امنا عائشة ملكة العفاف رضي االله عنها

This research aims to explore various features of Syyidah some highlights paper This. Personage noble s'(ر الله عا) ishah‘A’ distinguished aspects of her biographical account, her extraordinary intelligence, eminent rank amongst other wives of the Prophet Muḥammad (r), excellence over women of the world and being mentioned in the Qur’ān for her praiseworthy character. The scholarly areas, in which, she outshone others in Islamic intellectual heritage, include the Quranic exegesis, Ḥadīth narratives, jurisprudence, scholastic reasoning: analysis of jurisprudential methodology and principles of deduction of rulings and opinions. She adopted multifarious modes in this regards including explicit, direct, generalized, inductive and analogous approaches in the Islamic intellectual scholarship. By acknowledged was (ر الله عا) ishah‘A of grandeur The the companions of the Prophet, her scholarly point of view in conciliation of the problems of the Qur'ān and Ḥadīth made her stand out amongst all. She was a keen researcher, peaceful preacher and the greatest scholar of Islām of all times. Her knowledge and command over religious and social matters made her one of the most reliable experts of the Qur'ān, Ḥadīth and Fiqh. Such was her stature as the archangel Jibrā’īl presented his salutation to her.

Nonlinear Perturbation in Complex Plasma

In this chapter, we recapitulate the main outcomes of the whole thesis and give a glimpse of the future possible extension of the present work. The present thesis investigated a low frequency linear and nonlinear electrostatic waves in complex plasma. We have studied di¤erent plasma mediums through which these waves propagates. We have derived di¤erent integrable nonlinear evolution equations by using Sagdeev potential approach and reductive perturbation technique. The solitary wave solutions exists due to these nonlinear evolution equations. We have considered pair ion, pair ion-electrons and ambiplasma to studied the characteristics of di¤erent solitonic nature. The presence of third or more species in the two component plasma such as electronion or pair ion plasmas leads to some new interesting phenomena in complex plasma. The wider and broader ranges of frequencies wave spectrum are available in complex plasmas /multi-component plasmas than in the usual two component plasmas. Therefore, the researchers has taken an interest to study the linear and nonlinear propagation of waves in complex plasma during the last two decades. To understand the nonlinear behavior of such complex plasmas or multi component plasmas, the nonlinear structures such as vortices [118], solitons [119, 120, 121], shocks and envelope solitons [119, 121] have been studied by many authors. The mathematics of complex plasmas becomes little complicated due to the addition of multi species in a usual two component plasmas. Experimentally in laboratories the pair plasma consisting of electrons and positrons have been generated [10, 11, 12, 13, 14, 15, 16] by injecting a low-energy electron beam into a positron plasma. Once a pair plasma is produced then the identi…cation of the collective modes are di¢ cult in electron-positron plasmas due to the short annihilation time as compared to the plasma period and because of low plasma density. The …rst experimental evidence of pair ion plasma was given by Oohara et.al in 2005 [8]. He experimentally observed three di¤erent types of electrostatic modes along the magnetic …eld lines in a pair-ion fullerene plasma. According to his observation the …rst mode is an ion acoustic wave (IAW), the second mode is an ion plasma wave (IPW) and he labeled the third mode as an intermediate frequency wave (IFW). In 2007 Oohara et.al revised his experiment [9] by introducing fullerene into potassium plasma and generating alkali fullerene plasma (K+, e, C60 ). In this process electrons interact with fullerenes causing the production of positive and negative ions. In the same year H. Saleem [17] claimed that the research in pure pair ion plasma has become ambiguite because the experimental results are not in agreement with the existence theories. The author claimed that the ion acoustic waves measuring frequency is much larger than the theoretical values i.e. the ion acoustic frequency larger value indicates that the plasma contains signi…cant concentration of electrons. He also pointed out that the ion acoustic waves in pair-ion-electron plasma is reduced. Dusty plasma is generally an electron-ion plasma with additional charged components of micron or submicron sized particulates. The complexity of the system increases due to this extra component of macro particles. Due to this reason dusty plasma is referred to as complex plasma. In chapter 2, we have investigated the e¤ects of pair ion temperature, background dust density concentration, and nonlinear resonance e¤ect such as trapping of electrons on the arbitrary amplitude of dust ion acoustic soliton in PI-electron plasmas. The electrons are assumed to follow the vortex like Schamel distribution. The counter balance between nonlinearity induced by ion ‡uid convection and electrons trapping and dispersion due to temperature e¤ects admits solitary waves. We have derived an energy integral equation involving Sagdeev potential by using pseudopotential approach. We 89, C60 ). In this process electrons interact with fullerenes causing the production of positive and negative ions. In the same year H. Saleem [17] claimed that the research in pure pair ion plasma has become ambiguite because the experimental results are not in agreement with the existence theories. The author claimed that the ion acoustic waves measuring frequency is much larger than the theoretical values i.e. the ion acoustic frequency larger value indicates that the plasma contains signi…cant concentration of electrons. He also pointed out that the ion acoustic waves in pair-ion-electron plasma is reduced. Dusty plasma is generally an electron-ion plasma with additional charged components of micron or submicron sized particulates. The complexity of the system increases due to this extra component of macro particles. Due to this reason dusty plasma is referred to as complex plasma. In chapter 2, we have investigated the e¤ects of pair ion temperature, background dust density concentration, and nonlinear resonance e¤ect such as trapping of electrons on the arbitrary amplitude of dust ion acoustic soliton in PI-electron plasmas. The electrons are assumed to follow the vortex like Schamel distribution. The counter balance between nonlinearity induced by ion ‡uid convection and electrons trapping and dispersion due to temperature e¤ects admits solitary waves. We have derived an energy integral equation involving Sagdeev potential by using pseudopotential approach. We have shown for negative ions case that for all the model supports both positive and negative potential solitons, where the Mach number for positive (negative) soliton is limited from the above condition at which positive ion density becomes complex. It has been shown that for given plasma parameters our plasma model exhibits compressive solitons. By using the travelling wave assumption(xMt) along with its general solution we have derived a generalized S-KdV equation with mixed orders of nonlinearity from energy integral equation. We have also derived a solution involving lower and higher order nonlinearity. It is found that the above mentioned e¤ects changes the solitary wave pro…le quite signi…cantly. The presence of background dust gives strong and enhancing e¤ects on the existence domains of positive solitons, while the PI temperatures have an e¤ects which weekend the solitons amplitude. Here we are not discussing the existence domain and the criterion for rarefractive or negative solitons. It is found that due to the contributions of electrons the characteristics of nonlinear dust ion acoustic wave vary substantially. The solitary waves width and amplitude are di¤erent for some non-isothermal electrons than the case when they all are isothermal. This means that the transformation from non-isothermal to isothermal phase is nonuniform. It is also found that for isothermal electrons the amplitude of soliton is larger than that for trapped electron distributions, which means some kind of inertia in the propagation of nonlinear electrostatic perturbation is induced by the trapped electrons. From our numerical analysis, we have also deduced that the temperature e¤ects due to both types of ions are destructive and reduces the speed of the soliton. These results should be useful in understanding the wave phenomena and associated nonlinear electrostatic structures in doped pair-ion-electron plasmas, which are enriched with an extra massive charged component i.e. defect or dust, which may occur in space and laboratory level. In chapter 3, we have studied the e¤ects of streaming motion of ions on linear and nonlinear properties of unmagnetized, collisionless plasma by using ‡uid model. We have also studied how an increase in free energy of the plasma which is given by the stream of beams of ions leads to perturbations. Perturbation waves are growing due to these beams and instabilities called streaming instabilities occur. We have investi- gated the positive and negative ions streaming in pair ion plasma in the presence of non-isothermal electrons. We have investigated the growth rates from these streaming instabilities. The source of free energy is the motion of positive ions beam, if it is not supported continuously the instability quenches itself after the beam has slowed down to certain threshold which is determined by the background particles. We have discussed the growth rates as a function of wave vector, and the instabilities enhances as the growth increases otherwise damping of waves occur. We have also shown that the growth rate decreases with nonthermality . We have also investigated the e¤ects of the streaming of ions (v ), nonthermality and electron density concentration ( ) on the behavior of ion acoustic solitary waves in PI-electrons plasma. The electrons are assumed to follow the distribution with the population of fast particles [27, 28]. The counter balance between nonlinearity induced due to streaming motion and dispersion due to temperature e¤ects admits solitary waves. We have derived an energy integral equation involving Sagdeev potential by using pseudopotential approach. We have shown that the solitary wave structure is signi…cantly a¤ected by the streaming term in energy integral equation. We have also observed that the positive and negative ions streaming are acting opposite to each other on soliton pro…le. Also, we have observed that the nonthermality of electrons is de-energizing the soliton. In this investigation, our analysis has shown a compressive soliton, where the positive ions play a dominating role in its formation. These results should be useful in understanding the wave phenomena and associated nonlinear electrostatic structures in laboratory pair-ion-electron plasmas. In chapter 4, We have studied baryonic acoustic solitary waves in ambiplasma having baryons (positive and negative ions) and nonextensive leptons (electrons and positrons). Our results reveal that localized baryonic acoustic solitary structures, the features of which depends on leptons nonextensivity, concentration and temperature can exist in such multi component plasma. From our analysis we have shown that the present plasma model supports both compressive and rarefractive baryon acoustic (BA) solitons. We have shown from our results that the spatial patterns of BA solitons are signi…cantly modi…ed by the leptons nonextensive e¤ects. We have observed that in the ranges of 1 < q < 0:25 and 0 < q < 0:25 rarefractive solitons are formed while for q & 0:25 compressive solitons are observed. We have also observed that both the amplitude and width of the solitons increases for decreasing values of entropic index q, because of the presence of the high energetic leptons in the tails of the distribution function which energize soliton structures. Further we have investigated that increasing the number of leptons in the presence of nonextensivity diminish both the width and amplitude of solitons. We have also been observed that the nonextensive behavior of leptons a¤ected both the compressive and rarefractive solitons quite signi…cantly. Our present results may help to understand the salient features of baryonic acoustic waves in multi-component plasmas which may occur in astronomical, space and laboratory plasmas such as capsule implosion, shock tube, star formation and supernova explosion etc." xml:lang="en_US