Open access peer-reviewed chapter

Approximate Analytical Solution of Nonlinear Evolution Equations

Written By

Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee

Reviewed: 12 June 2020 Published: 16 September 2020

DOI: 10.5772/intechopen.93176

From the Edited Volume

Selected Topics in Plasma Physics

Edited by Sukhmander Singh

Chapter metrics overview

577 Chapter Downloads

View Full Metrics

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 c 2 = g h + 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 3 u x 3 = 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.

Advertisement

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:

N i T + N i U i X = 0 E1
U i T + U i U i X = e m i ψ X E2
ε 0 2 ψ X 2 = e N e N i E3

where the electrons obey Maxwell distribution, i.e., N e = en 0 e K B T e . N i , N e , U i , m i are the ion density, electron density, ion velocity and ion mass, respectively. ψ is the electrostatic potential, K B is the Boltzmann constant, T e 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 = X λ D , t = ω p T , ϕ = KT e , n i = N i n 0 , u i = U i c s , E4

where λ D = ε 0 K B T e / n 0 e 2 is the Debye length, c s = K B T e / m i is the ion acoustic speed, ω pi = n 0 e 2 / ε 0 m i is the ion plasma frequency and n 0 is the unperturbed density of ions and electrons. Hence using (4) in (1)–(3) we obtain the normalized set of equations as

n i t + n i u i x = 0 E5
u i t + u i u i x = ϕ x E6
2 ϕ x 2 = e ϕ n i E7

To linearized (5)–(7), let us write the dependent variable as sum of equilibrium and perturbed parts, so that we write n i = 1 + n ¯ i , u i = u ¯ i , ϕ = ϕ ¯ . Putting n i = 1 + n ¯ i where the values of parameters at equilibrium position is given by n 1 = 1 , u 1 = 0 and ϕ i = 0 in Eq. (5), we get

t 1 + n ¯ i + x u ¯ i + n ¯ i u ¯ i = 0 E8

neglecting the nonlinear term n ¯ i u ¯ i x from (8), we get

n ¯ i t ¯ + u ¯ i x ¯ = 0 E9

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

Putting u i = u ¯ i , ϕ = ϕ ¯ in Eq. (6), we get

u ¯ i t + u ¯ i u ¯ i x = ϕ ¯ x E10

Neglecting the nonlinear term from (10), we get

u ¯ i t + ϕ ¯ x ¯ = 0 E11

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

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

2 ϕ ¯ x = 1 + ϕ ¯ 1 n ¯ i 2 ϕ ¯ x = ϕ ¯ n ¯ i E12

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 e i kx ωt and of the form

n ¯ = n 0 e i kx ωt E13
u ¯ = u 0 e i kx ωt E14
ϕ ¯ = ϕ 0 e i kx ωt E15

So,

n ¯ t = in 0 ω e i kx ωt E16
n ¯ x = ikn 0 e i kx ωt E17
u ¯ t = iu 0 ω e i kx ωt E18
u ¯ x = iku 0 e i kx ωt E19
ϕ ¯ x = ik ϕ 0 e i kx ωt E20
2 ϕ ¯ x 2 = ik 2 ϕ 0 e i kx ωt E21

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

n 0 + iku 0 = 0 E22
u 0 + ik ϕ 0 = 0 E23
n 0 k 2 + 1 ϕ 0 = 0 E24

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

ik 0 0 ik 1 0 k 2 + 1 = 0 E25
i 2 ω 2 k 2 + 1 + i 2 k 2 = 0
ω 2 k 2 + 1 = i 2 k 2
ω 2 = k 2 k 2 + 1

This is the dispersion relation.

For small k , i.e., for weak dispersion we can expand as

ω = k 1 + k 2 1 2 = k 1 2 K 3 + E26

The phase velocity as

V p = ω k = 1 1 + k 2 E27

so that V p 1 as k 0 and V p 0 as k . The group velocity V g = dw dk is given by

V g = 1 1 + k 2 3 / 2 E28

In this case, we have V g < V p 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 c s . The long wave dispersion is weak, i.e., k λ D < < 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 1 2 k 3 + O k 5 as k 0 E29

The O k 5 correction corresponds to weak KdV type long wave dispersion. For short wave ( k ), the frequency ω = 1 , corresponding to the ion plasma frequency ω pi = c s λ D . Hence the ions oscillate in the fixed background of electrons.

Now the phase of the waves can be written as

kx ωt = k x t + 1 2 k 3 t E30

Here k x t and k 3 t have same dynamic status (dimension) in the phase. Assuming k to be small order of ε 1 / 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 = ε 1 / 2 x λt , τ = ε 3 / 2 t E31

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 n i = 1 , u i = 0 , ϕ = 0 and n e = e ϕ = e 0 = 1 at equilibrium)

n i = 1 + ε n i 1 + ε 2 n i 2 + ε 3 n i 3 + , E32
u i = 0 + ε u i 1 + ε 2 u i 2 + ε 3 u i 3 + , E33
ϕ = 0 + εϕ 1 + ε 2 ϕ 2 + ε 3 ϕ 3 + . 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

x = ξ ξ x + τ τ x , x = ε 1 2 ξ E35
t = ξ ξ t + τ τ t , t = ε 1 2 ξ + ε 3 2 τ E36
2 x 2 = x ε 1 2 ξ , 2 x 2 = ε 2 ξ 2 E37

We can express (5)–(7) in terms of ξ and τ as

ε 3 / 2 n i τ ε 1 / 2 λ n i ξ + ε 1 / 2 n i u i ξ = 0 E38
ε 3 / 2 u i τ ε 1 / 2 λ u i ξ + ε 1 / 2 u i u i x = ε 1 / 2 ϕ x E39
ε 2 ϕ ξ 2 = e ϕ n i E40

Substituting the Eqs. (31)(34) in Eqs. (38)(40) and collecting the lowest order O ε 3 / 2 terms we get

λ n i 1 ξ + u i 1 ξ = 0 , E41
λ u i 1 ξ = ϕ 1 ξ , E42
ϕ 1 n i 1 = 0 . E43

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

n i 1 = u i 1 λ , E44
u i 1 = ϕ 1 λ , E45
ϕ 1 = n i 1 . E46

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

λ 2 = ± 1 E47

Substituting the Eqs.(31)(34) in Eqs. (38)(40) and collecting order O ε 5 / 2 , we get

n i 1 τ λ n i 2 ξ + n i 1 u i 1 ξ + u i 2 ξ = 0 , E48
u i 1 τ λ u i 2 ξ + u i 1 u i 1 ξ = ϕ 2 ξ 2 , E49
ϕ 1 ξ 2 = ϕ 2 + 1 2 ϕ 1 2 n i 1 . E50

Differentiating Eq. (50) With respect to ξ and substituting for n i 2 ξ from Eq. (48) and for u i 2 ξ from Eq. (49), we finally obtain

ϕ 1 τ + ϕ 1 ϕ 1 ξ + 1 2 3 ϕ 1 ξ 3 = 0 . E51

Eq. (51) is known as KdV equation. ϕ 1 ϕ 1 ξ is the nonlinear term and 1 2 3 ϕ 1 ξ 3 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 u 0 τ where u 0 is a constant velocity normalized by c s .

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

ϕ 1 = ϕ m sech 2 η Δ E52

where ϕ m = 3 u 0 and Δ are the amplitude and width of the solitary waves. It is clear that height, width and speed of the pulse propotional to u 0 , 1 u 0 , and u 0 respectively. As ϕ m the amplitude is equal to 3 u 0 so u 0 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).

Figure 1.

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

Advertisement

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

n i t + n i u i x = 0 , E53
u i t + u i u i x = ϕ x ν id u i , E54
2 ϕ x 2 = 1 μ n e n + μ , E55

where n j (j = i,e for ion, electron), u i , ϕ are the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ = Z d n d 0 n 0 , ν id is the dust ion collisional frequency and the term S x t [4, 5], is a charged density source arising from experimental conditions for a single definite purpose. n 0 , Z d , n d 0 are the

3.1 Normalization

n i n i n 0 , u i u i C s , ϕ K B T e , x x λ D , t ω pi t E56

where C s = K B T e m i is the ion acoustic speed, T e as electron temperature, K B as Boltzmann constant, e as magnitude of electron charge and m i as mass of ions. λ D = T e 4 π n e 0 e 2 1 2 is the Debye length and ω pi = 4 π n e 0 e 2 m i 1 2 as ion-plasma frequency.

The normalized electron density is given by

n e = e ϕ . 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

ν id ε 3 / 2 ν id 0 . E58
S ε 2 S 2 . 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 = 1 1 μ . E60

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

ϕ 1 τ + A ϕ 1 ϕ 1 ξ + B 3 ϕ 1 ξ 3 + C ϕ 1 = B S 2 ξ , E61

where A = 3 λ 2 2 λ , B = λ 3 2 , C = ν id 0 2 .

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 sech 2 ξ τ and sech 4 ξ τ functions) and trigonometric forcing term [6] (in the form of sin ξ τ and cos ξ τ functions). Motivated by these work we assume that S 2 is a linear function of ξ such as S 2 = f 0 ξ B cos ωτ + P , where P is some constant and f 0 , ω denote the strength and the frequency of the source respectively. Put the expression of S 2 in Eq. (61) we get,

ϕ 1 τ + A ϕ 1 ϕ 1 ξ + B 3 ϕ 1 ξ 3 + C ϕ 1 1 = f 0 cos ωτ , E62

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

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

ϕ 1 = ϕ m sech 2 ξ W , E63

where ϕ m = 3 M A and W = 2 B M , with M as the Mach number.

In this case, it is well established that

I = ϕ 1 2 , E64

is a conserved. For small values of C and f 0 , let us assume that the solution of Eq. (62) is of the form

ϕ 1 = ϕ m τ sech 2 x M τ τ W τ , E65

where M τ is an unknown function of τ and ϕ m τ = 3 M τ A , W τ = 2 B / M τ .

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

dI + 2 CI = 2 f 0 cos ωτ ϕ 1 , dI + 2 CI = 24 f 0 B A M τ cos ωτ . E66

Again,

I = ϕ 1 2 , I = ϕ m 2 τ sech 4 ξ M τ τ W τ ,
I = 24 B A 2 M 3 / 2 τ . E67

Using Eq. (66) and (67) the expression of M τ is obtained as

M τ = M 8 ACf 0 16 C 2 + 9 ω 2 e 4 3 + 6 Af 0 16 C 2 + 9 ω 2 4 3 Ccos ωτ + ωsin ωτ .

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

ϕ 1 = ϕ m τ sech 2 ξ M τ τ W τ , E68

where ϕ m τ = 3 M τ A and W τ = 2 B M τ . The effect of the parameters, i.e., ion collision frequency parameter ν id 0 , strength of the external force f 0 on 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 f 0 . The values of other parameters are M 0 = 0.2 , ω = 1 , τ = 1 , μ = 0.2 , ν id 0 = 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 f 0 increases. In Figure 4, damp force KdV equation is plotted from Eq. (65) for different values of the dust ion collision frequency parameter ( ν id 0 ). The values of other parameters are M 0 = 0.2 , ω = 1 , τ = 1 , μ = 0.2 , f 0 = 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 ν id 0 increases and width of the solitary waves increases for increasing value of ν id 0 .

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 .

Advertisement

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

n i t + n i u i x = 0 , E69
u i t + u i u i x = ϕ x + η 2 u i x 2 ν id u i , E70
2 ϕ x 2 = 1 μ n e n i + μ , E71
n e = e ϕ , E72

where n i , n e , u i , 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

n i n i n 0 , u i u i C s , ϕ K B T e , x x λ D , t ω pi t

C s = K B T e m i is the ion acoustic speed, T e as electron temperature, K B as Boltzmann constant and m i as mass of ions, e as magnitude of electron charge. λ D = T e 4 π n e 0 e 2 1 2 is the Debye length and ω pi = m i 4 π n e 0 e 2 1 2 as ion-plasma frequency. Here, ν id is the dust-ion collisional frequency and μ = n 0 e n 0 i , where n 0 e and n 0 i 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 = ε 1 / 2 η 0 , E73
ν id ε 3 / 2 ν id 0 . 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 = 1 1 μ . E75

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

ϕ 1 τ + A ϕ 1 ϕ 1 ξ + B 3 ϕ 1 ξ 3 + C 2 ϕ 1 ξ 2 + D ϕ 1 = 0 , E76

where A = 3 λ 2 2 λ , B = v 3 2 , C = η 10 2 and D = ν id 0 2 .

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

ϕ 1 = ϕ m sech 2 ξ M 0 τ W , E77

where amplitude of the solitary waves ϕ m = 3 M 0 A and width of the solitary waves W = 2 B M 0 , with M 0 is the speed of the ion-acoustic solitary waves or Mach number.

It is well established for the KdV equation that,

I = ϕ 1 2 , E78

is a conserved quantity [7].

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

ϕ 1 = ϕ m τ sech 2 ξ M τ τ W τ , E79

where the amplitude ϕ m τ = 3 M τ A , width W τ = 2 B / M τ and velocity M τ have to be determined.

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

dI + 2 DI = 2 C ϕ 1 ξ 2 , dI + 2 DI = 2 C × 24 5 M 5 / 2 τ A 2 B . E80

where,

ϕ 1 ξ 2 = 24 5 M 5 / 2 τ A 2 B E81

and

I = ϕ 1 2 , I = ϕ m 2 τ sech 4 ξ M τ τ W τ , I = 24 B A 2 M 3 / 2 τ . E82

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

dM τ + PM τ = QM 2 τ , E83

which is the Bernoulli’s equation, where P = 4 3 D and Q = 4 15 C B .

The solution of the Eq. (83) is

M τ = PM 0 M 0 Q 1 e + Pe

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

M τ = PM 0 M 0 Q 1 e + Pe and M 0 = M 0 for τ = 0 .

Advertisement

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

n i t + n i u i x = 0 , E84
u i t + u i u i x = ϕ x ν id u , E85
2 ϕ x 2 = 1 μ n e n i + μ + S x t E86

where n j (j = i,e for ion, electron), u i , ϕ are the number density, ion fluid velocity and the electrostatic wave potential respectively. Here μ = Z d n d 0 n 0 , ν id is the dust-ion collisional frequency and the term S x t [4, 5], is a charged density source arising from experimental conditions for a single definite purpose. n 0 , Z d , n d 0 are the normalization:

n i n i n 0 , u i u i C s , ϕ K B T e , x x λ D , t ω pi t E87

where C s = K B T e m i is the ion acoustic speed, T e as electron temperature, K B as Boltzmann constant, e as magnitude of electron charge and m i as mass of ions. λ D = T e 4 π n e 0 e 2 1 2 is the Debye length and ω pi = 4 π n e 0 e 2 m i 1 2 as ion-plasma frequency.

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

n e = n e 0 1 + q 1 ϕ q + 1 2 q 1 E88

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 = 1 a 1 μ , E89

with a = q + 1 2 . Now taking the coefficients of next higher order of ε [i.e., coefficient of ε 5 / 2 from Eqs. (84) and (85) and coefficient of ε 2 from Eq. (86)], we obtain the DFKdV equation

ϕ 1 τ + A ϕ 1 ϕ 1 ξ + B 3 ϕ 1 ξ 3 + C ϕ 1 = B S 2 ξ , E90

where A = 3 2 λ a , B = λ 3 2 and C = ν id 0 2 , with b = q + 1 3 q 8 .

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 = ε x λt , τ = ε 3 t , E91

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

ν id ε 3 ν id 0 , E92
S ε 3 S 2 . 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 ε 2 from Eq. (84) and (85) and coefficients of ε from Eq. (86)], we obtain the following relations:

n i 1 = u i 1 λ , E94
u i 1 = ϕ 1 λ , E95
n i 1 = a 1 μ ϕ 1 . E96

Equating the coefficients of next higher order of ε , [i.e., coefficients of ε 3 from Eq. (84) and (85) and coefficients of ε from Eq. (86)],we obtain the following relations:

n i 2 = 1 λ u i 2 + n i 1 u i 1 E97
u i 1 ξ = 1 λ u i 1 u i 1 ξ + ϕ 2 ξ E98
n i 2 = a 1 μ a ϕ 2 + b ϕ 1 2 E99

Equating the coefficients of next higher order of ε , [i.e., coefficients of ε 4 from Eq. (84) and (85) and coefficients of ε from Eq. (86)], we obtain the following relations:

n i 1 τ λ n i 3 ξ + u i 3 ξ + n i 1 u i 2 ξ + n i 2 u i 1 ξ = 0 E100
u i 1 τ λ u i 3 ξ + ϕ 3 ξ + u i 1 u i 2 ξ + ν id 0 u i 1 = 0 E101
2 ϕ 1 ξ 2 = 1 μ a ϕ 3 + 2 b ϕ 1 ϕ 2 + c ϕ 1 3 n i 3 + S 2 E102

where a = 1 + q 2 , b = 1 + q 3 q 8 and c = 1 + q 3 q 5 3 q 48 .

From Eq. (94)(96), one can obtain the Phase velocity as λ 2 = 1 a 1 μ and from Eqs. (94)–(102), one can obtain the following nonlinear evaluation equation as:

ϕ 1 τ + A 1 ϕ 1 2 ϕ 1 ξ + B 1 3 ϕ 1 ξ 3 + C 1 ϕ 1 = B 1 S 2 ξ , E103

where A 1 = 15 4 λ 3 3 λ 3 c 1 μ 2 , B 1 = λ 3 2 and C 1 = ν id 0 2 .

It has been noticed that the behavior of nonlinear waves changes significantly in the presence of external periodic force. For simplicity, we assume that S 2 is a linear function of ξ such as S 2 = f 0 ξ cos ωτ + P , where P is some constant and f 0 , ω denote the strength and the frequency of the source respectively. Put the expression of S 2 in the Eq. (103) we get,

ϕ 1 τ + A 1 ϕ 1 2 ϕ 1 ξ + B 1 3 ϕ 1 ξ 3 + C 1 ϕ 1 = B 1 f 0 cos ωτ . 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 C 1 and f 0 , i.e., for C 1 = 0 and f 0 = 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,

ϕ 1 = ϕ m τ sech ξ M τ τ W τ , E105

where M τ is given by equation

M τ = π f 0 B 1 A 1 / 6 2 ω ω 2 + 4 C 1 2 sin ωτ + 2 C 1 ω cos ωτ + M π f 0 B 1 A 1 / 24 2 C 1 ω 2 + 4 C 1 2 e ν id 0 τ 2 .

The amplitude and width are as follows:

ϕ m τ = 1 A 6 π f 0 B 1 A 1 / 6 2 ω ω 2 + 4 C 1 2 sin ωτ + 2 C 1 ω cos ωτ + M π f 0 B 1 A 1 / 24 2 C 1 ω 2 + 4 C 1 2 e ν id 0 τ
W τ = B 1 W 1 + W 2

where

W 1 = π f 0 B 1 A 1 / 6 2 ω ω 2 + 4 C 1 2 sin ωτ + 2 C 1 ω cos ωτ W 2 = M π f 0 B 1 A 1 / 24 2 C 1 ω 2 + 4 C 1 2 e ν id 0 τ
Advertisement

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 = y ̂ B 0 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 z u = ϕ x Ω i ω pi w , E107
v t + u x + v y + w z v = ϕ y ν id v , E108
w t + u x + v y + w z w = ϕ z + Ω i ω pi u , E109
2 ϕ x 2 + 2 ϕ y 2 + 2 ϕ z 2 = δ 1 + δ 2 n e n E110

The normalized electron density is given by

n e = e ϕ , E111

where n , n e , u i = u v w , T e , m i , e , ϕ , Ω i , ω pi , νid and λ D 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 n n 0 , n e n e n e 0 , u i u i C s , ϕ T e , x x λ D , t ω pi t

Here δ 1 = n d 0 n i 0 , δ 2 = n e 0 n i 0 with the condition δ 1 + δ 2 = 1 . λ D = T e 4 π n e 0 e 2 1 / 2 , ω pi 1 = m i 4 π n e 0 e 2 1 / 2 , C s = T e m i .

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

ξ = ε 1 / 2 x ζ = ε 1 / 2 x λt , η = ε 1 / 2 y , τ = ε 3 / 2 t E112

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 + ε n 1 + ε 2 n 2 + E113
u = 0 + ε 3 / 2 u 1 + ε 2 u 2 + E114
v = 0 + ε v 1 + ε 2 v 2 + E115
w = 0 + ε 3 / 2 w 1 + ε 2 w 2 + E116
ϕ = 0 + εϕ 1 + ε 2 ϕ 2 + E117
ν id ε 3 / 2 ν id 0 E118
S x y z ε 2 S 2 x y z E119

Substituting the equations (112)-(119) into the system of Eqs. (106)-(110) equating the coefficient of ε , we get

ϕ 1 = n 1 δ 2 . E120

Equating the coefficient of ε 3 / 2 , we get

n 1 = v 1 λ E121
w 1 = ω pi Ω i ϕ 1 ξ , E122
v 1 = ϕ 1 λ , E123
u 1 = ω pi Ω i ϕ 1 η . E124

Considering the coefficient of ε 2 , the following relationships are obtained

w 2 = λ ω pi Ω i u 1 ζ , E125
u 2 = λ ω pi Ω i w 1 ζ , E126
2 ϕ 1 ξ 2 + 2 ϕ 1 ζ 2 + 2 ϕ 1 η 2 = δ 1 1 ϕ 2 + ϕ 1 2 2 n 2 + S 2 . E127

Comparing the coefficients of ε 5 / 2 , we obtain

n 1 τ λ n 2 ζ + u 2 ξ + ζ n 1 v 1 + v 2 ζ + w 2 η = 0 E128
v 1 τ λ v 2 ζ + v 1 v 1 ζ + ϕ 2 ζ ν id 0 v 1 = 0 . E129

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

1 λ 2 δ 2 = 0 E130

Expressing all the perturbed quantities in terms of ϕ 1 from Eq. (125)(129), the damped forced ZK equation is obtained as

ϕ 1 τ + A ϕ 1 ϕ 1 ζ + B 3 ϕ 1 ζ 3 + D ϕ 1 + C ζ 2 ϕ 1 ξ 2 + 2 ϕ 1 η 2 + B S 2 ζ = 0 E131

where

A = 3 2 λ λ 2 , B = λ 2 δ 2 , C = λ 2 δ 2 1 + ω pi 2 Ω i 2 , D = ν id 0 2 .

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 S 2 = f 0 B + + cos ωτ ,where f 0 and ω denote the strength and frequency of the source term respectively. Then Eq. (131) is of the form,

ϕ 1 τ + A ϕ 1 ϕ 1 ζ + B 3 ϕ 1 ζ 3 + D ϕ 1 + C ζ 2 ϕ 1 ξ 2 + 2 ϕ 1 η 2 = F 0 cos ωτ E132

where F 0 = ef 0 B . 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 , n are the direction cosines of the line of wave propagation, with l 2 + m 2 + n 2 = 1 . Substituting Eqs. (133) into the Eq. (132), we get

ϕ 1 τ + Al ϕ 1 ϕ 1 ξ + Bl 3 3 ϕ 1 ξ 3 + Cl m 2 + n 2 3 ϕ 1 ξ 3 + D ϕ 1 = F 0 cos ωτ ϕ 1 τ + P ϕ 1 ϕ 1 ξ + Q 3 ϕ 1 ξ 3 + D ϕ 1 = F 0 cos ωτ E134

where, P = Al , Q = Bl 3 + Cl m 2 + n 2 ,

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

ϕ 1 = ϕ m τ sech 2 ξ M τ τ W τ E135

where ϕ m τ = 3 M τ P and W τ = 2 Q M τ , with

M τ = M 8 PF 0 16 D 2 + 9 ω 2 e 4 3 + 6 PF 0 16 D 2 + 9 ω 2 4 3 Dcos ωτ + ωsin ωτ . E136
Advertisement

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 M τ which involve the forcing term F 0 cos ωτ 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 F 0 , 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.

References

  1. 1. Russell JS. Report on waves. In: Murray J, editor. Report of the British Association for the Advancement of Science. 1944. pp. 311-390
  2. 2. Korteweg DJ, de Vries G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philosophical Magazine. 1985;39:422
  3. 3. Zabusky NJ, Kruskal MD. Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Physical Review Letters. 1965;15:240-243
  4. 4. Sen A, Tiwary S, Mishra S, Kaw P. Nonlinear wave excitations by orbiting charged space debris objects. Advances in Space Research. 2015;56(3):429
  5. 5. Chowdhury S, Mandi L, Chatterjee P. Effect of externally applied periodic force on ion acoustic waves in superthermal plasmas. Physics of Plasmas. 2018;25:042112
  6. 6. Aslanov VS, Yudintsev VV. Dynamics, analytical solutions and choice of parameters for towed space debris with flexible appendages. Advances in Space Research. 2015;55:660
  7. 7. Israwi S, Kalisch H. Approximate conservation laws in the KdV equation. Physics Letters A. 2019;383:854
  8. 8. Tsallis CJ. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics. 1988;52:479
  9. 9. Zakharov VE, Kuznetsov EA. Three-dimensional solitons. Soviet Physics JETP. 1974;39:285

Written By

Laxmikanta Mandi, Kaushik Roy and Prasanta Chatterjee

Reviewed: 12 June 2020 Published: 16 September 2020