Propagation of dust-ion-acoustic solitary waves for damped modified Kadomtsev–Petviashvili–Burgers equation in dusty plasma with a q-nonextensive nonthermal electron velocity distribution

Abstract

The nonlinear dust ion acoustic solitary waves (DIAW) in a magnetized collisional dusty plasma comprising with negatively charged dust grain, positively charged ions along with q-nonextensive nonthermal electrons and neutral particles in the presence of small damping force is studied analytically through the framework of damped modified Kadomtsev-Petviashvili-Burgers (DMKPB) equation. Reductive perturbation technique (RPT) is employed to derive the DMKPB equation. It is observed that there is a critical point for the plasma parameters where the amplitude of the solitary wave of damped KP Burgers equation diverges. The DMKPB equation is derived from there and the soliton like solutions with finite amplitude is extracted. The influence of various plasma parameters like entropic index, dust ion collisional frequency, ion kinematic viscosity, speed of the traveling wave and the parameter indicating the ratio between unperturbed dust ion density and electron are investigated on the propagation of dust ion acoustic wave (DIAW). A significant effect on the wave structures due to the variation of present plasma parameters has been observed. Finally, the temporal evolution of a solitary wave solution is depicted through a numerical standpoint.

Appendix

Assuming that conservation property (3.31) holds in the system we write,

$$\begin{aligned} I= & {} \int \limits _{-\infty }^{\infty }\phi _{1}^{2}d\zeta \nonumber \\= & {} \phi _{m}^{2}(\tau )\int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta \,\, ({\text {using the result}}\, (3.32))\nonumber \\= & {} 2\phi _{m}^{2}(\tau )W\left( \tau \right) \nonumber \\= & {} 2.\frac{6\left( M\left( \tau \right) -C\right) }{Al}\sqrt{\frac{Bl^{3}}{ \left( M\left( \tau \right) -C\right) }} \nonumber \\= & {} \frac{12\sqrt{Bl}}{A}\left( M\left( \tau \right) -C\right) ^{\frac{1}{2}} \end{aligned}$$

(6.1)

where

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M( \tau ) \tau }{W( \tau ) }\right) d\zeta =2W( \tau ) \end{aligned}$$

Differentiating (3.31) with respect to $\tau$ and using the Eq. (3.29) we get

$$\begin{aligned} \frac{dI}{d\tau }= & {} 2\int \limits _{-\infty }^{\infty }\phi _{1}\frac{ \partial \phi _{1}}{\partial \tau }d\zeta \nonumber \\= & {} -2A\int \limits _{-\infty }^{\infty }\phi _{1}^{3}\frac{\partial \phi _{1}}{ \partial \xi }d\zeta -2B\int \limits _{-\infty }^{\infty }\phi _{1}\frac{ \partial ^{3}\phi _{1}}{\partial \xi ^{3}}d\zeta -2E\int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial ^{2}\phi _{1}}{\partial \xi ^{2}}d\zeta \nonumber \\&-2D\int \limits _{-\infty }^{\infty }\phi _{1}^{2}d\zeta -2C\int \limits _{-\infty }^{\infty }\phi _{1}\left( \int \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}d\xi \right) d\zeta \end{aligned}$$

(6.2)

From (3.32), we have

$$\begin{aligned} \phi _{1}=\phi _{m}\left( \tau \right) \sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \end{aligned}$$

So

$$\begin{aligned} \frac{\partial \phi _{1}}{\partial \xi }=-\frac{l\phi _{m}\left( \tau \right) }{W\left( \tau \right) }\sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \end{aligned}$$

Therefore

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\phi _{1}^{3}\frac{\partial \phi _{1}}{ \partial \xi }d\zeta= & {} -\frac{l\phi _{m}^{4}\left( \tau \right) }{W\left( \tau \right) }\int \limits _{-\infty }^{\infty }\sec h^{4}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta \nonumber \\= & {} 0 \end{aligned}$$

(6.3)

Similarly

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial ^{3}\phi _{1}}{ \partial \xi ^{3}}d\zeta =0 \end{aligned}$$

(6.4)

Using (6.3) and (6.4), we have from (6.2),

$$\begin{aligned} \frac{dI}{d\tau }= & {} -2E\int \limits _{-\infty }^{\infty }\phi _{1}\frac{ \partial ^{2}\phi _{1}}{\partial \xi ^{2}}d\zeta -2D\int \limits _{-\infty }^{\infty }\phi _{1}^{2}d\zeta -2C\int \limits _{-\infty }^{\infty }\phi _{1}\left( \int \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}d\xi \right) d\zeta \;\, or \nonumber \\ \frac{dI}{d\tau }+ 2DI= & {} -2E\int \limits _{-\infty }^{\infty }\phi _{1}\frac{ \partial ^{2}\phi _{1}}{\partial \xi ^{2}}d\zeta -2C\int \limits _{-\infty }^{\infty }\phi _{1}\left( \int \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}d\xi \right) d\zeta \end{aligned}$$

(6.5)

Again

$$\begin{aligned} \frac{\partial ^{2}\phi _{1}}{\partial \xi ^{2}}=\frac{l^{2}\phi _{m}\left( \tau \right) }{W^{2}\left( \tau \right) }\left\{ \sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{ \zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) -\sec h^{3}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) \right\} \end{aligned}$$

and

$$\begin{aligned} \phi _{1}\frac{\partial ^{2}\phi _{1}}{\partial \xi ^{2}}=\frac{l^{2}\phi _{m}^{2}\left( \tau \right) }{W^{2}\left( \tau \right) }\left\{ \sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) -\sec h^{4}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \right\} \end{aligned}$$

Therefore

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\phi _{1}\frac{\partial ^{2}\phi _{1}}{ \partial \xi ^{2}}d\zeta= & {} \frac{l^{2}\phi _{m}^{2}\left( \tau \right) }{ W^{2}\left( \tau \right) }\int \limits _{-\infty }^{\infty }\{\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) -\sec h^{4}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \}d\zeta \nonumber \\= & {} \frac{l^{2}\phi _{m}^{2}\left( \tau \right) }{W^{2}\left( \tau \right) } \left\{ 2\int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta -\int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta \right\} \nonumber \\= & {} -\frac{2l^{2}}{3}\frac{\phi _{m}^{2}\left( \tau \right) }{W\left( \tau \right) } \nonumber \\= & {} -\frac{2l^{2}}{3}\frac{6\left( M\left( \tau \right) -C\right) }{Al}\sqrt{ \frac{\left( M\left( \tau \right) -C\right) }{Bl^{3}}} \nonumber \\= & {} -\frac{4}{A\sqrt{Bl}}\left( M\left( \tau \right) -C\right) ^{\frac{3}{2}} \end{aligned}$$

(6.6)

where

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta =\frac{2}{3 }W\left( \tau \right) \end{aligned}$$

and

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta =2W\left( \tau \right) \end{aligned}$$

Again

$$\begin{aligned} \frac{\partial \phi _{1}}{\partial \chi }= & {} -\frac{m\phi _{m}\left( \tau \right) }{W\left( \tau \right) }\sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \\ \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}= & {} \frac{m^{2}\phi _{m}\left( \tau \right) }{W^{2}\left( \tau \right) }\left\{ \sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \tan h^{2}\left( \frac{ \zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) -\sec h^{3}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) \right\} \\ \int \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}d\xi= & {} \frac{ m^{2}\phi _{m}\left( \tau \right) }{W^{2}\left( \tau \right) }\int \left\{ \sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) \tan h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) -\sec h^{3}\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \right\} d\xi \\= & {} \frac{m^{2}\phi _{m}\left( \tau \right) }{lW\left( \tau \right) }\left\{ -\sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) \tan h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) +F\left( \chi ,\tau \right) \right\} \end{aligned}$$

Therefore

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\phi _{1}\left( \int \frac{\partial ^{2}\phi _{1}}{\partial \chi ^{2}}d\xi \right) d\zeta= & {} \frac{m^{2}\phi _{m}^{2}\left( \tau \right) }{lW\left( \tau \right) }\int \limits _{-\infty }^{\infty }\left\{ -\sec h^{2}\left( \frac{\zeta -M\left( \tau \right) \tau }{ W\left( \tau \right) }\right) \tan h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) \right. \nonumber \\&\left. +F\left( \chi ,\tau \right) \sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) } \right) \right\} d\zeta \nonumber \\= & {} \frac{m^{2}\phi _{m}^{2}\left( \tau \right) }{lW\left( \tau \right) } F\left( \chi ,\tau \right) \int \limits _{-\infty }^{\infty }\sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta \nonumber \\= & {} \frac{\pi m^{2}\phi _{m}^{2}\left( \tau \right) }{l}F\left( \chi ,\tau \right) \end{aligned}$$

(6.7)

where

$$\begin{aligned} \int \limits _{-\infty }^{\infty }\sec h\left( \frac{\zeta -M\left( \tau \right) \tau }{W\left( \tau \right) }\right) d\zeta =\pi W\left( \tau \right) \end{aligned}$$

Using (6.6) and (6.7), we have from (6.5)

$$\begin{aligned} \frac{dI}{d\tau }+2DI=\frac{8E}{A\sqrt{Bl}}\left( M\left( \tau \right) -C\right) ^{\frac{3}{2}}-\frac{2C\pi m^{2}}{l}\phi _{m}^{2}( \tau )F( \chi ,\tau ) \end{aligned}$$

Combining the equations (3.35) and (3.36) we finally get

$$\begin{aligned} \frac{6\sqrt{Bl}}{A}\left( M\left( \tau \right) -C\right) ^{-\frac{1}{2}} \frac{dM\left( \tau \right) }{d\tau }+2D\frac{12\sqrt{Bl}}{A}\left( M\left( \tau \right) -C\right) ^{\frac{1}{2}}= & {} \frac{8E}{A\sqrt{Bl}}\left( M\left( \tau \right) -C\right) ^{\frac{3}{2}}\nonumber \\ \text {or,}\,\,\frac{dM\left( \tau \right) }{d\tau }+4D\left( M\left( \tau \right) -C\right)= & {} \frac{4E}{3Bl}\left( M\left( \tau \right) -C\right) ^{2}\nonumber \\ \text {i.e.,}\,\,\frac{d\left( M\left( \tau \right) -C\right) }{d\tau }+4D\left( M\left( \tau \right) -C\right)= & {} \frac{4E}{3Bl}\left( M\left( \tau \right) -C\right) ^{2} \end{aligned}$$

(6.8)