distributions, both in the solar wind and in planetary magnetospheres and magnetosheaths, have revealed that non-Maxwellian distributions are quite common. Tsallis distribution is thought to be useful generalization of BG statistics and to be appropriate for the statistical description of the long-range interaction systems, characterizing the non-equilibrium stationary state [14-16]. In Tsallis distribution, q is the nonextensive parameter and for q ≠ 1, it gives power law distribution functions. For q < 1, high energy states are more probable than in the extensive case. The distribution corresponds to power law behaviour for large energy. On the other hand, for q > 1 high energy states are less probable than in the extensive case, there is a cutoff beyond which no states exist. Boltzmann-Gibbs entropy is obtained from Tsallis entropy Sq if the parameter q →1 [12].