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Analytical solitary wave solution of the dust ion acoustic waves (DIAWs) is studied in the frame-work of Korteweg-de Vries (KdV), damped force Korteweg-de Vries (DFKdV), damped force modified Korteweg-de Vries (DFMKdV) and damped forced Zakharov-Kuznetsov (DFZK) equations in an unmagnetized collisional dusty plasma consisting of negatively charged dust grain, positively charged ions, Maxwellian distributed electrons and neutral particles. Using reductive perturbation technique (RPT), the evolution equations are obtained for DIAWs.
Department of Mathematics, Gushkara Mahavidyalaya, India
Kaushik Roy
Beluti M.K.M. High School, India
Prasanta Chatterjee*
Department of Mathematics, Visva-Bharati, India
*Address all correspondence to: prasantacvb@gmail.com
1. Introduction
In the field of physics and applied mathematics research getting an exact solution of a nonlinear partial differential equation is very important. The elaboration of many complex phenomena in fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, etc. is possible if analytical solutions can be obtained. Most of the differential equation arises in these field has no explicit solution as popularly known. This problem creates hindrances in the study of nonlinear phenomena and makes it time-consuming in the research of nonlinear models in the plasma and other science. However recent researches in nonlinear differential equations have seen the development of many approximate analytical solutions of partial and ordinary differential equations.
The history behind the discovery of soliton is not only interesting but also significant. In 1834 a Scottish scientist and engineer—John Scott-Russell first noticed the solitary water wave on the Edinburgh Glasgow Canal. In 1844 [1] in “Report on Waves” he accounted his examinations to the British Association. He wrote “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished and after a chase of one or two miles I lost it in the windings of the channel. Such in the month of August 1834 was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.” He coined the word “solitary wave.” The solitary wave is called so because it often occurs as a single entity and is localized. The most important characteristics of solitary waves were unearthed after thorough study along with extensive wave-tank experiments. The following are the properties of solitary waves:
(a) These localized bell-shaped waves travel with enduring form and velocity. The speed of these waves are given by c2=gh+a, where g, a, h are respectively represent the acceleration of the gravity, amplitude of the wave and the undisturbed depth of the water. (b) Solitary waves can cross each other without any alteration.
John Scott-Russell’s study created a stir in the scientific community. His study not only initiated a debate with the prevailing knowledge of the theories of waves but also challenged the antecedent knowledge of waves. The previous study claimed that a periodic wave of finite amplitude and permanent shape are feasible only in deep water unlike Russell’s observation that the permanent profile is also possible in shallow water. Finally the stable form of solitary waves was received in scientific community with the aid of nonlinearity and dispersion. An ideal equilibrium between nonlinearity and dispersion can generate such waves.
Diederik Johannes Korteweg in 1895 [2] along with his PhD student Gustav De Vries obtained an equation from the primary equation of hydrodynamics. This equation explains shallow water waves where the existence of solitary waves was mathematically recognized. This equation is called KdV equation which is of the form ∂u∂t+Au∂u∂x+B∂3u∂x3=0. One of the most popular equations of soliton theory, this equation helps in explaining primary ideas that lie behind the soliton concept. Martin Zabusky and Norman Kruskal [3] in 1965 solved KdV equation numerically and noticed that the localized waves retain their shape and momentum in collisions. These waves were known as “solitons.” Soliton are solitary waves with the significant property that the solitons maintain the form asymptotically even when it experiences a collision. The fundamental “microscopic” properties of the soliton interaction; (i) the interaction does not change the soliton amplitudes; (ii) after the interaction, each soliton gets an additional phase shift; (iii) the total phase shift of a soliton acquired during a certain time interval can be calculated as a sum of the elementary phase shifts in pair wise collisions of this soliton with other solitons during this time interval is of importance. Solitons are mainly used in fiber optics, optical computer etc. which has really generated a stir in today’s scientific community. The conventional signal dispensation depends on linear system and linear systems. After all in this case nonlinear systems create more well-organized algorithms. The optical soliton is comparatively different from KdV solitons. Unlike the KdV soliton that illustrates the wave in a solitary wave, the optical soliton in fibers is the solitary wave of an envelope of a light wave. In this regard, the optical soliton in a fiber is treated as an envelope soliton.
This chapter will discuss the analytical solitary wave solution of the KdV and KdV-like equations. In the study of nonlinear dispersive waves, these equations are generally seen. The KdV equation, a generic equation, is important in the study of weakly nonlinear long waves. This equation consists of a single humped wave characterized by several unique properties. The Soliton solutions of the KdV equation have been quite popular but it also not devoid of problems. The problems not only restrict to dispersion but also dissipation and interestingly these are not dominated by the KdV equation. The standard KdV equation fails to explain the development of small-amplitude solitary waves in case the particles collide in a plasma system. KdV equation with an additional damping term or the damped Korteweg-de Vries (DKdV) equation becomes handy in explaining this issue of elaborating the character of the wave. But in the presence of any critical physical situation (critical point) nonlinearity of the KdV equation disappears and the amplitude of the waves reaches infinity. To control this situation, a new nonlinear partial differential equation has to be derived that can explain the system at that critical point. This is known as the modified Korteweg-de Vries (MKdV) equation. In the presence of collisions, this equation is not also adequate and a damped MKdV equation is necessary. Also in the presence of force source term then the equation will be further modified and become DFKdV/DFMKdV.
Now we will derive the KdV equation from a classic plasma model, in which we consider a collision-free unmagnetized plasma consists of electrons and ions, in which ions are mobile and electrons obey the Maxwell distribution. The basic equation will be given as:
∂Ni∂T+∂NiUi∂X=0E1
∂Ui∂T+Ui∂Ui∂X=−emi∂ψ∂XE2
ε0∂2ψ∂X2=eNe−NiE3
where the electrons obey Maxwell distribution, i.e., Ne=en0eeϕKBTe. Ni, Ne, Ui, mi are the ion density, electron density, ion velocity and ion mass, respectively. ψ is the electrostatic potential, KB is the Boltzmann constant, Te is the electron temperature and e is the charge of the electrons.
To write Eqs. (1)–(3) in dimensionless from we introduce the following dimensionless variables
x=XλD,t=ωpT,ϕ=eψKTe,ni=Nin0,ui=Uics,E4
where λD=ε0KBTe/n0e2 is the Debye length, cs=KBTe/mi is the ion acoustic speed, ωpi=n0e2/ε0mi is the ion plasma frequency and n0 is the unperturbed density of ions and electrons. Hence using (4) in (1)–(3) we obtain the normalized set of equations as
∂ni∂t+∂niui∂x=0E5
∂ui∂t+ui∂ui∂x=−∂ϕ∂xE6
∂2ϕ∂x2=eϕ−niE7
To linearized (5)–(7), let us write the dependent variable as sum of equilibrium and perturbed parts, so that we write ni=1+n¯i,ui=u¯i,ϕ=ϕ¯. Putting ni=1+n¯i where the values of parameters at equilibrium position is given by n1=1,u1=0 and ϕi=0 in Eq. (5), we get
∂∂t1+n¯i+∂∂xu¯i+n¯iu¯i=0E8
neglecting the nonlinear term ∂n¯iu¯i∂x from (8), we get
Since the system (22)–(24) is a system of linear homogeneous equation so for nontrivial solutions we have
−iωik00−iωik10−k2+1=0E25
⇒−i2ω2k2+1+i2k2=0
⇒ω2k2+1=−i2k2
⇒ω2=k2k2+1
This is the dispersion relation.
For small k, i.e., for weak dispersion we can expand as
ω=k1+k2−12=k−12K3+⋯E26
The phase velocity as
Vp=ωk=11+k2E27
so that Vp→1 as k→0 and Vp→0 as k→∞. The group velocity Vg=dwdk is given by
Vg=11+k23/2E28
In this case, we have Vg<Vp for all k>0. The group velocity is more important as energy of a medium transfer with this velocity.
For long-wave as k→0, the leading order approximation is ω=k, corresponding to non-dispersive acoustic waves with phase speed ω/k=1. Hence this speed is the same as the speed of the ion-acoustic waves cs. The long wave dispersion is weak, i.e., kλD<<1. This means that the wavelength is much larger than the Debye length. In these long waves, the electrons oscillate with the ions. The inertia of the wave is provided by the ions and the restoring pressure force by the electrons. At the next order in k, we find that
ω=k−12k3+Ok5ask→0E29
The Ok5 correction corresponds to weak KdV type long wave dispersion. For short wave (k→∞), the frequency ω=1, corresponding to the ion plasma frequency ωpi=csλD. Hence the ions oscillate in the fixed background of electrons.
Now the phase of the waves can be written as
kx−ωt=kx−t+12k3tE30
Here kx−t and k3t have same dynamic status (dimension) in the phase. Assuming k to be small order of ε1/2, ε being a small parameter measuring the weakness of the dispersion, Here x−t is the traveling wave form and time t is the linear form.
Let us consider a new stretched coordinates ξ,τ such that
ξ=ε1/2x−λt,τ=ε3/2tE31
where ε is the strength of nonlinearity and λ is the Mach number (phase velocity of the wave). ε may be termed as the size of the perturbation. Let the variables be perturbed from the stable state in the following way (considering ni=1,ui=0,ϕ=0 and ne=eϕ=e0=1 at equilibrium)
ni=1+εni1+ε2ni2+ε3ni3+⋯,E32
ui=0+εui1+ε2ui2+ε3ui3+⋯,E33
ϕ=0+εϕ1+ε2ϕ2+ε3ϕ3+⋯.E34
where x and t are function of ξ and τ so partial derivatives with respect to x and t can be transform into partial derivative in terms of ξ and τ so
Differentiating Eq. (50) With respect to ξ and substituting for ∂ni2∂ξ from Eq. (48) and for ∂ui2∂ξ from Eq. (49), we finally obtain
∂ϕ1∂τ+ϕ1∂ϕ1∂ξ+12∂3ϕ1∂ξ3=0.E51
Eq. (51) is known as KdV equation. ϕ1∂ϕ1∂ξ is the nonlinear term and 12∂3ϕ1∂ξ3 is the dispersive terms. Only nonlinearity can impose energy into the wave and the wave breaks but in presence of both nonlinearity and dispersive a stable wave profile is possible.
The steady-state solution of this KdV equation is obtained by transforming the independent variables ξ and τ to η=ξ−u0τ where u0 is a constant velocity normalized by cs.
The steady state solution of the KdV Eq. (51) can be written as
ϕ1=ϕmsech2ηΔE52
where ϕm=3u0 and Δ are the amplitude and width of the solitary waves. It is clear that height, width and speed of the pulse propotional to u0,1u0, and u0 respectively. As ϕm the amplitude is equal to 3u0 so u0 specify the energy of the solitary waves. So the larger the energy, the greater the speed, larger the amplitude and narrower the width (Figure 1).
Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by
∂ni∂t+∂niui∂x=0,E53
∂ui∂t+ui∂ui∂x=−∂ϕ∂x−νidui,E54
∂2ϕ∂x2=1−μne−n+μ,E55
where nj (j = i,e for ion, electron), ui,ϕ are the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ=Zdnd0n0, νid is the dust ion collisional frequency and the term Sxt [4, 5], is a charged density source arising from experimental conditions for a single definite purpose. n0,Zd,nd0 are the
3.1 Normalization
ni→nin0,ui→uiCs,ϕ→eϕKBTe,x→xλD,t→ωpitE56
where Cs=KBTemi is the ion acoustic speed, Te as electron temperature, KB as Boltzmann constant, e as magnitude of electron charge and mi as mass of ions. λD=Te4πne0e212 is the Debye length and ωpi=4πne0e2mi12 as ion-plasma frequency.
The normalized electron density is given by
ne=eϕ.E57
3.2 Phase velocity and nonlinear evolution equation
We introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables also considered as (32)–(34) with
νid∼ε3/2νid0.E58
S∼ε2S2.E59
Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (53)–(55) and equating the coefficients of lowest order of ε, we get the phase velocity as
λ=11−μ.E60
Taking the coefficients of next higher order of ε, we obtain the damped force KdV equation
∂ϕ1∂τ+Aϕ1∂ϕ1∂ξ+B∂3ϕ1∂ξ3+Cϕ1=B∂S2∂ξ,E61
where A=3−λ22λ, B=λ32,C=νid02.
It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. It is paramount to note that the source term or forcing term due to the presence of space debris in plasmas may be of different kind, for example, Gaussian forcing term [4], hyperbolic forcing term [4], (in the form of sech2ξτ and sech4ξτ functions) and trigonometric forcing term [6] (in the form of sinξτ and cosξτ functions). Motivated by these work we assume that S2 is a linear function of ξ such as S2=f0ξBcosωτ+P, where P is some constant and f0, ω denote the strength and the frequency of the source respectively. Put the expression of S2 in Eq. (61) we get,
∂ϕ1∂τ+Aϕ1∂ϕ1∂ξ+B∂3ϕ1∂ξ3+Cϕ11=f0cosωτ,E62
which is termed as damped and forced KdV (DFKdV) equation.
In absence of C and f0, i.e., for C=0 and f0=0 the Eq.(62) takes the form of well-known KdV equation with the solitary wave solution
ϕ1=ϕmsech2ξ−MτW,E63
where ϕm=3MA and W=2BM, with M as the Mach number.
In this case, it is well established that
I=∫−∞∞ϕ12dξ,E64
is a conserved. For small values of C and f0, let us assume that the solution of Eq. (62) is of the form
ϕ1=ϕmτsech2x−MττWτ,E65
where Mτ is an unknown function of τ and ϕmτ=3MτA, Wτ=2B/Mτ.
Differentiating Eq. (64) with respect to τ and using Eq. (62), one can obtain
where ϕmτ=3MτA and Wτ=2BMτ. The effect of the parameters, i.e., ion collision frequency parameterνid0, strength of the external force f0on the solitary wave solution of the damp force KdV Eq. (62) have been numerically studied. In Figure 2, the soliton solution of (62) is plotted from (63)in the absence of external periodic force and damping.
In Figure 3, the soliton solution of the damp force KdV equation is plotted from Eq. (65) for different values of the strength of the external periodic force f0. The values of other parameters are M0=0.2,ω=1,τ=1,μ=0.2,νid0=0.01. It is observed that the solution produces solitary waves and the amplitude of the solitary waves increases as the value of the parameter f0 increases. In Figure 4, damp force KdV equation is plotted from Eq. (65) for different values of the dust ion collision frequency parameter (νid0). The values of other parameters are M0=0.2,ω=1,τ=1,μ=0.2,f0=0.01. It is observed that the solution produces solitary waves and the amplitude of the solitary waves decreases as the value of the parameter νid0 increases and width of the solitary waves increases for increasing value of νid0.
To obtain damped KdV Burgers equation we considered an unmagnetized collisional dusty plasma which contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations are as follows
∂ni∂t+∂niui∂x=0,E69
∂ui∂t+ui∂ui∂x=−∂ϕ∂x+η∂2ui∂x2−νidui,E70
∂2ϕ∂x2=1−μne−ni+μ,E71
ne=eϕ,E72
where ni,ne,ui,ϕ, are the number density of ions, the number density of electrons, the ion fluid velocity and the electrostatic wave potential, respectively.
Here normalization is taken as follows
ni→nin0,ui→uiCs,ϕ→eϕKBTe,x→xλD,t→ωpit
Cs=KBTemi is the ion acoustic speed, Te as electron temperature, KB as Boltzmann constant and mi as mass of ions, e as magnitude of electron charge. λD=Te4πne0e212 is the Debye length and ωpi=mi4πne0e212 as ion-plasma frequency. Here, νid is the dust-ion collisional frequency and μ=n0en0i, where n0e and n0i are the unperturbed number densities of electrons and ions, respectively.
4.1 Perturbation
To obtain damped KdV burger we introduced the same stretched coordinates use in Eq.(31). The expansion of the dependent variables are also considered same as (32)–(34) with
η=ε1/2η0,E73
νid∼ε3/2νid0.E74
4.2 Phase velocity and nonlinear evolution equation
Substituting the above expansions (32)-(34) and (73)–(74) along with stretching coordinates (31) into Eqs. (69)–(71) and equating the coefficients of lowest order of ε, the phase velocity is obtained as
λ=11−μ.E75
Taking the coefficients of next higher order of ε, we obtain the DKdVB equation
∂ϕ1∂τ+Aϕ1∂ϕ1∂ξ+B∂3ϕ1∂ξ3+C∂2ϕ1∂ξ2+Dϕ1=0,E76
where A=3−λ22λ, B=v32, C=−η102 and D=νid02 .
In absence of C and D, i.e., for C=0 and D=0 the Eq.(76) takes the form of well-known KdV equation with the solitary wave solution
ϕ1=ϕmsech2ξ−M0τW,E77
where amplitude of the solitary waves ϕm=3M0A and width of the solitary waves W=2BM0, with M0 is the speed of the ion-acoustic solitary waves or Mach number.
For small values of C and D, let us assume that amplitude, width and velocity of the dust ion acoustic waves are dependent on τ and the slow time dependent solution of Eq. (76) is of the form
ϕ1=ϕmτsech2ξ−MττWτ,E79
where the amplitude ϕmτ=3MτA, width Wτ=2B/Mτ and velocity Mτ have to be determined.
Differentiating Eq. (78) with respect to τ and using Eq. (76), one can obtain
Let us consider an unmagnetized collisional dusty plasma that contains cold inertial ions, stationary dusts with negative charge and Maxwellian distributed electrons. The normalized ion fluid equations which include the equation of continuity, equation of momentum balance and Poisson’s equation, governing the DIAWs, are given by
∂ni∂t+∂niui∂x=0,E84
∂ui∂t+ui∂ui∂x=−∂ϕ∂x−νidu,E85
∂2ϕ∂x2=1−μne−ni+μ+SxtE86
where nj (j = i,e for ion, electron), ui,ϕ are the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ=Zdnd0n0, νid is the dust-ion collisional frequency and the term Sxt [4, 5], is a charged density source arising from experimental conditions for a single definite purpose. n0,Zd,nd0 are the normalization:
ni→nin0,ui→uiCs,ϕ→eϕKBTe,x→xλD,t→ωpitE87
where Cs=KBTemi is the ion acoustic speed, Te as electron temperature, KB as Boltzmann constant, e as magnitude of electron charge and mi as mass of ions. λD=Te4πne0e212 is the Debye length and ωpi=4πne0e2mi12 as ion-plasma frequency.
The normalized q-nonextensive electron number density takes the form [8]:
ne=ne01+q−1ϕq+12q−1E88
Phase velocity and nonlinear evolution equation
We introduced the same stretched coordinates use in Eq. (31). The expansion of the dependent variables also considered same as (32)–(34) and (58)–(59). Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (84)–(86) and equating the coefficients of lowest order of ε, we get the phase velocity as
λ=1a1−μ,E89
with a=q+12. Now taking the coefficients of next higher order of ε [i.e., coefficient of ε5/2 from Eqs. (84) and (85) and coefficient of ε2 from Eq. (86)], we obtain the DFKdV equation
∂ϕ1∂τ+Aϕ1∂ϕ1∂ξ+B∂3ϕ1∂ξ3+Cϕ1=B∂S2∂ξ,E90
where A=32λ−bλa, B=λ32 and C=νid02, with b=q+13−q8.
Now at the certain values, for example q=0.6 and μ=0.5, there is a critical point at which A=0, which imply the infinite growth of the amplitude of the DIASW solution as nonlinearity goes to zero. Therefore, at the critical point at which A=0 the stretching (31) is not valid. For describing the evolution of the nonlinear system at or near the critical point we introduce the new stretched coordinate as
ξ=εx−λt,τ=ε3t,E91
and expand of the dependent variables same as Eqs. (32)–(34) with
νid∼ε3νid0,E92
S∼ε3S2.E93
Now substituting Eq. (32)–(34) and (91)–(93) into the basic Eqs. (84)–(86) and equating the coefficients of lowest order of ε, [i.e., coefficients of ε2 from Eq. (84) and (85) and coefficients of ε from Eq. (86)], we obtain the following relations:
ni1=ui1λ,E94
ui1=ϕ1λ,E95
ni1=a1−μϕ1.E96
Equating the coefficients of next higher order of ε, [i.e., coefficients of ε3 from Eq. (84) and (85) and coefficients of ε from Eq. (86)],we obtain the following relations:
ni2=1λui2+ni1ui1E97
∂ui1∂ξ=1λui1∂ui1∂ξ+∂ϕ2∂ξE98
ni2=a1−μaϕ2+bϕ12E99
Equating the coefficients of next higher order of ε, [i.e., coefficients of ε4 from Eq. (84) and (85) and coefficients of ε from Eq. (86)], we obtain the following relations:
∂ni1∂τ−λ∂ni3∂ξ+∂ui3∂ξ+∂ni1ui2∂ξ+∂ni2ui1∂ξ=0E100
∂ui1∂τ−λ∂ui3∂ξ+∂ϕ3∂ξ+∂ui1ui2∂ξ+νid0ui1=0E101
∂2ϕ1∂ξ2=1−μaϕ3+2bϕ1ϕ2+cϕ13−ni3+S2E102
where a=1+q2, b=1+q3−q8 and c=1+q3−q5−3q48.
From Eq. (94)–(96), one can obtain the Phase velocity as λ2=1a1−μ and from Eqs. (94)–(102), one can obtain the following nonlinear evaluation equation as:
∂ϕ1∂τ+A1ϕ12∂ϕ1∂ξ+B1∂3ϕ1∂ξ3+C1ϕ1=B1∂S2∂ξ,E103
where A1=154λ3−3λ3c1−μ2, B1=λ32 and C1=νid02.
It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. For simplicity, we assume that S2 is a linear function of ξ such as S2=f0ξcosωτ+P, where P is some constant and f0, ω denote the strength and the frequency of the source respectively. Put the expression of S2 in the Eq. (103) we get,
∂ϕ1∂τ+A1ϕ12∂ϕ1∂ξ+B1∂3ϕ1∂ξ3+C1ϕ1=B1f0cosωτ.E104
Such a form of this source function is observed in experimental situations or conditions for a particular device. Eq. (104) is termed as damped force modified Korteweg-de Varies (DFMKdV) equation.
In absence of C1 and f0, i.e., for C1=0 and f0=0 the Eq.(104) takes the form of well-known MKdV equation.
The slow time dependence form of the ion acoustic waves solution of the DFMKdV Eq. (104) is given by,
Let us consider a plasma model [9] consisting of cold ions, Maxwellian electrons in the presence of dust particles and the external static magnetic field B=ŷB0 along the y-axis. The normalized continuity, momentum and Poisson’s equations are as follows
∂n∂t+∂nu∂x+∂nv∂y+∂nw∂z=0,E106
∂u∂t+u∂∂x+v∂∂y+w∂∂zu=−∂ϕ∂x−Ωiωpiw,E107
∂v∂t+u∂∂x+v∂∂y+w∂∂zv=−∂ϕ∂y−νidv,E108
∂w∂t+u∂∂x+v∂∂y+w∂∂zw=−∂ϕ∂z+Ωiωpiu,E109
∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2=δ1+δ2ne−nE110
The normalized electron density is given by
ne=eϕ,E111
where n,ne,ui=uvw,Te,mi,e,ϕ,Ωi,ωpi,νid and λD are the ion number density, electron number density, ion velocity, electron temperature, ion mass, electron charge, electrostatic potential, ion cyclotron frequency, ion plasma frequency, dust ion collision frequency and Debye length respectively.
Here the normalization is done as follows:
n→nn0,ne→nene0,ui→uiCs,ϕ→eϕTe,x→xλD,t→ωpit
Here δ1=nd0ni0,δ2=ne0ni0 with the condition δ1+δ2=1. λD=Te4πne0e21/2,ωpi−1=mi4πne0e21/2,Cs=Temi.
To obtain the DFZK equation we introduce the new stretched coordinates as
ξ=ε1/2xζ=ε1/2x−λt,η=ε1/2y,τ=ε3/2tE112
where ε is the strength of nonlinearity and λ is the phase velocity of waves. The expression of the dependent variables as follows:
n=1+εn1+ε2n2+⋯E113
u=0+ε3/2u1+ε2u2+⋯E114
v=0+εv1+ε2v2+⋯E115
w=0+ε3/2w1+ε2w2+⋯E116
ϕ=0+εϕ1+ε2ϕ2+⋯E117
νid∼ε3/2νid0E118
Sxyz∼ε2S2xyzE119
Substituting the equations (112)-(119) into the system of Eqs. (106)-(110) equating the coefficient of ε, we get
ϕ1=n1δ2.E120
Equating the coefficient of ε3/2, we get
n1=v1λE121
w1=−ωpiΩi∂ϕ1∂ξ,E122
v1=ϕ1λ,E123
u1=ωpiΩi∂ϕ1∂η.E124
Considering the coefficient of ε2, the following relationships are obtained
w2=λωpiΩi∂u1∂ζ,E125
u2=−λωpiΩi∂w1∂ζ,E126
∂2ϕ1∂ξ2+∂2ϕ1∂ζ2+∂2ϕ1∂η2=δ11−ϕ2+ϕ122−n2+S2.E127
Comparing the coefficients of ε5/2, we obtain
∂n1∂τ−λ∂n2∂ζ+∂u2∂ξ+∂∂ζn1v1+∂v2∂ζ+∂w2∂η=0E128
∂v1∂τ−λ∂v2∂ζ+v1∂v1∂ζ+∂ϕ2∂ζ−νid0v1=0.E129
Using the relationships (120)–(124), one can obtain the linear dispersion relation as
1−λ2δ2=0E130
Expressing all the perturbed quantities in terms of ϕ1 from Eq. (125)–(129), the damped forced ZK equation is obtained as
Choudhury et al. [5] studied analytical electron acoustic solitary wave (EASW) solution in the presence of periodic force for an unmagnetized plasma consisting of cold electron fluid, superthermal hot electrons and stationary ions. Motivated by the these works, here we consider the source term as S2=f0Beζ+fξ+gηcosωτ,where f0 and ω denote the strength and frequency of the source term respectively. Then Eq. (131) is of the form,
where F0=−ef0B. To find the analytical solution of Eq. (132), we transform the damped-forced ZK equation to the KdV equation. We introduce new variable:
ξ=lζ+mξ+nη,E133
where l, m, n are the direction cosines of the line of wave propagation, with l2+m2+n2=1. Substituting Eqs. (133) into the Eq. (132), we get
It is clear from the structure of the solitary wave solution of the DFKdV, DFMKdV and DFZK that the soliton amplitude and width depends on the nonlinearity and dispersion of the evolution equations, which are the function of different plasma parameter involve in the consider plasma system. Also evident from the structure of the approximate analytical solution that the amplitude and the width of the soliton depends on the Mach number Mτ which involve the forcing term F0cosωτ and the damping parameter. Thus the amplitude and the width of the solitary wave structure changes with the different plasma parameters. Also they are changes with the change of strength of external force F0, frequency of the external force ω and the collisional frequency between the different plasma species. The effect of these parameter can be studied through numerical simulation.
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Written By
Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee
Reviewed: 12 June 2020Published: 16 September 2020