The invariant ion-acoustic waves in the plasma

The space plasmas have been found empirically to be separated into those residing far from the classical thermal equilibrium and those residing near equilibrium. The modern formalism of the kappa distributions explains this distinction under the value of the kappa index, the intensive parameter that characterizes thermodynamics together with temperature. Recent studies have suggested that by defining an invariant kappa index as the zero dimensionality spectral index, [Formula: see text], which is independent of the dimensionality, the degrees of freedom, or the numbers of particles, one may separately consider the physical and thermodynamic feature of the kappa index in space plasmas by utilizing [Formula: see text]. This study extends the mentioned idea to the ion-acoustic waves (IAWs) in the astrophysical plasmas in order to deriving an invariant formalism for the IAWs including the pure thermodynamic features of the background particles. This paper is based on the kinetic theory formalism and the hydrodynamic fluid description for extracting the characteristics of the invariant IAWs. Relying on the Vlasov-Poisson equations, considering a low-frequency band for the weakly damped ion oscillations, we have derived the most generalized formalism of the ion-sound speed in space plasmas in terms of the extended polytropic indices of the plasma species, [Formula: see text], and also the generalized formalism of Landau damping for the invariant IAWs in terms of [Formula: see text], wavelength, and temperatures of the plasma species. In the hydrodynamic description, we have normalized the fluid parameters in terms of the generalized quantities, including the extended formulations of the ion-sound speed and Debye length. Then, by using the perturbation expansion in linear and nonlinear regimes, we may find some other issues in the formalism of the invariant IAWs, such as the effect of the perturbed potential degrees of freedom, [Formula: see text], the isothermal/extended phase speed of the IAWs, and the combined effects of the wave steepening and dispersion of ion waves. We have also derived a generalized KdV equation and its solitary wave solutions in an invariant formalism. Based on the empirical evidences in space plasmas, the far-equilibrium plasmas are characterized by [Formula: see text] ([Formula: see text]), while the near-equilibrium plasmas are labeled with [Formula: see text] ([Formula: see text]). We have numerically analyzed our solutions from the anti-equilibrium states at [Formula: see text] ([Formula: see text]) towards the equilibrium states at [Formula: see text] ([Formula: see text]). Our theoretical study provides strong evidence, for the first time, about the distinction of plasmas under the value of the kappa index. Our analysis confirms the distinction of the involved IAWs diagrams in the two mentioned regions, where the transition from far-equilibrium states to the near-equilibrium states may occur in the vicinity [Formula: see text] ([Formula: see text]), denoting the escape state of the evolution.