Nonlinear waves in bipolar complex viscous astroclouds

Abstract

A theoretical evolutionary model to analyze the dynamics of strongly nonlinear waves in inhomogeneous complex astrophysical viscous clouds on the gravito-electrostatic scales of space and time is procedurally set up. It compositionally consists of warm lighter electrons and ions (Boltzmanian); and cold massive bi-polar dust grains (inertial fluids) alongside vigorous neutral dynamics in quasi-neutral hydrodynamic equilibrium. Application of the Sagdeev pseudo-potential method reduces the inter-coupled structure equations into a pair of intermixed forced Korteweg-de Vries-Burgers ($f$-KdVB) equations. The force-terms are self-consistently sourced by inhomogeneous gravito-electrostatic interplay. A numerical illustrative shape-analysis based on judicious astronomical parametric platform shows the electrostatic waves evolving as compressive dispersive shock-like eigen-modes. A unique transition from quasi-monotonic to non-monotonic oscillatory compressive shock-like patterns is found to exist. In contrast, the self-gravitational and effective perturbations grow purely as non-monotonic compressive oscillatory shock-like structures with no such transitory features. It is seen that the referral frame velocity acts as amplitude-reducing agent (stabilizing source) for the electrostatic fluctuations solely. A comparison in the prognostic light of various earlier satellite-based observations and in-situ measurements is presented. The paper ends up with synoptic highlights on the main implications and non-trivial applications in the interstellar space and cosmic plasma environments leading to bounded structure formation.

Appendices

Appendix A: Coefficients of the electrostatic $f$-KdVB equation

The involved coefficients in the electrostatic $f$-KdVB equation (Eq. (23)) are defined as follows

$$\begin{aligned} A_{1}& = \bigl[ ( n_{e0} - n_{i0} ) + \bigl\{ 4 ( 3\alpha_{2} )^{1 / 2}Z_{ -}^{3}n_{ - 0} \bigr\} \mu^{ - 5} \\ &\quad{} - \bigl\{ 4 \bigl( 3\alpha_{1} \delta_{ -, +}^{5} \bigr)^{1 / 2}Z_{ +}^{3}n_{ + 0} \bigr\} \mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + Z_{ -}^{2}n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6\alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + Z_{ +}^{2}n_{ + 0} ( 3 \alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2} \delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{2} &= \biggl[ \frac{1}{2} \bigl( Z_{ -}^{2}n_{ - 0} \kappa_{ -} \bigr)\mu^{ - 3} \bigl\{ 2 - 3\alpha_{2} \mu^{ - 2} \bigr\} \\ &\quad{} - \frac{1}{2} \bigl( Z_{ +}^{2}n_{0 +} \kappa_{ +} \bigr)\mu^{ - 3} \\ &\quad{}\times \bigl\{ - 2 - 6 \alpha_{1}\delta_{ -, +}^{2}\mu^{ - 2} + 9 \alpha_{1}\delta_{ -, +} \mu^{ - 2} \bigr\} \biggr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + Z_{ -}^{2}n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6 \alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + Z_{ +}^{2}n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2} \delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{3}& = - ( \rho_{0}Gm_{ -} ) e^{ - 2} \biggl[ ( n_{e0} + n_{i0} ) + Z_{ -}^{2}n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6 \alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + Z_{ +}^{2}n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{1} )^{ - 1}\mu^{ - 1} - \frac{3}{2} \delta_{ -, +} \mu^{ - 3} - 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{4} &= \biggl[ Z_{ -} n_{ - 0} ( 3 \alpha_{2} )^{1 / 2} \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6 \alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + Z_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{1}\delta_{ -, +} )^{ - 1} \mu^{ - 1} + \frac{3}{2}\mu^{ - 3} - 6\alpha_{1} \delta_{ -, +} \mu^{ - 5} \biggr\} \biggr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + Z_{ -}^{2}n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6 \alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + Z_{ +}^{2}n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2} \delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{5}& = \bigl[ 4 ( 3\alpha_{2} )^{1 / 2}Z_{ -}^{2} n_{ - 0} \mu^{ - 5} + 4 \bigl( 3\alpha_{1} \delta_{ -, +}^{3} \bigr)^{1 / 2}Z_{ +}^{2} n_{ + 0}\mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + ( 3\alpha_{2} )^{1 / 2}Z_{ -}^{2}n_{ - 0} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6\alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2}Z_{ +}^{2}n_{ + 0} \\ &\quad{}\times \biggl\{ ( 3\alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{6} &= \bigl[ 4 ( 3\alpha_{2} )^{1 / 2}Z_{ -}^{2} n_{ - 0}\mu^{ - 5} + 4 \bigl( 3\alpha_{1} \delta_{ -, +}^{3} \bigr)^{1 / 2}Z_{ +}^{2} n_{ + 0}\mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + Z_{ -}^{2}n_{ - 0} ( 3 \alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6 \alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + ( 3\alpha_{1} \delta_{ -, +} )^{1 / 2}Z_{ +}^{2}n_{ + 0} \\ &\quad{}\times \biggl\{ ( 3\alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \\ A_{7} &= \bigl[ - 4 ( 3\alpha_{2} )^{1 / 2}Z_{ -} n_{ - 0}\mu^{ - 5} + 4 ( 3\alpha_{1} \delta_{ -, +} )^{1 / 2}Z_{ +} n_{ + 0} \mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ ( n_{e0} + n_{i0} ) + ( 3 \alpha_{2} )^{1 / 2}Z_{ -}^{2}n_{ - 0} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + 6\alpha_{2} \mu^{ - 5} \biggr\} \\ &\quad{} + ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2}Z_{ +}^{2}n_{ + 0} \\ &\quad{}\times \biggl\{ ( 3\alpha_{1} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\delta_{ -, +} \mu^{ - 3} + 6\alpha_{1} \delta_{ -, +}^{2}\mu^{ - 5} \biggr\} \biggr]^{ - 1}, \end{aligned}$$

and

$$\begin{aligned} F_{E} ( \varPhi,\varPsi ) = A_{4}\frac{\partial \varPsi}{\partial \xi} + A_{5}\varPhi \frac{\partial \varPsi}{\partial \xi} + A_{6}\varPsi \frac{\partial \varPhi}{\partial \xi} + A_{7}\varPsi \frac{\partial \varPsi}{\partial \xi}. \end{aligned}$$

Appendix B: Coefficients of the self-gravitational $f$-KdVB equation

The involved coefficients in the self-gravitational $f$-KdVB equation (Eq. (31)) are given as

$$\begin{aligned} B_{1} &= \bigl[ - 4m_{ +} n_{ + 0} ( 3 \alpha_{1}\delta_{ -, +} )^{1 / 2}\mu^{ - 5} + 4m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \mu^{ - 5} \\ &\quad{} + 4m_{n}n_{n0} ( 3\alpha_{3} \delta_{ -,n} )^{1 / 2}\mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3\alpha_{1}\delta_{ -, +} )^{ - 1}\mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6 \alpha_{1}\delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2} \mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6 \alpha_{3}\delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{2} &= \biggl[ \frac{1}{2} ( m_{ +} n_{ + 0}\kappa_{ +} )\mu^{ - 3} \bigl\{ - 2 - 21 ( \alpha_{1}\delta_{ -, +} )\mu^{ - 2} \bigr\} \\ &\quad{} + \frac{1}{2} ( m_{ -} n_{ - 0}\kappa_{ -} ) \mu^{ - 3} \bigl\{ - 2 + 3\alpha_{2}\mu^{ - 2} \bigr\} \\ &\quad{}\times \frac{1}{2} ( m_{n}n_{n0}\kappa_{n} ) \mu^{ - 3} \bigl\{ - 2 + 3 ( \alpha_{3}\delta_{ -,n} ) \mu^{ - 2} \bigr\} \biggr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3 \alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3 \alpha_{1}\delta_{ -, +} )^{ - 1}\mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6\alpha_{1}\delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3 \alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + ( 6 \alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6\alpha_{3} \delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{3} &= - \rho_{0} \biggl[ m_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3\alpha_{1}\delta_{ -, +} )^{ - 1} \mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6\alpha_{1} \delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2} \mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3} \delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3} \delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6\alpha_{3} \delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{4} &= \biggl[ m_{ +} n_{ + 0} ( 3 \alpha_{1}\delta_{ -, +} )^{1 / 2}Z_{ +} \\ &\quad{}\times \biggl\{ ( 3\alpha_{1} )^{ - 1}\mu^{ - 1} - \frac{5}{2} \delta_{ -, +} \mu^{ - 3} - 6 \bigl( \alpha_{1} \delta_{ -, +}^{2} \bigr)\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2}Z_{ -} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \biggr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3 \alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3 \alpha_{1}\delta_{ -, +} )^{ - 1}\mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6\alpha_{1}\delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3 \alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + ( 6 \alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6 \alpha_{3}\delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{5} &= \bigl[ 4m_{ +} n_{ + 0} \bigl( 3 \alpha_{1}\delta_{ -, +}^{5} \bigr)^{1 / 2}Z_{ +}^{2} \mu^{ - 5} \\ &\quad{} - 4m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2}Z_{ -}^{2}\mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3\alpha_{1}\delta_{ -, +} )^{ - 1}\mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6 \alpha_{1}\delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2} \mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6 \alpha_{3}\delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{6} &= \bigl[ 4m_{ +} n_{ + 0} \bigl( 3 \alpha_{1}\delta_{ -, +}^{3} \bigr)^{1 / 2}Z_{ +} \mu^{ - 5} \\ &\quad{} + 4m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2}Z_{ -} \mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3\alpha_{1}\delta_{ -, +} )^{ - 1} \mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6\alpha_{1} \delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2} \mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6 \alpha_{3}\delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \\ B_{7} &= \bigl[ 4m_{ +} n_{ + 0} \bigl( 3 \alpha_{1}\delta_{ -, +}^{3} \bigr)^{1 / 2}Z_{ +} \mu^{ - 5} \\ &\quad{} + 4m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2}Z_{ -} \mu^{ - 5} \bigr] \\ &\quad{}\times \biggl[ m_{ +} n_{ + 0} ( 3\alpha_{1}\delta_{ -, +} )^{1 / 2} \\ &\quad{}\times \biggl\{ - ( 3\alpha_{1}\delta_{ -, +} )^{ - 1} \mu^{ - 1} + \frac{5}{2}\mu^{ - 3} + ( 6\alpha_{1} \delta_{ -, +} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{ -} n_{ - 0} ( 3\alpha_{2} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3 \alpha_{2} )^{1 / 2}\mu^{ - 1} + \frac{3}{2} \mu^{ - 3} + ( 6\alpha_{2} )\mu^{ - 5} \biggr\} \\ &\quad{} - m_{n}n_{n0} ( 3\alpha_{3}\delta_{ -,n} )^{1 / 2} \\ &\quad{}\times \biggl\{ ( 3\alpha_{3}\delta_{ -,n} )^{ - 1}\mu^{ - 1} + \frac{3}{2}\mu^{ - 3} + \bigl( 6 \alpha_{3}\delta_{ -,n}\mu^{ - 5} \bigr) \biggr\} \biggr]^{ - 1}, \end{aligned}$$

and

$$\begin{aligned} F_{G} ( \varPhi, \varPsi ) = B_{4}\frac{\partial \varPhi}{\partial \xi} + B_{5}\varPhi \frac{\partial \varPhi}{\partial \xi} + B_{6}\varPsi \frac{\partial \varPhi}{\partial \xi} + B_{7}\varPhi \frac{\partial \varPsi}{\partial \xi}. \end{aligned}$$