Non-stationary Solitary Wave Solution for Damped Forced Kadomtsev–Petviashvili Equation in a Magnetized Dusty Plasma with q-Nonextensive Velocity Distributed Electron

Abstract

Reductive perturbation method (RPM) is used to obtain damped forced Kadomtsev–Petviashvili (DFKP) equation for the ion acoustic waves (IAWs) in a magnetized dusty plasma comprising electrons abiding by q-nonextensive velocity distribution, in the presence of external periodic force along with a damping term. A nonstationary solitary wave solution of IAW under the influence of forcing and damping term is derived through the framework of KP equation. The influence of various plasma parameters such as electron velocity distribution parameter (q), collisional frequency ($\nu _{id0}$), initial wave velocity ($M_0$), external periodic forcing term ($f_0$) and periodicity of external force ($\omega$) etc. on solitary wave structures are studied from a numerical standpoint. Significant effects in the variation of width and amplitude of the soliton are observed due to the change of the parameters $f_0$, M and $\omega$. It is found that there is a parametric regime of $f_0$ for which the solitary structure exists in the form of a Gaussian pulse and beyond a cut-off value of $f_0$, the solitary structure collapses.

Appendix

We consider the Eq. (30) and differentiating with respect to $\tau$ we get

$$\begin{aligned} \frac{dI}{d\tau }= & {} \int \limits _{-\infty }^{\infty }2\phi _{1}\frac{ \partial \phi _{1}}{\partial Y }dY \nonumber \\= & {} -\int \limits _{-\infty }^{\infty }\left( 2\phi _{1}\left( Al\phi _{1}+\frac{Cm^{2}}{l}\right) \frac{\partial \phi _{1}}{\partial Y } + Bl^{3}\frac{\partial ^{3}\phi _{1}}{\partial Y ^{3}}\right) dY ~~~~(\text {using Eq.}\, {(28)})\nonumber \\= & {} -2Al\int \limits _{-\infty }^{\infty }{\phi _{1}}^2 \frac{\partial \phi _{1}}{\partial Y}\partial Y -2\frac{Cm^2}{l}\int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial \phi _{1}}{\partial Y }\partial Y -Bl^3\int \limits _{-\infty }^{\infty }2\phi _{1} \frac{ \partial ^{3} \phi _{1}}{\partial Y^{3} }\partial Y\nonumber \\= & {} -2AlI_1 - 2\frac{Cm^2}{l} I_2 -Bl^3 I_3. \end{aligned}$$

(40)

Now

$$\begin{aligned}&I_1= \int \limits _{-\infty }^{\infty }{\phi _{1}}^2 \frac{\partial \phi _{1}}{\partial Y }dY= \int \limits _{-\infty }^{\infty }{\phi _{1}}^3 d\phi _1 =\left[ \frac{\phi _{1}^{3}}{3}\right] _{-\infty }^{\infty } =0 ~~~~({\mathrm{Assuming} \,\,\phi _{1}\rightarrow 0\,\,\mathrm{as}\,\, Y\rightarrow \pm \infty }) \end{aligned}$$

(41)

$$\begin{aligned}&I_2= \int \limits _{-\infty }^{\infty }{\phi _{1}} \frac{\partial \phi _{1}}{\partial Y }dY= \int \limits _{-\infty }^{\infty }{\phi _{1}}^2 d\phi _1 = \left[ \frac{\phi _{1}^{2}}{2}\right] _{-\infty }^{\infty } =0 ~~~~ ({\mathrm{Assuming}\,\, \phi _{1}\rightarrow 0\,\,\mathrm{as}\,\, Y\rightarrow \pm \infty }) \end{aligned}$$

(42)

$$\begin{aligned} I_3= & {} 2\int \limits _{-\infty }^{\infty }\phi _{1} \frac{ \partial ^{3} \phi _{1}}{\partial Y^{3} }dY \nonumber \\= & {} 2\left[ \phi _{1}{\frac{\partial ^{2} \phi _{1}}{\partial Y^2}}\right] _{-\infty }^{\infty }-2\int \limits _{-\infty }^{\infty }\frac{\partial \phi _{1}}{\partial Y} \frac{\partial ^{2} \phi _{1}}{\partial Y^2} dY \nonumber \\= & {} 2.0 - \int \limits _{-\infty }^{\infty }d\bigg \{\bigg (\frac{\partial \phi _{1}}{\partial Y}\bigg )^2\bigg \}~~~~ ({\mathrm{Assuming}\,\, \mathrm{the} \,\,\mathrm{boundary} \,\,\mathrm{condition} \,\,\phi _{1}\rightarrow 0 \,\,\mathrm{as }\,\,Y\rightarrow \pm \infty }) \nonumber \\= & {} -\left[ \bigg ({\frac{\partial \phi _{1}}{\partial Y}}\bigg )^2\right] _{-\infty }^{\infty } \nonumber \\= & {} 0 ~~~~ ({\mathrm{Assuming}\,\, \mathrm{the}\,\, \mathrm{boundary}\,\, \mathrm{condition }\,\,{\frac{\partial \phi _{1}}{\partial Y}}\rightarrow 0 \,\,\mathrm{as}\,\, Y\rightarrow \pm \infty }) \end{aligned}$$

(43)

Hence Eq. (40) proves that (30) remains conserve for KdV-type equation.