Influence of External Periodic Force On Ion Acoustic Waves in a Magnetized Dusty Plasma Through Forced KP Equation and Modified Forced KP Equation

Abstract

The propagation properties of ion acoustic waves (IAW) have been studied in magnetized collisional dusty plasmas consisted by cold ions, charged dust and q-nonextensive velocity distributed electrons, in the presence of external periodic force within the framework of Kadomtsev–Petviashvili (KP) equation and modified Kadomtsev–Petviashvili (MKP) equation. The well-known reductive perturbation technique (RPT) has been utilized to derive forced Kadomtsev–Petviashvili (FKP) equation as well as modified forced Kadomtsev–Petviashvili (MFKP) equation from basic governing equation. Approximate analytical solutions for both the equations are explored, and it is found that positive potential compressive solitary wave solution as well as negative potential rarefactive soliton exists for both the models. It is found that external force plays an effective role for discretization of solitary structures in the same direction for FKP models, but the most remarkable fact is that a positive potential soliton of MFKP model is scattered alternatively in different directions under the application of external forces. Effect of several plasma parameters like initial wave velocity ($\mu$), force wave frequency ($\omega$), nonextensive parameter (q) and the forcing coefficient ($f_0$) on the wave propagation is demonstrated elaborately through numerical analysis. Finally, the effect of forcing term in FKP and MFKP models is discussed in a comparative view.

Appendix

We consider Equation (29), and differentiating with respect to $\tau$ we get

$$\begin{aligned} \frac{dI}{d\tau }= & {} \int \limits _{-\infty }^{\infty }2\phi _{1}\frac{ \partial \phi _{1}}{\partial Y }dY \nonumber \\= & {} -\int \limits _{-\infty }^{\infty }\left( 2\phi _{1}\left( Al\phi _{1}+\frac{Cm^{2}}{l}\right) \frac{\partial \phi _{1}}{\partial Y } + Bl^{3}\frac{\partial ^{3}\phi _{1}}{\partial Y ^{3}}\right) dY \;\;(\text {using equation }27)\nonumber \\= & {} -2Al\int \limits _{-\infty }^{\infty }{\phi _{1}}^2 \frac{\partial \phi _{1}}{\partial Y }dY -2\frac{Cm^2}{l}\int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial \phi _{1}}{\partial Y }dY -Bl^3\int \limits _{-\infty }^{\infty }2\phi _{1} \frac{ \partial ^{3} \phi _{1}}{\partial Y^{3} }dY\nonumber \\= & {} -2AlI_1 - 2\frac{Cm^2}{l} I_2 -Bl^3 I_3. \end{aligned}$$

(59)

Now

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

(60)

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

(61)

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

(62)

Hence Equation (59) shows that Equation (27) remains conserve for modified KdV-type equation. Now consider

$$\begin{aligned} I_4= \int \limits _{-\infty }^{\infty }{\phi _{1}}^3 \frac{\partial \phi _{1}}{\partial Y}dY= \int \limits _{-\infty }^{\infty }{\phi _{1}}^3 d\phi _1 =\left[ \frac{\phi _{1}^{}}{4}\right] _{-\infty }^{\infty } =0 ~~~~(\text {Assuming }\phi _{1}\rightarrow 0\text { as }Y\rightarrow \pm \infty ) \end{aligned}$$

(63)

To prove the conservation property of modified KdV-type equation, we consider again

$$\begin{aligned} \frac{dI}{d\tau }= & {} \int \limits _{-\infty }^{\infty }2\phi _{1}\frac{ \partial \phi _{1}}{\partial Y }dY \nonumber \\= & {} -\int \limits _{-\infty }^{\infty }\left( 2\phi _{1}\left( A_{1}l{\phi _{1}}^2+\frac{C_{1}m^{2}}{l}\right) \frac{\partial \phi _{1}}{\partial Y } + B_{1}l^{3}\frac{\partial ^{3}\phi _{1}}{\partial Y ^{3}}\right) dY ~~~~(\text {using equation }55)\nonumber \\= & {} -2A_{1}l\int \limits _{-\infty }^{\infty }{\phi _{1}}^3 \frac{\partial \phi _{1}}{\partial Y }dY -2\frac{C_{1}m^2}{l}\int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial \phi _{1}}{\partial Y }dY -B_{1}l^3\int \limits _{-\infty }^{\infty }2\phi _{1} \frac{ \partial ^{3} \phi _{1}}{\partial Y^{3} }dY\nonumber \\= & {} -2A_{1}lI_4 - 2\frac{Cm^2}{l} I_2 -Bl^3 I_3 \nonumber \\= & {} 0. ~~~~(\text {using the values of } I_2, I_3 \text { and }I_4.) \end{aligned}$$

(64)

This also proves that Equation (55) remains conserve for modified KdV-type equation.