Propagation and domains of the invariant ion-acoustic solitons in the plasmas

Abstract

Observational pieces of evidence of space probes Voyagers and IBEX to study the Sun’s heliosphere, the outer Solar system, and interstellar space beyond the Sun’s heliosphere indicate that perturbations in some regions may occur in situations out of the pure thermal equilibrium, e.g., in the outer heliosphere regions, the inner heliosheath regions, and in heliopause regions. The data analysis extracted from these probes also shows that the transitions between the near/far-equilibrium states may happen in some areas, e.g., the slow solar wind e-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {e^{-}}$$\\end{document} (Ulysses) plasmas and the fast solar wind He+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {He^{+}}$$\\end{document} plasmas. The modern formalism of the kappa distributions explains the distinction between the near/far-equilibrium states under the value of the kappa index, as an intensive thermodynamic parameter. For providing more clarity to this formalism, an invariant kappa index as the zero dimensionality spectral index κ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _{0}$$\\end{document} is determined to consider the physical and thermodynamic feature of the kappa index in space plasmas, where it is independent of the dimensionality, the degrees of freedom, or the numbers of particles. Recently, this idea has extended for studying the invariant ion-acoustic waves (IAWs) in the astrophysical plasmas. Then, we discussed the pure thermodynamic features of the background particles. By utilizing κ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa _{0}$$\\end{document}, we found the distinction of the involved IAWs diagrams in the near/far-equilibrium states and also the transition from far-equilibrium states to the near-equilibrium states in the vicinity of a critical spectral/polytropic index. This paper extends the invariant formalism of the ion waves to the propagation features and structure of the nonlinear perturbations in the outer Solar system and interstellar space beyond the Sun’s heliosphere relevant to the mentioned observational evidences. We study the propagation and allowed domains of the invariant ion-acoustic solitary waves (IASWs) by considering the advanced aspects of the kappa distribution formalism. The central parameters of our formalism for analysis of the allowed domains of the solitary waves and shocks are the polytropic (adiabatic) index associated with the kappa distributed electrons, γe\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _{e}$$\\end{document}, and a well-defined and extended Mach number Mγe\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {M}}_{\\gamma _{e}}$$\\end{document} (the fractional wave speed to the generalized ion-sound speed). We have used Sagdeev’s methodology for deriving the energy-integral equation of the IASWs, which describes the formation of the possible potential wells (pseudo-potentials) for trapping the arbitrary amplitude solitons (pseudo-particles). The analysis of the Mach number domains is developed by extracting (ϕ,Mγe)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\phi ,{\\mathcal {M}}_{\\gamma _{e}})$$\\end{document} domains for the possibility of the solitary wave solutions in the plasma. We also show variation of the relevant (γe,Mγe)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma _{e},{\\mathcal {M}}_{\\gamma _{e}})$$\\end{document} domains. The formalism of the energy-integral equation and the domains of invariant IASWs has illustrated in two cases. At first, we show the general aspects of the problem by considering Ti≪Te\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{i} \\ll T_{e}$$\\end{document} (the cold ion plasma limit), and then we extend it to a warm plasma with finite-temperature ions.