Chapter 1

1 Introduction

Plasma is the popular state of the matter in the universe. We live in a bubble of essentially non-ionized gas in the midst of an otherwise ionized environment. 99% of the matter in the universe is in the plasma state in the form of an electrified gas with the atoms dissociated into positive ions and negative electrons. The outer layers of the Sun and stars in general are made up of matter in an ionized state and from these regions winds blow through interstellar space contributing, along with stellar radiation, to the ionized state of the interstellar gas [1]. The Earth and its lower atmosphere is an exception, forming a plasma-free oasis in a plasma universe. The upper atmosphere on the other hand, stretching into the ionosphere and beyond to the magnetosphere, is rich in plasma effects.

Our technical age is unthinkable without plasma. Plasma arc switches are used in the distribution of electric energy; high-pressure lamps illuminate our streets and serve as light sources in modern data projectors; fluorescent tubes light our offices; computer chips are etched with plasma technologies; plasma-assisted deposition processes result in flat computer screens and large-area solar cells. The future energy supply may benefit from electricity produced by controlled nuclear fusion.

From a scientific point of view, matter in the known universe is often classified in terms of four states: solid, liquid, gaseous, and plasma. This fourth state of matter in 1879 was first identified by Sir William Crookes [kr]. The collective behavior of the charged ionized gases was described by Lord Rayleigh[ra], in 1906, in the analysis of electron oscillation in Thomson model of atom. However, the term plasma was first used by Langmuir [2] in 1928 to describe the ionized regions in gas discharges. In the 1940’s Hannes Alfvén[al ] developed a theory of hydromagnetic waves (now called Alfvén waves) and proposed that these waves would be important in astrophysical plasmas. This theory was explained in 1950 in the classical book Cosmical Electrodynamics [book]. In the early 1950’s large-scale plasma physics based magnetic fusion energy research started simultaneously in the USA, Britain and the then Soviet Union. The development of high-powered lasers after 1960 opened up a new field of plasma physics.

After this many scientists studied the properties of the new matter state.

1.1 plasma definitions

. Any ionized gas with specific conditions can be called plasma. A useful definition according to [1] is

“A plasma is a quasineutral gas of charged and neutral particles which exhibits collective behavior”

For more simplification we must define the two expressions: quasineutral and collective behavior. Quasineutral means the plasma is neutral enough so that one can take the positive ion is equal to the density of electrons i.e.n_i=n_e=n, where n is a common density called the plasma density, but not so neutral that all the interesting electromagnetic forces vanish [1].

The most important distinction between a plasma and a normal gas is the fact that mutual Coulomb interactions between charged particles are important in the dynamics of a plasma and cannot be disregarded.

1.2 Debye shielding

Plasma has its ability to shield out electric potentials that are applied to it. To illustrate that : two charged balls connected to a battery inserted into plasma medium then a cloud of ions would surround the negative ball and a cloud of electrons would surround the positive ball. If the temperature is finite, the “edge” of the cloud then occurs at the radius where the potential energy is approximately equal to the thermal energy k_B T of the particles, and the shielding is not complete. Potentials of the order of (k_B T)⁄e can leak into the plasma and cause finite electric fields to exist there. Solving poisson equation for the electrostatic potential Φ(x) in a fully ionized plasma in one dimension;

ᵋ_0 (d^2 Φ)/〖d x〗^2 = -e(n_i-n_e)

Assuming that the ion density n_i is given by n and the electron density is given by n_e=n exp〖(eΦ⁄(kT_e ))〗 , it gives the potential as follow:

Φ=Φ_0 exp〖((-x)⁄λ_D )〗

as λ_D is the Debye length which is characteristic parameter in plasma physics [1].

1.3 Criteria for plasmas

Although any plasma can be defined as ionized gas, not all ionized plasma can be defined as plasma since the basic parameters of plasma conditions are significantly different from those of ordinary neutral gases, such as Debye length〖 λ〗_D, plasma frequency ω_pe , and the average number of electrons in the Debye radius N_D.

•The Debye length λ_D

It is defined as the characteristic distance at which deviations from quasi-neutrality can occur. It is calculated as follows:

λ_D=√((ɛ_0 kT_e)/(n e^2 )).

For a plasma, it is required that the Debye length is much less than the length L of the tube containing the plasma

λ_D<< L.

• The plasma density parameter〖 N〗_D

In general, the shielding of small disturbances of the electric field is mainly driven by the more mobile electric charges where

N_D=4/3 nπλ^3.

The number of electric charges inside the sphere with the radius of the Debye length must be large enough so that the statistical approach to shielding is applicable and the collective behavior applies.

N_D<< L

This can be defined as the plasma density parameter.

• Plasma frequency ω_pe

Another important plasma parameter is the plasma frequency. It can be calculated as follows:

ω_pe=√((n e^2)/(ɛ_0 m))

The plasma frequency is the frequency of collective electrostatic oscillations. It must be lower than the collision frequency; if the collision frequency rate were higher than the plasma frequency, the plasma would decay and does not exist. The plasma frequency parameter can be define as follows:

ω_pe≪1⁄τ

where τ is the mean time between collisions with neutral atoms and 1⁄τ is the collision frequency.

So there are three critical conditions for plasma. Once these conditions have been verified, the plasma can be established

λ≪L .

〖 N〗_D≪L.

ω_pe≪1⁄(τ ).

1.4 Plasma parameters

Three fundamental parameters characterize a plasma:

1. The particle density n (measured in particles per cubic meter),

2. The temperature T of each species (usually measured in eV),

3. The steady state magnetic field B (measured in Tesla).

A host of subsidiary parameters (e.g., Debye length, Larmor radius, plasma frequency, cyclotron frequency, thermal velocity) can be derived from these three fundamental parameters. Plasma applications cover an extremely wide range of n and T_e : n varies over 28 orders of magnitude from 〖10〗^6 to 〖10〗^34 m^(-3) , and k_B T can vary over seven orders from 0.1 to 〖10〗^6 e V. Figure 1.1 shows the ranges of temperature, electron density and Debye length for typical plasmas found in nature and in technological applications. Also the separation between the quantum and classical regimes is shown in Figure 1.1. Quantum effects need to be taken into account when the uncertainty in an electron’s position is comparable to the average distance to the nearest electron [4].

Figure 1.1 Range of temperature, electron density, and Debye length for typical plasma phenomena in nature and in technological applications. Only phenomena to the left of the quantum degeneracy line are considered plasmas and can be treated with formulations from classical physics.

1.5 Pair-ion plasma

Pair-ion plasma is consisting of positive and negative ions with the equal mass. It occurs in nature as the (H^+,O_2^-) and (H^+,H^- ) plasmas have been found in the D-region and F-region of the earth’s ionosphere and experimentally [6, 7].

Let us take fullerene pair ion as example. There are two methods to produce fullerene pair ion plasma in laboratory. First one is Pair-ion plasma generation using central electron Beam. The second is Pair-ion plasma generation using hollow electron beam.

1.5.1 Pair-ion plasma generation using central electron Beam.

A pair-ion plasma source using fullerenes requires dc discharge plasma source with a vacuum chamber of 15.7 cm in diameter and 260 cm in length, as shown schematically in Fig.1.2. A uniform magnetic field is applied (B=0.3 T) by solenoid coils and the background gas pressure is 2 x 〖10〗^(-4) Pa. A copper cylinder (10 cm in diameter and 30 cm in length) with two copper annuli is fixed in a cylindrical ceramic furnace and heated to 500 c^o. A tungsten spiral-wire cathode (1.2 cm in diameter) is installed at left side of the chamber in Fig1.2 and heated to 2000 c^o or more. There is a ground grid in front of the cathode and the distance between the cathode and the grid can be minutely adjusted, and is less than 1 cm.

The electron beam flows along magnetic field lines, passes through the cylinder, and terminates at an endplate. The copper cylinder and the annuli are grounded. Region (II) in the cylinder is a fullerene-ion production region. The cylinder and the ceramic furnace have a hole (3 cm in diameter) on the sidewall and an oven for fullerene sublimation is set there. A fullerene sample, which is commercially available 〖co〗_60 powder of 99.5% purity, is heated in the oven. Typical oven temperature under operating conditions ranges between 400 and 600c^o. The fullerene vapor produced by sublimation effuses through a 0.3-cmdiameter hole under molecular flow conditions. The electron beam crosses the fullerene vapor. Fullerene ions (〖co〗_60^+,〖co〗_60^- ) are produced by an electron-impact ionization.

The produced fullerene ions pass through the hole of the downstream-side annulus and enter the experimental region; Region (III). The plasma density in Region (III) depends on the electron beam energy. Fig.1.3 shows the characteristic of the positive-saturation current I_q of the Langmuir probe depending on the electron-beam energy. Here, I_q is considered to be in proportion to the positive-ion (plasma) density [8].

Fig 1.2 Schematic drawings of the experimental setup. An electron beam for ionization exists in the central region. Cathode consists of tungsten spiral-wire. Pair-ion plasma using fullerene is generated in the periphery region of the downstream.

Fig.1.3. Characteristics of fullerene discharge dependence on electron beam energy.

1.5.2 Pair-ion plasma generation using hollow electron Beam

A pair-ion plasma using a hollow electron beam is generated by the improvement of the pair-ion plasma source as aforementioned. The experimental setup is schematically shown in Fig.1.4. A copper annulus is installed at the downstream-side exit of a copper cylinder heated to 500 c^o. A ceramic cathode is radiatively heated from behind to approximately 1500c^o by a tungsten wire heater. Low-energy electrons (0.1 eV), thermionically emitted from the cathode, are accelerated by an electric field between the cathode and the ground grid (Region(I)), forming an electron beam with the energy of 0–150 eV, which is controlled by changing the cathode bias voltage Vk. A stainless-steel disk (6 cm in diameter) is installed on the grid; therefore, the shape of the electron beam becomes a hollow tube. The hollow electron beam flows along magnetic field lines and is terminated at the annulus. The fullerene vapor produced by sublimation is introduced from the side wall of the cylinder and crossed by the electron beam in Region (II). Positive fullerene ions (〖co〗_60^+) are produced by the electron-impact ionization, being accompanied by the simultaneous production of low-energy electrons (12 eV). These electrons attach to neutral fullerene molecules, and negative fullerene ions (〖co〗_60^-) are produced.

Fig1. 4. Schematic drawings of the experimental setup. Shape of an electron beam for ionization is hollow. Pure pair-ion plasma using fullerene is generated in the downstream.

1.6 Nonlinear Evolution Equations

1.6.1 Korteweg-de Vries (KdV) Equation

The Korteweg-de Vries (KdV) equation is the following nonlinear partial differential equation for U(X,T) :

∂U/∂T+A U ∂U/∂X+B (∂^2/(∂X^2 )) ∂U/∂X=0

Where X and T are independent variables and A and B are real, nonzero constants depending on the plasma parameters.This equation was first derived by Korteweg and de Vries (1895) for shallow water waves. In this equation two of the most important properties of a plasma nonlinearity and dispersion are represented. kdv is nonlinear through the convective term U ∂U/∂X , dispersive through the term(∂^2/(∂X^2 )) ∂U/∂X .Kdv equation is essential in solving a broad class of weakly nonlinear dispersive systems in plasma physics [12]

Modified Korteweg-de Vries (MKdV) Equation

To include the effect of higher order nonlinearity in an investigation, it is customary to assume the nonlinear coefficient A= 0 of the KdV equation to derive the MKdV equation with a different stretching.

Sometimes, higher order terms of the perturbative parameter in the expansion of the variables are incorporated to deduce modified equation or evolution equation

1.6.2 Zakharov-Kuznetsov(ZK) Equation

The standard form of Zakharov-Kuznetsov (ZK) equation is

∂U/∂T+U ∂U/∂X+(∂^2/(∂X^2 )+∂^2/(∂Y^2 )+∂^2/(∂Z^2 )) ∂U/∂X=0

The ZK equation describes the evolution of weakly nonlinear long waves in dispersive media in which the transverse coordinates are also taken into account for a strongly magnetized plasma. Zakharov and Kuznetsov [13] used this equation to study the behavior of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field[14].

The ZK equation is one of the better studied canonical three/two-dimensional extensions of the KdV equation. It is noted that for the nonlinear mode of the electrostatic wave (like ion acoustic wave) the ZK equation is valid for a magnetized and rotating plasma . Moreover,ZK-equation supports stable lump solitary waves. This makes the ZK equation a very attractive model equation for the study of vortex soliton in plasmas and fluid physics [35]

Chapter 2

Rogue wave behaviour in a pair-ion plasma

2.1 Introduction

In 2003 Oohara and Hatakeyama[15] have developed a novel method for generating pure pair-ion plasma. They used the fullerene as an ion source through the processes of hollow-electron-beam impact ionization, electron attachment, preferential radial diffusion of ions and resultant electron separation in the presence of an axial magnetic field. Also Schermann and Majorthe studied the Characteristics of electron-free plasma consisting of an unequal mass negative and positive ions as (I- and TI+) produced by photo-dissociation of molecular TII and confined in a paul rf quadrupole electric field[16]. Pair plasmas have their own thermodynamical properties, in a pure pair-ion plasma. It is found that both of the ions species have a single time scale to relax to thermodynamical equilibrium owing to their equal masses [17].

A pair-ion plasma is similar to an electron-positron plasma; however, the problem of annihilation is absent in a pair-ion plasma. The short life time of electron-positron plasma and low density production of positrons in laboratory experiments make it difficult to analyze various collective modes. One of the major obstacles in generating a stable electrone-positron plasma is the weak source of positrons (106 positrons/s) obtained using the radioactive sources[18] and(108-109 positrons/s)with accelerator based source[19]. Therefore the attention is focused on the stable generation of pair-ion plasmas in laboratory such as fullerene C±60 and hydrogen H± plasmas. Experimental observations[20] of a pair-ion fullerene C±60 plasma have invoked a great deal of interest in these topics. It has been reported that pure PI fullerene plasmas can support three kinds of electrostatic waves propagating parallel to an external static magnetic field. These waves are the ion plasma waves (IPW), the ion acoustic wave (IAW), and the third one has been named as the intermediate frequency wave (IFW). However, there are two observations on the above experiment[20].

First the IAW has frequency larger than the theoretically calculated one. The second is that the IFW has a mysterious feature that the group velocity is negative but the phase velocity is positive i.e. the mode looks like a backward wave. However, the IPW shows no special features in a pair-ion plasma. Some theoretical investigations have shown that the acoustic speed becomes larger in a pair-ion plasma if it is not pure and contains a significant concentration of electrons[21]. Saleem [22]put a criterion to define a pure pair-ion plasma and suggested that the lighter elements (like H and He) are more suitable to produce pair-ion plasmas.

The nonlinear waves in plasma described either by the Korteweg-de Vries equation KdV family equations or the nonlinear Schrődinger equation (NLSE). The KdV equation describes the evolution of non modulated wave, i.e. a bare pulse with no fast oscillations inside the packet, which is usually called the KdV soliton. The NLSE governs the dynamics of a modulated wave packet. In the NLSE, the nonlinearities are in balance with the wave group dispersion and the resulting stationary solutions of the NLSE have an envelope structure, called envelope soliton[23]. The NLSE, where the modulational instability (MI) phenomenon could be studied, is considered as one of the most important equations which governs the movement of the nonlinear structures in many branches of physics; condensed matter, nonlinear optics, plasma, and even biophysics[24]. When the wave packet is modulationaly unstable, it may dissociate into smaller wave packets or to single waves. One of the most interested type of waves which could be formed in the case of the modulation instability is the rogue waves.

Rogue waves are nonlinear waves which are short-lived phenomena appearing suddenly out of nowhere, so they can be quite unexpected and mysterious [25]. The average height of the rogue waves can be two or even more times the height of the surrounding waves. Importantly, rogue waves have been observed in many fields; mid-ocean and coastal waters [26], optical systems [27], fiber optics [28], parametrically driven capillary waves [29], Bose-Einstein condensates [30,31], super fluids[32], optical cavities[33], atmospheric physics and plasma physics[34].

The ion thermal rogue waves in a pair-ion plasma system without electrons has not been studied yet. So we try here to cover this important subject and study how the formed wave may be affected by the presence of stationary dust particles. It is well known that in the presence of two opposite polarity species as fluids, there were some critical density of negative ions at which the nonlinear coefficient of the KdV equation vanishes and this leads to terminate the nonlinear term of the KdV equation and the KdV equation becomes inadequate to describe the system. Moreover to study the evolution equation of the system in the vicinity of such critical density, new stretched independent variables are required. Introducing these new variables into the system of equations will lead to Extended Modified Korteweg-de Vries equation (EMKdV). Using the derivative expansion technique method, this new equation could be transformed into the corresponding NLSE which is valid at a small wave number, from which we can study the MI and so the possibility of forming the rogue waves[35], which is the motive of this study. This paper is organized as follows: in Sec.2 the model equations are introduced and the derivation of the evolution equations and the EMKdV, is derived and then transformed into the corresponding NLSE. In Sec.3 the numerical results are discussed and finally Sec.4 is devoted for the conclusion.

2.2 Model Equations And Derivation Of NLS Equation

We consider a warm unmagnatized plasma system consisting of positive and negative ions as fluids. The system of normalized equations describing such system is given by:

(∂n_p)/∂t+∂/∂x (n_p u_p )=0 (2.1)

(∂/∂t+u_p ∂/∂x) u_p+1/n_p (∂n_p)/∂x+∂ɸ/∂x=0 (2.2)

(∂n_n)/∂t+∂/∂x (n_n u_n )=0 (2.3)

(∂/∂t+u_n ∂/∂x) u_n+σ/n_n (∂n_n)/∂x-∂ɸ/∂x=0 (2.4)

And the poission equation

∂²φ/∂x²=n_n-n_p (2.5)

In Eqs.(2.1)-(2.5), the subscripts p, n for positive ions and negative ions , respectively. All number densities are normalized with respect to the unperturbed number density of the positive ions n_p^((0)). The velocities u_p and u_n are normalized with respect to positive ion-thermal speed〖〖(T〗_p⁄(m_p))〗^(1⁄2) . The potential ɸ is normalized by the thermal potential T_p⁄e . The space and time are normalized by positive ion debye length 〖〖(T〗_p⁄( 4πe^2 n_p^((0))))〗^(1⁄2) and the inverse of the positive ion plasma period〖〖(m〗_p⁄( 4πe^2 n_p^((0))))〗^(1⁄2) , respectively. Here, e is the electron charge, T_pthe positive ions temperature in energy unit, σ=T_n⁄T_p is the negative to positive ions temperature ratio.The quasineutrality condition for this system is . and are the unperturbed densities of negative ions and unperturbed positive ions density respectively.

To find the evolution equation of this system we follow the standard method of reductive perturbation technique[36] by introducing the following stretched coordinates:

X=ε^(1⁄2) (x-λt) and τ=〖 ε〗^(3⁄2) t. (2.6)

where is the phase velocity and is a small parameter. Furthermore, we expand all the physical dependent quantities in Eqs.(2.1)-(2.5) as follows :

n_p=1+εn_p^((1))+ɛ^2 n_p^((2))+ɛ^3 n_(p )^((3)), (2.7)

n_n=N_n+εn_n^((1))+ɛ^2 n_n^((2))+ɛ^3 n_(n )^((3)), (2.8)

u_p=εn_p^((1))+ɛ^2 u_p^((2))+ɛ^3 u_(p )^((3)), (2.9)

u_n=εu_n^((1))+ɛ^2 u_n^((2))+ɛ^3 u_(n )^((3)), (2.10)

And

ɸ=εɸ^((1))+ɛ^2 ɸ^((2))+ɛ^3 ɸ^((3)) . (2.11)

Substituting the Eq.(2.6) and the Eqs.(2.7)-(2.11) into the set of the basic equations and separating the different orders of ɛ , we get from the first order

n_p^((1))=1/((λ^2-1)) ɸ^((1)),n_n^((1))=〖-N〗_n/((λ^2-σ)) ɸ^((1)) (2.12)

u_p^((1))=λ/((λ^2-1)) ɸ^((1)),u_n^((1))=(-λ)/((λ^2-σ)) ɸ^((1)) (2.13)

λ=〖((N_n+σ)⁄(N_n+1))〗^(1⁄2) . (2.14)

The set of second order equations

1/((λ^2-1)) (∂ɸ^((1)))/∂τ-λ (∂n_p^((2) ))/∂X+(∂u_p^((2) ))/∂X+2λ/(λ^2-1)^2 ɸ^((1) ) (∂ɸ^((1) ))/∂X=0 (2.15)

λ/((λ^2-1)) (∂ɸ^((1)))/∂τ+(∂n_p^((2) ))/∂X-λ (∂u_p^((2) ))/∂X+1/((λ^2-1) ) ɸ^((1) ) (∂ɸ^((1) ))/∂X+(∂ɸ^((2) ))/∂X=0 (2.16)

〖-N〗_n/((λ^2-σ)) (∂ɸ^((1)))/∂τ-λ (∂n_n^((2) ))/∂X+N_n (∂u_n^((2) ))/∂X+(2λN_n)/(λ^2-σ)^2 ɸ^((1) ) (∂ɸ^((1) ))/∂X=0 (2.17)

〖-λN〗_n/((λ^2-σ)) (∂ɸ^((1)))/∂τ+σ (∂n_n^((2) ))/∂X-λN_n (∂u_n^((2) ))/∂X+N_n/(λ^2-σ)^2 ɸ^((1) ) (∂ɸ^((1) ))/∂X-N_n (∂ɸ^((2) ))/∂X=0

2.18))

Proceeding to the second order equations(2.15)-(2.18) with the aid of first order quantities it is easy to obtain the KdV equation as

∂ɸ/∂τ+Aɸ ∂ɸ/∂X+1/2 B (∂^3 ɸ)/(∂X^3 )=0 (2.19)

Where

A=((3λ^2-1) (λ^2-σ)^3-N_n [(3λ^2-σ) (λ^2-1)^3])/(2λ(λ^2-1)(λ^2-σ)[N_n (λ^2-1)^2+(λ^2-σ)^2]) (2.20)

B=((λ^2-σ)^2 (λ^2-1)^2)/(2λ[N_n (λ^2-1)^2+(λ^2-σ)^2]) (2.21)

There is a critical density value of the negative ions at which the nonlinear coefficient of this equation vanishes. In this case the stretched variables (2.6) are not valid to use any more, so we assume the new stretched variables[37,38]

X=ɛ(x-λt) and τ=ɛ^3 t. (2.22)

Using this expansion (22) with the previously assumed expansion in Eqs.(7)-(11) and following the method in Refs.37 and 38 we obtain from The second order quantities is as following

n_n^((2))=〖-N〗_n/((λ^2-σ)) ɸ^((2))+(N_n (3λ^2-σ))/〖2(λ^2-σ)〗^3 〖ɸ^((1))〗^2, (2.24)

u_p^((2))=λ/((λ^2-1)) ɸ^((2))+(λ(λ^2+1))/〖2(λ^2-1)〗^3 〖ɸ^((1))〗^2, (2.25)

u_n^((2))=〖-N〗_n/((λ^2-σ)) ɸ^((2))+(N_n (3λ^2-σ))/〖2(λ^2-σ)〗^3 〖ɸ^((1))〗^2, (2.26)

Proceeding to the higher order of ɛ we can easily obtain the modified KdV equation.

(2. 27)

where

(2.28)

In the vicinity of the critical density of negative ions, neither the KdV equation (2.19) nor the modified KdV equation (2.27) is adequate for describing the evolution of the system. So we have to look for a new evolution equation. Following the work done by Watanabe[39] and El-labany and El-Sheikh[37] we obtain

(2.29)

which describes the evolution equation in the vicinity of the critical density. The nonlinear term is a combination of the nonlinear term of the KdV equation (2.19) and the nonlinear term of the modified KdV equation (2.27).

Now in order to study the possibility of generation of rogue waves in this system, we should determine the regions of modulational instability in which the rogue waves could be formed. This could be achievable by converting Eq. (2.29) into the corresponding NLSE equation which is valid only at small wave numbers. To do so we apply the derivative expansion technique[40].

We assume a solution of Eq. (2.29) in the form of a weakly modulated sinusoidal wave by expanding ɸ as:

(2.30)

where is the carrier wavenumber and is the frequency for the given wave. The stretched variables and are

(2.31)

where v_g is the group velocity, will be determined later(kX-ωτ) assume that all perturbed quantities depend on the fast scales via the phase only, while the slow scales enter the arguments of the th harmonic amplitude . Since must be real, must satisfy the reality ndition , where the asterisk indicates the complex conjugate.The derivative operators appearing in the system of the basic equations become

( 2.32)

Using Eqs. (2.30)-(2.32) into (2.29), we obtain

(2.33)

The first-order approximation (m=1) with (l=1) yields the linear dispersion relation

(2.34) ω=-Bk^3

For the first-order harmonic (l=1) of the second-order approximation (m=2), we find that

(2.35)

which corresponds to the group velocity.((∂ω)⁄(∂k))

For the second harmonic ( ) with we have

(2.36) ,

whereas the zero harmonic ( ) for this order, give

(2.37)

Proceeding to the third-order approximation ( )and solving for the first harmonic equations ( ),an explicit compatibility condition will be found, which take the form of the NLSE

(2.38)

where the dispersion coefficient is given by

(2.39)

and the nonlinear coefficient is given by

(2.40)

There are many solutions to the Eq(2.38). One of these solutions is the rational solution which is an appropriate form for describing the rogue wave solution for > zero [41] which is a localized solution in both space and time and is given by

(2.41)

and is represented in Fig.2.1, which describes the variation of rogue waves in a three dimensions.

Figure 2.1: The Amplitude of the rogue waves Φ with ξ and η at =0.8 and k =0.05.

2.3 ESULTS AND DISCUSSION

The NLSE (38) supports many kinds of solutions that governs the modulation instability process in the system which happens due to the interchange of energy between the formed wave packet and the surrounding medium. If this balance is sustained then we will have a case of modulation stability (the sign of the product of is less than zero) and in this case the system may form what is called a dark soliton. Otherwise, the wave packet dissociates and may form grey solitons, bright solitons and many other types. One of them is the rogue waves. So before studying the behavior of these waves and how they are varying with different system parameters we have first to determine the modulation instability regions and determine how they vary with the system parameters.

Fig. 2.2 represents the variation of the product with the wave number at different values of temperature ratio It is noticed that decreasing the values of enhances the modulation stability of the system and so the decrease of the probability of forming rogue waves.

Figure2.2 The variation of PQ with k at different values of σ where

σ =0.65, σ =0.75, σ =0.83.

Fig.2.3 shows the variation of the maximum amplitude of the rogue waves with . It is obvious that the value the maximum amplitude decreases with the increase of the temperature ratio.

Figure 2.3: The variation of Φ with σ at k=0.132.

2.4 SYMEREY

The possibility of forming the ion thermal rogue wave is studied through studying the modulation instability of the system at small wave numbers. It is found that the system pararmeters affect both the probability of forming the rogue waves and its amplitude.

Chapter 3

Formation of solition in electronegative plasma with two streaming beam.

3.1 introduction

Plasma has wide variety in its component it could be simple plasma consist of full ionized ions and electrons or multicomponent plasma which is a mixture of ionized particle in addition to neutral atoms and may be dust. One example of multicomponent plasma is negative ion plasma. The plasma consisting of negative ions, positive ions and Electrons is called negative ion plasma. Negative ions plasmas have been found in space environments and laboratory. Laboratory plasma with sufficient faction of negative ion was available since von Goeler [42] produced plasma consisting of〖 cs〗^+, 〖cl〗^- and electrons using a beam of cscl onto the hot tungsten plate of a Q machine. Wong [43] introduced SF_6 gas into an argon discharge. Electron attachment to SF, molecules produced a plasma with SF_6^- negative ions. Later song report on experimental Investigation of ion-acoustic waves in a Q-machine plasma, consisting of K+ positive ions, SF, negative ions, and electrons[44].

Negative ions are found in the Earth’s ionosphere and mesosphere [45] .The dominant negative ion in the ionosphere down to 85km is O_2^- , with O^- the principal negative ion species above that altitude. Negative ions have also been found in the ionospheres of Mercury, the Earth’s moon and Jupiter’s moons, as well as installer atmospheres [46]. In comets like Hally based on measurements with the Giotto spacecraft, according to [47] various negative ion species in coma of comet Halley are identified.

The presence of negative ions in a plasma can be strongly effective in the collective behavior of the plasma, as well as excite new modes. Also it could significantly determines the charged species balance and the electron energy distribution function. presence of negative ions in a plasma causes a reduction of the electron shielding; therefore, the propagation characteristic of these nonlinear waves in negative ion plasma is different than simple two component electron ion plasmas.[48–50] In the presence of negative ions, the ion waves exhibit two modes of propagation a “fast” mode and a “slow” mode. Gill et al. [51] studied the ion acoustic soliton and double layers in plasma consisting of positive and negative ions with non thermal electron. Therefore, many works have been done for plasmas with negative ions [52, 53].

Soliton is a special nonlinear structure in the evolution of nonlinear wave formed when dispersive spreading balances nonlinear steepening, has been actively investigated in many areas such as nonlinear optics, solid state physics, fluids, and plasmas. Solitary waves are frequently observed in laboratory plasmas as well as in naturally occurring plasmas. Ion acoustic solitary waves in negative ion plasmas with Maxwellian electron distribution has been studied by many researchers [54-56].

Several years ago, Zakharov and Kuznetsov[57] derived an equation for nonlinear IAWs in a magnetized plasma composed of cold ions and hot isothermal electrons. The Zakharov-Kuznetsov ZK equation has also been derived for other physical scenarios. The existence and instability of the propagating solitary wave solutions of the ZK equation have been invesrigated.

In this chapter we study the probability of soliton formations in the presence of negative oxygen ions using ZK equation. We tried here to cover this important subject and studying how the formed wave may be affected by the presence of the negative oxygen ions. This chapter is organized as follows: in Sec3.2 the model equations are introduced the derivation of ZK and the transformation used to get soltions solution from ZK equation .In Sec.3.3 the numerical results are discussed.

3.2 Model Equations And Derivation of ZK Equation:

We consider a warm magnetized plasma system consisting of positive and negative oxygen ions as fluids and stream of hydrogen and electron beam. The plasma is confined in a magnetic field B=B_0 x ̅ where( x) ̅ is the unit vector along the x-axis. The system of normalized equations describing such system is given by:

For positive oxygen ion

(∂n_(o+))/∂t+∇.(n_(o+) u_(o+) )=0 , (3.1)

(∂/∂t+u_(o+) ∇) u_(o+)+ 1/n_(o+) ∇n_(o+)+Z_(O+) ∇ɸ -( u_(O+)× Ωx ̅ )=0 , (3.2)

For negative oxygen ion:

(∂n_(o-))/∂t+∇(n_(o-) u_(o-) )=0 , (3.3)

(∂/∂t+u_(o-) ∇) u_(o-)+ (μ_(o-) σ_(o-))/n_(o-) ∇n_(o-)-μ_(o-) Z_(O-) ∇ɸ+μ_(o-) z_(o-)/z_(o+) (u_(o-)×Ωx ̅ )=o, (3.4)

For hydrogen stream

(∂n_h)/∂t+∇(n_h u_h )=0 , (3.5)

(∂/∂t+u_h ∇) u_h+ (μ_h σ_h)/n_h ∇n_h+μ_h Z_h ∇ɸ-μ_h 1/z_(o+) (u_h×Ωx ̅ )=o,

(3.6)

For electron beam

(∂n_e)/∂t+∇(n_e u_e )=0 , (3.7)

(∂/∂t+u_e ∇) u_e+ (μ_e σ_e)/n_e ∇n_e-μ_e Z_e ∇ɸ-μ_e 1/z_(o+) (u_e×Ωx ̅ )=o, (3.8)

And the poission equation

∇^2 ɸ=1/z_(o+) (〖z_(o-) n〗_(o-)-n_(o+)-n_h+n_e). (3.9)

In Eqs. (3.1)-(3.9), the subscripts o+, o- , h , e for positive oxygen ions, negative oxygen ions, hydrogen beam and electron beam, respectively. The densities n_(o+),n_(o-),n_h and n_e are the densities of positive oxygen ions, negative oxygen ions, hydrogen and electron, respectively. All number densities are normalized with respect to the unperturbed number density of the positive oxygen ions n_(o+)^0. The velocities u_(o+),u_(o-),u_h and u_(e )are normalized with respect to positive oxygen ion thermal speed 〖〖(T〗_(O+)⁄(m_(o+)))〗^(1⁄2).The potential ɸ is normalized by the thermal potential〖 T〗_(O+)⁄e . The space and time are normalized by oxygen positive ion debye length 〖〖(T〗_(O+)⁄( 4πe^2 z_(o+) n_(o+)^((0))))〗^(1⁄2)and the inverse of the positive ion plasma period〖〖 (m〗_(O+)⁄( 4πe^2 z_(o+) n_(o+)^((0))))〗^(1⁄2), respectively. Here, e is the electron charge, T_(O+) the positive oxygen ions temperature in energy unit. Also σ_i=T_i⁄T_(o+) (i=o-,h and e ) are the ratios of the temperature of oxygen negative ions, hydrogen and electron to oxygen positive ions temperature. Also μ_(o-) , μ_h and μ_e are the ratios of mass of oxygen negative ions, hydrogen and electron to oxygen positive ions mass. The quasineutrality condition for this system is z_(o-) N_(O-)+N_e-N_h=z_(o+) .where N_(o-), N_e and N_h are the unperturbed densities of oxygen negative ions, electrons and hydrogen unperturbed density respectively.

This system is governed by the full set of equations (3.1)-(3.9). For small but finite amplitude, these equations can be analyzed using a reductive perturbation theory. We introduce stretched space-time variable X, Y, Z and T as

X=ε^(1⁄2) (x-λ t), Y=ε^(1⁄2) y,Z=ε^(1⁄2) z, and T=ε^(1⁄2) t. (3.10)

where λ is the phase velocity and ɛ is a small parameter. Furthermore, we expand all the physical dependent quantities in Eqs.(3.1)-(3.9) as follows : n_(0+)=1+ɛn_(0+)^((1))+ɛ^2 n_(o+)^((2))+ɛ^3 n_(o+)^((3))…, (3.11)

n_(0-)=N_(O-)+ɛn_(0-)^((1))+ɛ^2 n_(o-)^((2))+ɛ^3 n_(o-)^((3))+.., (3.12)

n_h=N_h+ɛn_h^((1))+ɛ^2 n_h^((2))+ɛ^3 n_h^((3))+.., (3.13)

n_h=N_h+ɛn_h^((1))+ɛ^2 n_h^((2))+ɛ^3 n_h^((3))+.., (3.14)

Velocities for positive and negative oxygen ions

u_jx=ɛn_jx^((1))+ɛ^2 n_jx^((2))+ɛ^3 n_jx^((3))+⋯, (3.15)

u_(j(y,z))=ɛ^(3⁄2) n_(j(y,z))^((1))+ɛ^2 n_(j(y,z))^((2))+ɛ^(5⁄2) n_(j(y,z))^((3))+⋯, (3.16)

Velocities for hydrogen and electron stream

u_kx=v_k+ɛn_kx^((1))+ɛ^2 n_kx^((2))+ɛ^3 n_kx^((3))+⋯, (3.17)

u_(k(y,z))=v_k+ɛ^(3⁄2) n_(k(y,z))^((1))+ɛ^2 n_(k(y,z))^((2))+ɛ^(5⁄2) n_(k(y,z))^((3))+⋯, (3.18)

and the potential ɸ

〖 ɸ〗^((1))+ɛ^2 ɸ^((2))+ɛ^3 ɸ^((3))+⋯, (3.19)ɛ= ɸ

where j=o+,o- and k=h,e .

Substituting the Eqs.(3.10)-(3.19) into the set of the basic equations and separating the different orders of ɛ , we get from the first order:

n_(0+)^((1))=z_(o+)/((λ^2-1)) 〖 ɸ〗^((1)), n_(0-)^((1))=(-Z_(O-) μ_(O-) N_(o-))/((λ^2-μ_(o-) σ_(o-))) 〖 ɸ〗^((1)), (3.20)

n_h^((1))=(μ_h N_h)/((λ_h^2-μ_h σ_h)) 〖 ɸ〗^((1)),n_e^((1))=(-μ_e N_e)/((λ_e^2-μ_e σ_e)) 〖 ɸ〗^((1)), (3.21)

u_(0+,x)^((1))=(z_(o+ ) λ)/((λ^2-1)) 〖 ɸ〗^((1)), u_(0-,x)^((1))=〖(-Z_(O-) μ_(O-) λ)/((λ^2-μ_(o-) σ_(o-))) ɸ〗^((1)), (3.22)

u_(h,x)^((1))=(-μ_h λ_h)/((〖λ_h〗^2-μ_h σ_h)) 〖 ɸ〗^((1)),u_(e,x)^((1))=(-μ_e λ_e)/((〖λ_e〗^2-μ_e σ_e)) 〖 ɸ〗^((1)), (3.23)

u_(0+,y)^((1))=(-Z_(O+ ) λ^2)/(Ω(λ^2-1)) (∂ɸ^((1)))/∂Z,u_(0-,y)^((1))=(-Z_(O+) λ^2)/(Ω(λ^2-μ_(o-) σ_(o-))) (∂ɸ^((1)))/∂Z, (3.24)

u_(h,y)^((1))=(-Z_(O+) λ_h^2)/(Ω(λ_h^2-μ_h σ_h)) (∂ɸ^((1)))/∂Z,u_(e,y)^((1))=(-Z_(O+) λ_e^2)/(Ω(λ_e^2-μ_e σ_e))

u_(0+,z)^((1))=(Z_(O+ ) λ^2)/(Ω(λ^2-1)) (∂ɸ^((1)))/∂Y , u_(0-,z)^((1))=(Z_(O+) λ^2)/(Ω(λ^2-μ_(o-) σ_(o-))) (∂ɸ^((1)) )/∂Y (3.26)

u_(h,z)^((1))=(Z_(O+) λ_h^2)/(Ω(λ_h^2-μ_h σ_h)) (∂ɸ^((1)))/∂Z , u_(e,z)^((1))=(-Z_(O+) λ_e^2)/(Ω(λ_e^2-μ_e σ_e)) (∂ɸ^((1)))/∂Z , (3.27)

and poisson equation gives the linear dispersion realation

(z_(o-)^2 μ_(o-) N_(o-))/((λ^2-μ_(o-) σ_(o-)))+(μ_h N_h)/((λ_h^2-μ_h σ_h))+(μ_e N_e)/((λ_e^2-μ_e σ_e))+(z_(o+)^2)/((λ^2-1))=0 . (3.28)

Proceeding to the next orders it is easy to obtain the ZK equation as

∂ɸ/∂τ+Aɸ ∂ɸ/∂X+B (∂^3 ɸ)/(∂X^3 )+C ∂/∂X ( (∂^2 ɸ)/(∂Y^2 )+(∂^2 ɸ)/(∂Z^2 ))=0, (3.29)

As

A=⌈(z_(o+)^3 (〖3λ〗^2-1 ))/〖(λ^2-1)〗^3 -(z_(o-)^3 μ_(O-)^2 N_(o-) (〖3λ〗^2-μ_(o-) σ_(o-) ))/(λ^2-μ_(o-) σ_(o-) )^3 +( μ_h^2 N_h (3λ_h^2-μ_h σ_h ))/(λ_h^2-μ_h σ_h )^3 -( μ_e^2 N_e (3λ_e^2-μ_e σ_e ))/〖(λ_e^2-μ_e σ_e )〗^3 ⌉*⌈(〖2z〗_(o+)^2 λ)/(λ^2-1)^2 -(2z_(o-)^2 μ_(o-) N_(o-) λ_(o-))/(λ^2-μ_(o-) σ_(o-) )^2 +(2μ_h N_h λ_h)/(λ_h^2-μ_h σ_h )^2 -( 2μ_e N_e λ_e)/(λ_e^2-μ_e σ_e )^2 ⌉^(-1)

(3.30)

B=Z_(O+) ⌈(〖2z〗_(o+)^2 λ)/(λ^2-1)^2 -(2z_(o-)^2 μ_(o-) N_(o-) λ_(o-))/(λ^2-μ_(o-) σ_(o-) )^2 +(2μ_h N_h λ_h)/(λ_h^2-μ_h σ_h )^2 -( 2μ_e N_e λ_e)/(λ_e^2-μ_e σ_e )^2 ⌉^(-1), (3.31)

C=B(1+(Z_(O+ ) λ^4)/〖Ω^2 (λ^2-1)〗^2 +(Z_(O+ ) λ^4)/(μ_(o-) Ω^2 (λ^2-μ_(o-) σ_(o-) )^2 )+(Z_(O+ ) λ_h^3 λ)/〖 μ_h Ω^2 (λ_h^2-μ_h σ_h )〗^2 + (Z_(O+ ) λ_e^3 λ)/〖μ_e Ω^2 (λ_e^2-μ_e σ_e )〗^2 ) , (3.32)

We are seeking for Stationary solitary wave solutions derived from ZK equation. So let us introduce the variable

η=L_x X+L_y Y+L_z Z-τ, (3.33)

Where〖 L〗_x ,L_y and〖 L〗_yare the directions cosines

Introduce equation (3.33) into ZK equation (3.34) it gives the solution of solitary wave as

φ= 3/(L_x A) 〖Sech〗^2 (1/2 √(1/(L_x^3 B+L_x C(1-L_x^2 ) )) η), (3.34)ϕ

= φ_0 〖Sech〗^2 ( η⁄W). (3.35)φ φ_0 and W are the amplitude and the width of soliton respectivel

3.3 the numerical results are discussed

Fig.3.1 describes the variation of soliton waves in a three dimensions as function on density of hydrogen . Fig 3.2 represents the variation of the soliton width W with streaming velocity of the hydrogen beam v_h for certain value of hydrogen to oxygen positive ion temperature σ_h .It is noticed that decreasing the values of σ_h leading to increase the width of soliton waves. In fig 3.3 we introduce relation between the amplitude of soliton wave and streaming velocity of the hydrogen beam v_h for certain value of hydrogen to oxygen positive ion temperature σ_h , it show that as the streaming velocity of the hydrogen beam v_h increases the amplitude of solition decreases also the same effect for increasing hydrogen to oxygen positive ion temperature σ_h .

So the precence of streaming hydrogen beam decrease the probality of solitions formation.

Fig. 3.1 the variation of soliton waves in a three dimensions as U= 0.001,μ_(o-)=1,μ_e=29450 ,μ_h=16,

Fig3.2 represents the variation of the soliton width W with streaming velocity of the hydrogen beam v_h for certain value of hydrogen to oxygen positive ion temperature〖 σ〗_h

Fig3.2 represents the variation of the soliton amplitude with streaming velocity of the hydrogen beam v_h for certain value of hydrogen to oxygen positive ion temperature〖 σ〗_h

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