**Author details**

where *f* <sup>0</sup> and *ω* denote the strength and frequency of the source term respectively.

*∂ζ*

damped-forced ZK equation to the KdV equation. We introduce new variable:

where *l*, *m*, *n* are the direction cosines of the line of wave propagation, with

*∂*2 *ϕ*1 *∂ξ*<sup>2</sup> þ *∂*2 *ϕ*1 *∂η*<sup>2</sup>

¼ *F*<sup>0</sup> *cos*ð Þ *ωτ* (132)

(134)

(135)

*:* (136)

� �

*<sup>B</sup>* . To find the analytical solution of Eq. (132), we transform the

*ϕ*1

*<sup>∂</sup>ξ*<sup>3</sup> <sup>þ</sup> *<sup>D</sup>ϕ*<sup>1</sup> <sup>¼</sup> *<sup>F</sup>*<sup>0</sup> *cos*ð Þ *ωτ*

*W*ð Þ*τ* � �

> 4 3

, with

*ξ* ¼ ð Þ *lζ* þ *mξ* þ *nη* , (133)

*<sup>∂</sup>ξ*<sup>3</sup> <sup>þ</sup> *<sup>D</sup>ϕ*<sup>1</sup> <sup>¼</sup> *<sup>F</sup>*<sup>0</sup> *cos*ð Þ *ωτ*

*Dcos*ð Þþ *ωτ ωsin*ð Þ *ωτ* � �

Then Eq. (131) is of the form,

*Selected Topics in Plasma Physics*

where *<sup>F</sup>*<sup>0</sup> ¼ � *ef* <sup>0</sup>

*∂ϕ*<sup>1</sup> *<sup>∂</sup><sup>ζ</sup>* <sup>þ</sup> *<sup>B</sup> <sup>∂</sup>*<sup>3</sup>

*∂ϕ*<sup>1</sup> *<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *Bl*<sup>3</sup> *<sup>∂</sup>*<sup>3</sup>

where, *<sup>P</sup>* <sup>¼</sup> *Al*, *<sup>Q</sup>* <sup>¼</sup> *Bl*<sup>3</sup> <sup>þ</sup> *Cl m*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> ð Þ,

<sup>16</sup>*D*<sup>2</sup> <sup>þ</sup> <sup>9</sup>*ω*<sup>2</sup> � �

) *∂ϕ*<sup>1</sup> *<sup>∂</sup><sup>τ</sup>* <sup>þ</sup> *<sup>P</sup>ϕ*<sup>1</sup>

where *<sup>ϕ</sup>m*ð Þ¼ *<sup>τ</sup>* <sup>3</sup>*M*ð Þ*<sup>τ</sup>*

*<sup>M</sup>*ð Þ¼ *<sup>τ</sup> <sup>M</sup>* � <sup>8</sup>*PF*<sup>0</sup>

**7. Conclusions**

**130**

*ϕ*1

*<sup>∂</sup>ζ*<sup>3</sup> <sup>þ</sup> *<sup>D</sup>ϕ*<sup>1</sup> <sup>þ</sup> *<sup>C</sup> <sup>∂</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>m</sup>*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> <sup>¼</sup> 1. Substituting Eqs. (133) into the Eq. (132), we get

*<sup>∂</sup>ξ*<sup>3</sup> <sup>þ</sup> *Cl m*<sup>2</sup> <sup>þ</sup> *<sup>n</sup>*<sup>2</sup> � � *<sup>∂</sup>*<sup>3</sup>

*ϕ*1

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

*<sup>ϕ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ϕ</sup>m*ð Þ*<sup>τ</sup> sech*<sup>2</sup> *<sup>ξ</sup>* � *<sup>M</sup>*ð Þ*<sup>τ</sup> <sup>τ</sup>*

ffiffiffiffiffiffiffi *Q M*ð Þ*τ* q

6*PF*<sup>0</sup> <sup>16</sup>*D*<sup>2</sup> <sup>þ</sup> <sup>9</sup>*ω*<sup>2</sup>

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 ð Þ *M*ð Þ*τ* which involve the forcing term *F*<sup>0</sup> cosð Þ *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.

*ϕ*1

*∂ϕ*<sup>1</sup> *<sup>∂</sup><sup>ξ</sup>* <sup>þ</sup> *<sup>Q</sup> <sup>∂</sup>*<sup>3</sup>

*<sup>P</sup>* and *W*ð Þ¼ *τ* 2

*e* �4 <sup>3</sup>*D<sup>τ</sup>* <sup>þ</sup>

*∂ϕ*<sup>1</sup> *<sup>∂</sup><sup>τ</sup>* <sup>þ</sup> *<sup>A</sup>ϕ*<sup>1</sup>

*∂ϕ*<sup>1</sup>

*<sup>∂</sup><sup>τ</sup>* <sup>þ</sup> *Alϕ*<sup>1</sup>

*l*

Laxmikanta Mandi1,2† , Kaushik Roy3† and Prasanta Chatterjee<sup>1</sup> \*

```
1 Department of Mathematics, Visva-Bharati, Santiniketan, India
```
2 Department of Mathematics, Gushkara Mahavidyalaya, Guskhara, India


† These authors contributed equally.

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