Damped force KdV equation - Selected Topics in Plasma Physics

solitary waves. So the larger the energy, the greater the speed, larger the amplitude and narrower the width (Figure 1).

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

∂ni

∂t þ∂ðniuiÞ

∂x ¼0, (53)

∂ui

∂t þu_i∂ui

∂x ¼ �∂ϕ

∂x�ν_idu_i, (54)

∂²ϕ

∂x²¼ð1�μÞne�nþμ, (55) where n_j(j = i,e for ion, electron), u_i,ϕare the number density, ion fluid velocity and the electrostatic wave potential respectively. Hereμ¼^Z^d_nⁿ₀^d0,ν_idis the dust ion collisional frequency and the term S x, tð Þ[4, 5], is a charged density source arising from experimental conditions for a single definite purpose.

n0, Zd, nd0are the 3.1 Normalization

n_i!ni

n₀, u_i! ui

C_s,ϕ! eϕ

K_BT_e, x! x

λ_D, t!ω_pit (56)

Figure 1.

Solitary wave solution of Eq. (52) for the parameter value t¼1, u0¼0:2.

where Cs ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

KBTe

� �

is the ion acoustic speed, Teas electron temperature, KB

as Boltzmann constant, e as magnitude of electron charge and m_ias mass of ions.

λ_D ¼ _4πn^T_e0^e_e2

� �¹₂

� �

is the Debye length andω_pi�¼�⁴^πⁿ_m^e0_i^e²�¹₂�

as ion-plasma frequency.

The normalized electron density is given by

ne¼e^ϕ: (57)

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�ε³⁼²ν_id0: (58)

S�ε²S2: (59)

Substituting (31)–(34) and (58)–(59) along with stretching coordinates into Eqs. (53)–(55) and equating the coefficients of lowest order ofε, we get the phase velocity as

λ¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�μ ð Þ

p : (60)

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

∂ϕ^{ð Þ}¹

∂τ þAϕ^{ð Þ}¹ ∂ϕ^{ð Þ}¹

∂ξ þB∂³ϕ^{ð Þ}¹

∂ξ³ þCϕ^{ð Þ}¹ ¼B∂S₂

∂ξ , (61)

where A¼^3�₂λ^λ², B¼^λ₂³,C¼^ν^id0₂.

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 sech²ðξ,τÞand sech⁴ðξ,τÞfunctions) and trigonometric forcing term [6] (in the form of sinðξ,τÞand cosðξ,τÞfunctions). Motivated by these work we assume that S2is a linear function ofξsuch as S2¼ ^f_B⁰^ξcosð Þ þωτ 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,

∂ϕ^{ð Þ}¹

∂τ þAϕ^{ð Þ}¹ ∂ϕ^{ð Þ}¹

∂ξ þB∂³ϕ^{ð Þ}¹

∂ξ³ þCϕ^{ð Þ}¹1¼ f₀cosð Þ,ωτ (62) which is termed as damped and forced KdV (DFKdV) equation.

In absence of C and f₀, i.e., for C¼0 and f₀¼0 the Eq.(62) takes the form of well-known KdV equation with the solitary wave solution

ϕ₁¼ϕ_msech² ξ�Mτ W

� �

, (63)

whereϕ_m¼^3M_A and W¼2 ffiffiffiffi

q , with M as the Mach number.

In this case, it is well established that I¼

ð_∞

�∞ϕ²₁dξ, (64)

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

ϕ₁¼ϕ_mð Þτ sech² x�Mð Þτ τ Wð Þτ

� �

, (65)

where Mð Þτ is an unknown function ofτandϕ_mð Þ ¼τ ^3M_A^{ð Þ}^τ, Wð Þ ¼τ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B=Mð Þτ

p .

Differentiating Eq. (64) with respect toτand using Eq. (62), one can obtain dI

dτþ2CI¼2 f₀cosð Þωτ ð_∞

�∞ϕ₁dξ, )dI

dτþ2CI¼24 f₀ ffiffiffi pB A

ffiffiffiffiffiffiffiffiffiffiffi Mð Þτ

p cosð Þωτ :

(66)

Again,

I¼ ð_∞

�∞

ϕ²₁dξ, I¼

ð_∞

�∞

ϕ_m²ð Þτ sech⁴ ξ�Mð Þτ τ Wð Þτ

� �

dξ, I¼24 ffiffiffi

A² M³⁼²ð Þ:τ (67)

Using Eq. (66) and (67) the expression of Mð Þτ is obtained as Mð Þ ¼τ M� 8ACf₀

16C²þ9ω²

� �

e^�⁴³^Cτþ 6Af₀ 16C²þ9ω²

3Ccosð Þ þωτ ωsinð Þωτ

� �

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

ϕ₁¼ϕ_mð Þsechτ ² ξ�Mð Þτ τ Wð Þτ

� �

, (68)

whereϕ_mð Þ ¼τ ^3M_A^{ð Þ}^τ and Wð Þ ¼τ 2 ffiffiffiffiffiffiffiffi

MBð Þτ

q . The effect of the parameters, i.e., ion collision frequency parameterðν_id0Þ, 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.

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 M₀¼0:2,ω¼1,τ¼1,μ¼0:2,ν_id0¼0:01. 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 Selected Topics in Plasma Physics

frequency parameter (ν_id0). The values of other parameters are M₀¼0:2,ω¼1, τ¼1,μ¼0:2, f₀¼0:01. It is observed that the solution produces solitary waves and the amplitude of the solitary waves decreases as the value of the parameterν_id0 increases and width of the solitary waves increases for increasing value ofν_id0.

Figure 2.

Solitary wave solution of Eq. (62) in the absence of damping(ν_id0¼0) and external force( f₀¼0) with the parameter value M0¼0:2,ω¼1,τ¼1,μ¼0:2.

Figure 3.

Variation of solitary wave from Eq. (62) for the different values of f₀with M₀¼0:2,ω¼1,τ¼1,μ¼0:2, ν_id0¼0:01.

Approximate Analytical Solution of Nonlinear Evolution Equations DOI: http://dx.doi.org/10.5772/intechopen.93176

whereϕ_m¼^3M_A and W ¼2 ffiffiffiffi