Open access peer-reviewed chapter

Approximate Analytical Solution of Nonlinear Evolution Equations

By Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee

Submitted: March 6th 2020Reviewed: June 12th 2020Published: September 16th 2020

DOI: 10.5772/intechopen.93176

Downloaded: 65

Abstract

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.

Keywords

  • solitary wave
  • soliton
  • KdV
  • DKdV
  • DFZK

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, hare 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 ut+Auux+B3ux3=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.

2. The Korteweg-de Vries equation

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:

NiT+NiUiX=0E1
UiT+UiUiX=emiψXE2
ε02ψX2=eNeNiE3

where the electrons obey Maxwell distribution, i.e., Ne=en0eKBTe. Ni, Ne, Ui, miare the ion density, electron density, ion velocity and ion mass, respectively. ψis the electrostatic potential, KBis the Boltzmann constant, Teis the electron temperature and eis the charge of the electrons.

To write Eqs. (1)(3) in dimensionless from we introduce the following dimensionless variables

x=XλD,t=ωpT,ϕ=KTe,ni=Nin0,ui=Uics,E4

where λD=ε0KBTe/n0e2is the Debye length, cs=KBTe/miis the ion acoustic speed, ωpi=n0e2/ε0miis the ion plasma frequency and n0is the unperturbed density of ions and electrons. Hence using (4) in (1)–(3) we obtain the normalized set of equations as

nit+niuix=0E5
uit+uiuix=ϕxE6
2ϕx2=eϕniE7

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¯iwhere the values of parameters at equilibrium position is given by n1=1,u1=0and ϕi=0in Eq. (5), we get

t1+n¯i+xu¯i+n¯iu¯i=0E8

neglecting the nonlinear term n¯iu¯ixfrom (8), we get

n¯it¯+u¯ix¯=0E9

which is the linearized form of Eq. (5).

Putting ui=u¯i,ϕ=ϕ¯in Eq. (6), we get

u¯it+u¯iu¯ix=ϕ¯xE10

Neglecting the nonlinear term from (10), we get

u¯it+ϕ¯x¯=0E11

This is the linearized form of Eq. (6).

Putting ni=1+n¯i,ϕ=ϕ¯in Eq. (7), we get

2ϕ¯x=1+ϕ¯1n¯i2ϕ¯x=ϕ¯n¯iE12

Hence Eqs. (9), (11), (12) are the linearized form of Eq. (5)(7) respectively.

To get dispersion relation for low frequency wave let us assume that the perturbation is proportional to eikxωtand of the form

n¯=n0eikxωtE13
u¯=u0eikxωtE14
ϕ¯=ϕ0eikxωtE15

So,

n¯t=in0ωeikxωtE16
n¯x=ikn0eikxωtE17
u¯t=iu0ωeikxωtE18
u¯x=iku0eikxωtE19
ϕ¯x=ikϕ0eikxωtE20
2ϕ¯x2=ik2ϕ0eikxωtE21

Putting these value in Eqs. (9), (11) and (12), we get,

n0+iku0=0E22
u0+ikϕ0=0E23
n0k2+1ϕ0=0E24

Since the system (22)–(24) is a system of linear homogeneous equation so for nontrivial solutions we have

ik00ik10k2+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+k212=k12K3+E26

The phase velocity as

Vp=ωk=11+k2E27

so that Vp1as k0and Vp0as k. The group velocity Vg=dwdkis given by

Vg=11+k23/2E28

In this case, we have Vg<Vpfor all k>0. The group velocity is more important as energy of a medium transfer with this velocity.

For long-wave as k0, 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=k12k3+Ok5ask0E29

The Ok5correction 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

kxωt=kxt+12k3tE30

Here kxtand k3thave same dynamic status (dimension) in the phase. Assuming kto be small order of ε1/2, εbeing a small parameter measuring the weakness of the dispersion, Here xtis the traveling wave form and time tis the linear form.

Let us consider a new stretched coordinates ξ,τsuch that

ξ=ε1/2xλt,τ=ε3/2tE31

where εis the strength of nonlinearity and λis the Mach number (phase velocity of the wave). εmay 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=0and ne=eϕ=e0=1at 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 ξand τso partial derivatives with respect to xand tcan be transform into partial derivative in terms of ξand τso

x=ξξx+ττx,x=ε12ξE35
t=ξξt+ττt,t=ε12ξ+ε32τE36
2x2=xε12ξ,2x2=ε2ξ2E37

We can express (5)–(7) in terms of ξand τas

ε3/2niτε1/2λniξ+ε1/2niuiξ=0E38
ε3/2uiτε1/2λuiξ+ε1/2uiuix=ε1/2ϕxE39
ε2ϕξ2=eϕniE40

Substituting the Eqs. (31)(34) in Eqs. (38)(40) and collecting the lowest order Oε3/2terms we get

λni1ξ+ui1ξ=0,E41
λui1ξ=ϕ1ξ,E42
ϕ1ni1=0.E43

Integrating Eqs. (41)(43) and all the variables tend to zero as ξ. We get

ni1=ui1λ,E44
ui1=ϕ1λ,E45
ϕ1=ni1.E46

From Eq. (44)(46) we get the phase velocity as

λ2=±1E47

Substituting the Eqs.(31)(34) in Eqs. (38)(40) and collecting order Oε5/2, we get

ni1τλni2ξ+ni1ui1ξ+ui2ξ=0,E48
ui1τλui2ξ+ui1ui1ξ=ϕ2ξ2,E49
ϕ1ξ2=ϕ2+12ϕ12ni1.E50

Differentiating Eq. (50) With respect to ξand substituting for ni2ξfrom Eq. (48) and for ui2ξfrom Eq. (49), we finally obtain

ϕ1τ+ϕ1ϕ1ξ+123ϕ1ξ3=0.E51

Eq. (51) is known as KdV equation. ϕ1ϕ1ξis the nonlinear term and 123ϕ1ξ3is 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 ξand τto η=ξu0τwhere u0is a constant velocity normalized by cs.

The steady state solution of the KdV Eq. (51) can be written as

ϕ1=ϕmsech2ηΔE52

where ϕm=3u0and Δare the amplitude and width of the solitary waves. It is clear that height, width and speed of the pulse propotional to u0,1u0,and u0respectively. As ϕmthe amplitude is equal to 3u0so u0specify the energy of the solitary waves. So the larger the energy, the greater the speed, larger the amplitude and narrower the width (Figure 1).

Figure 1.

Solitary wave solution of Eq. (52) for the parameter value t = 1 , u 0 = 0.2 .

3. Damped force KdV equation

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

nit+niuix=0,E53
uit+uiuix=ϕxνidui,E54
2ϕx2=1μnen+μ,E55

where nj(j = i,e for ion, electron), ui,ϕare the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ=Zdnd0n0, νidis 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,nd0are the

3.1 Normalization

ninin0,uiuiCs,ϕKBTe,xxλD,tωpitE56

where Cs=KBTemiis the ion acoustic speed, Teas electron temperature, KBas Boltzmann constant, eas magnitude of electron charge and mias mass of ions. λD=Te4πne0e212is the Debye length and ωpi=4πne0e2mi12as 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

νidε3/2νid0.E58
Sε2S2.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=11μ.E60

Taking the coefficients of next higher order of ε, we obtain the damped force KdV equation

ϕ1τ+Aϕ1ϕ1ξ+B3ϕ1ξ3+Cϕ1=BS2ξ,E61

where A=3λ22λ, 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 sech2ξτand sech4ξτfunctions) and trigonometric forcing term [6] (in the form of sinξτand cosξτfunctions). Motivated by these work we assume that S2is a linear function of ξsuch as S2=f0ξBcosωτ+P, where P is some constant and f0, ωdenote the strength and the frequency of the source respectively. Put the expression of S2in Eq. (61) we get,

ϕ1τ+Aϕ1ϕ1ξ+B3ϕ1ξ3+Cϕ11=f0cosωτ,E62

which is termed as damped and forced KdV (DFKdV) equation.

In absence of Cand f0, i.e., for C=0and f0=0the Eq.(62) takes the form of well-known KdV equation with the solitary wave solution

ϕ1=ϕmsech2ξW,E63

where ϕm=3MAand W=2BM, with Mas the Mach number.

In this case, it is well established that

I=ϕ12,E64

is a conserved. For small values of Cand f0, let us assume that the solution of Eq. (62) is of the form

ϕ1=ϕmτsech2xMττWτ,E65

where Mτis an unknown function of τand ϕmτ=3MτA, Wτ=2B/Mτ.

Differentiating Eq. (64) with respect to τand using Eq. (62), one can obtain

dI+2CI=2f0cosωτϕ1,dI+2CI=24f0BAMτcosωτ.E66

Again,

I=ϕ12,I=ϕm2τsech4ξMττWτ,
I=24BA2M3/2τ.E67

Using Eq. (66) and (67) the expression of Mτis obtained as

Mτ=M8ACf016C2+9ω2e43+6Af016C2+9ω243Ccosωτ+ωsinωτ.

Therefore, the solution of the Eq. (62) is

ϕ1=ϕmτsech2ξMττWτ,E68

where ϕmτ=3MτAand 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.

Figure 2.

Solitary wave solution of Eq. (62) in the absence of damping( ν id 0 = 0 ) and external force( f 0 = 0 ) with the parameter value M 0 = 0.2 , ω = 1 , τ = 1 , μ = 0.2 .

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 f0increases. 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 νid0increases and width of the solitary waves increases for increasing value of νid0.

Figure 3.

Variation of solitary wave from Eq. (62) for the different values of f 0 with M 0 = 0.2 , ω = 1 , τ = 1 , μ = 0.2 , ν id 0 = 0.01 .

Figure 4.

Variation of solitary wave from Eq. (62) for the different values of collisional frequency ν id 0 with M 0 = 0.2 , ω = 1 , τ = 1 , μ = 0.2 , f 0 = 0.01 .

4. Damped KdV Burgers equation

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

nit+niuix=0,E69
uit+uiuix=ϕx+η2uix2νidui,E70
2ϕx2=1μneni+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

ninin0,uiuiCs,ϕKBTe,xxλD,tωpit

Cs=KBTemiis the ion acoustic speed, Teas electron temperature, KBas Boltzmann constant and mias mass of ions, eas magnitude of electron charge. λD=Te4πne0e212is the Debye length and ωpi=mi4πne0e212as ion-plasma frequency. Here, νidis the dust-ion collisional frequency and μ=n0en0i, where n0eand n0iare 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
νidε3/2νid0.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=11μ.E75

Taking the coefficients of next higher order of ε, we obtain the DKdVB equation

ϕ1τ+Aϕ1ϕ1ξ+B3ϕ1ξ3+C2ϕ1ξ2+Dϕ1=0,E76

where A=3λ22λ, B=v32, C=η102and D=νid02.

In absence of Cand D, i.e., for C=0and D=0the Eq.(76) takes the form of well-known KdV equation with the solitary wave solution

ϕ1=ϕmsech2ξM0τW,E77

where amplitude of the solitary waves ϕm=3M0Aand width of the solitary waves W=2BM0, with M0is the speed of the ion-acoustic solitary waves or Mach number.

It is well established for the KdV equation that,

I=ϕ12,E78

is a conserved quantity [7].

For small values of Cand D, let us assume that amplitude, width and velocity of the dust ion acoustic waves are dependent on τand the slow time dependent solution of Eq. (76) is of the form

ϕ1=ϕmτsech2ξMττWτ,E79

where the amplitude ϕmτ=3MτA, width Wτ=2B/Mτand velocity Mτhave to be determined.

Differentiating Eq. (78) with respect to τand using Eq. (76), one can obtain

dI+2DI=2Cϕ1ξ2,dI+2DI=2C×245M5/2τA2B.E80

where,

ϕ1ξ2=245M5/2τA2BE81

and

I=ϕ12,I=ϕm2τsech4ξMττWτ,I=24BA2M3/2τ.E82

Substituting Eq. (81) and (82) into Eq. (80), we obtain

dMτ+PMτ=QM2τ,E83

which is the Bernoulli’s equation, where P=43Dand Q=415CB.

The solution of the Eq. (83) is

Mτ=PM0M0Q1e+Pe

Therefore, the slow time dependence form of the ion acoustic solitary wave solution of the DKdVB Eq. (76) is given by (79)where.

Mτ=PM0M0Q1e+Peand M0=M0for τ=0.

5. Damped force MKdV equation

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

nit+niuix=0,E84
uit+uiuix=ϕxνidu,E85
2ϕx2=1μneni+μ+SxtE86

where nj(j = i,e for ion, electron), ui,ϕare the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ=Zdnd0n0, νidis 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,nd0are the normalization:

ninin0,uiuiCs,ϕKBTe,xxλD,tωpitE87

where Cs=KBTemiis the ion acoustic speed, Teas electron temperature, KBas Boltzmann constant, eas magnitude of electron charge and mias mass of ions. λD=Te4πne0e212is the Debye length and ωpi=4πne0e2mi12as ion-plasma frequency.

The normalized q-nonextensive electron number density takes the form [8]:

ne=ne01+q1ϕq+12q1E88

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=1a1μ,E89

with a=q+12. Now taking the coefficients of next higher order of ε[i.e., coefficient of ε5/2from Eqs. (84) and (85) and coefficient of ε2from Eq. (86)], we obtain the DFKdV equation

ϕ1τ+Aϕ1ϕ1ξ+B3ϕ1ξ3+Cϕ1=BS2ξ,E90

where A=32λa, B=λ32and C=νid02, with b=q+13q8.

Now at the certain values, for example q=0.6and 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=0the 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=εxλt,τ=ε3t,E91

and expand of the dependent variables same as Eqs. (32)(34) with

νidε3νid0,E92
Sε3S2.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 ε2from Eq. (84) and (85) and coefficients of εfrom Eq. (86)], we obtain the following relations:

ni1=ui1λ,E94
ui1=ϕ1λ,E95
ni1=a1μϕ1.E96

Equating the coefficients of next higher order of ε, [i.e., coefficients of ε3from Eq. (84) and (85) and coefficients of εfrom Eq. (86)],we obtain the following relations:

ni2=1λui2+ni1ui1E97
ui1ξ=1λui1ui1ξ+ϕ2ξE98
ni2=a1μaϕ2+bϕ12E99

Equating the coefficients of next higher order of ε, [i.e., coefficients of ε4from Eq. (84) and (85) and coefficients of εfrom Eq. (86)], we obtain the following relations:

ni1τλni3ξ+ui3ξ+ni1ui2ξ+ni2ui1ξ=0E100
ui1τλui3ξ+ϕ3ξ+ui1ui2ξ+νid0ui1=0E101
2ϕ1ξ2=1μaϕ3+2bϕ1ϕ2+cϕ13ni3+S2E102

where a=1+q2, b=1+q3q8and c=1+q3q53q48.

From Eq. (94)(96), one can obtain the Phase velocity as λ2=1a1μand from Eqs. (94)–(102), one can obtain the following nonlinear evaluation equation as:

ϕ1τ+A1ϕ12ϕ1ξ+B13ϕ1ξ3+C1ϕ1=B1S2ξ,E103

where A1=154λ33λ3c1μ2, B1=λ32and 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 S2is a linear function of ξsuch as S2=f0ξcosωτ+P, where P is some constant and f0, ωdenote the strength and the frequency of the source respectively. Put the expression of S2in the Eq. (103) we get,

ϕ1τ+A1ϕ12ϕ1ξ+B13ϕ1ξ3+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 C1and f0, i.e., for C1=0and f0=0the 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,

ϕ1=ϕmτsechξMττWτ,E105

where Mτis given by equation

Mτ=πf0B1A1/62ωω2+4C12sinωτ+2C1ωcosωτ+Mπf0B1A1/242C1ω2+4C12eνid0τ2.

The amplitude and width are as follows:

ϕmτ=1A6πf0B1A1/62ωω2+4C12sinωτ+2C1ωcosωτ+Mπf0B1A1/242C1ω2+4C12eνid0τ
Wτ=B1W1+W2

where

W1=πf0B1A1/62ωω2+4C12sinωτ+2C1ωcosωτW2=Mπf0B1A1/242C1ω2+4C12eνid0τ

6. Damped force Zakharov-Kuznetsov equation

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=ŷB0along the y-axis. The normalized continuity, momentum and Poisson’s equations are as follows

nt+nux+nvy+nwz=0,E106
ut+ux+vy+wzu=ϕxΩiωpiw,E107
vt+ux+vy+wzv=ϕyνidv,E108
wt+ux+vy+wzw=ϕz+Ωiωpiu,E109
2ϕx2+2ϕy2+2ϕz2=δ1+δ2nenE110

The normalized electron density is given by

ne=eϕ,E111

where n,ne,ui=uvw,Te,mi,e,ϕ,Ωi,ωpi,νidand λDare 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:

nnn0,nenene0,uiuiCs,ϕTe,xxλD,tωpit

Here δ1=nd0ni0,δ2=ne0ni0with the condition δ1+δ2=1. λD=Te4πne0e21/2,ωpi1=mi4πne0e21/2,Cs=Temi.

To obtain the DFZK equation we introduce the new stretched coordinates as

ξ=ε1/2xζ=ε1/2xλt,η=ε1/2y,τ=ε3/2tE112

where εis the strength of nonlinearity and λis 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
νidε3/2νid0E118
Sxyzε2S2xyzE119

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=ωpiΩiϕ1ξ,E122
v1=ϕ1λ,E123
u1=ωpiΩiϕ1η.E124

Considering the coefficient of ε2, the following relationships are obtained

w2=λωpiΩiu1ζ,E125
u2=λωpiΩiw1ζ,E126
2ϕ1ξ2+2ϕ1ζ2+2ϕ1η2=δ11ϕ2+ϕ122n2+S2.E127

Comparing the coefficients of ε5/2, we obtain

n1τλn2ζ+u2ξ+ζn1v1+v2ζ+w2η=0E128
v1τλv2ζ+v1v1ζ+ϕ2ζνid0v1=0.E129

Using the relationships (120)–(124), one can obtain the linear dispersion relation as

1λ2δ2=0E130

Expressing all the perturbed quantities in terms of ϕ1from Eq. (125)(129), the damped forced ZK equation is obtained as

ϕ1τ+Aϕ1ϕ1ζ+B3ϕ1ζ3+Dϕ1+Cζ2ϕ1ξ2+2ϕ1η2+BS2ζ=0E131

where

A=32λλ2,B=λ2δ2,C=λ2δ21+ωpi2Ωi2,D=νid02.

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=f0B++cosωτ,where f0and ωdenote the strength and frequency of the source term respectively. Then Eq. (131) is of the form,

ϕ1τ+Aϕ1ϕ1ζ+B3ϕ1ζ3+Dϕ1+Cζ2ϕ1ξ2+2ϕ1η2=F0cosωτE132

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=++,E133

where l, m, nare the direction cosines of the line of wave propagation, with l2+m2+n2=1. Substituting Eqs. (133) into the Eq. (132), we get

ϕ1τ+Alϕ1ϕ1ξ+Bl33ϕ1ξ3+Clm2+n23ϕ1ξ3+Dϕ1=F0cosωτϕ1τ+Pϕ1ϕ1ξ+Q3ϕ1ξ3+Dϕ1=F0cosωτE134

where, P=Al, Q=Bl3+Clm2+n2,

The analytical solitary wave solution of the Eq. (134) as obtained in (68), is

ϕ1=ϕmτsech2ξMττWτE135

where ϕmτ=3MτPand Wτ=2QMτ, with

Mτ=M8PF016D2+9ω2e43+6PF016D2+9ω243Dcosωτ+ωsinωτ.E136

7. Conclusions

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 Mτwhich involve the forcing term F0cosωτand 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 ωand the collisional frequency between the different plasma species. The effect of these parameter can be studied through numerical simulation.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee (September 16th 2020). Approximate Analytical Solution of Nonlinear Evolution Equations, Selected Topics in Plasma Physics, Sukhmander Singh, IntechOpen, DOI: 10.5772/intechopen.93176. Available from:

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