Analytic Solutions of the Schamel-KdV Equation by Using Different Methods: Application to a Dusty Space Plasma

Year 2017, Volume: 21 Issue: 1, 208 - 215, 15.04.2017

Orhan Dönmez Durmuş Dağhan

https://doi.org/10.19113/sdufbed.04724

Cited By: 2

Abstract

The wave properties in a dusty space plasma consisting of positively and negatively charged  dust as well as distributed nonisothermal electrons are investigated by using the exact traveling wave solutions of the Schamel-KdV equation. The analytic solutions are obtained by the  different types $(G'/G)$-expansion methods and direct integration. The nonlinear dynamics of ion-acoustic waves for the various values of phase speed $V_p$, plasma parameters  $\alpha$, $\sigma$, and $\sigma_d$,  and the source term $\mu$ are studied. We have observed different types of waves from the different analytic solutions obtained from the different methods. Consequently, we have found the discontinuity, shock or solitary waves. It is also concluded  that these parameters play an important role in the presence of solitary waves inside the plasma. Depending on plasma parameters, the discontinuity wave turns  into solitary wave solution for the  certain values of the phase speed and plasma parameters. Additionally, exact solutions of  the Schamel-KdV equation may also be used to understand the wave types and properties in the different plasma systems.

Keywords

Schamel-KdV equation, Dusty space plasma, Shock wave, Soliton

References

• [1] Davidson, R.C. 1972. Methods in Nonlinear Plasma Theory. Academic Press, New York, NY, USA.
• [2] Whitham, G.B. 1974. Linear and NonlinearWaves, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA.
• [3] Tagare, S.G., Chakrabarti, A. 1974. Solution of a generalized Korteweg-de Vries equation. Phys. Fluids, 17(1974), 1331.
• [4] Das, G.C., Tagare, S.G., Sarma, J. 1998. Quasipotential analysis for ion-acoustic solitary wave and double layers in plasmas. Planet. Space Sci., 46(4), 417.
• [5] Hassan, M.M. 2010. New Exact Solutions of Two Nonlinear Physical Models. Commun. Theor. Phys. (Beijing, China), 53, 596-604.
• [6] Lee, J., Sakthivel, R. 2011. Exact traveling wave solutions of the Schamel-Korteweg-de Vries equation. Reports on Mathematical Physics, 68(2), 153-161.
• [7] Schamel, H. 1972. Stationary Solitary, Snoidal and Sinusoidalion-acoustic Waves. Plasma Phys., 14, 905-924.
• [8] Schamel, H. 1973. A modified Korteweg-de Vries equation for ion-acoustic waves due to resonant electrons. Journal of Plasma Physics, 9(3), 377-387.
• [9] Daghan, D., Donmez, O., Tuna, A. 2010. Explicit solutions of the nonlinear partial differential equations. Nonlinear Anal.: Real World App., 11(3), 2152-2163.
• [10] Daghan, D., Donmez, O. 2016. Analytic solutions and parametric studies of the Schamel equation for two different ion-acoustic waves in plasmas. Submitted to the Journal.
• [11] Liu, S.K., Fu, Z.T., Liu, S.D., Zhao, Q. 2001. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A, 289, 69-74.
• [12] Fu, Z.T., Liu, S.K., Liu, S.D., Zhao, Q. 2001. New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A, 290, 72-76.
• [13] Hassan, M.M. 2004. Exact solitary wave solutions for a generalized KdVBurgers equation. Chaos, Solitons and Fractals, 19, 1201-1206.
• [14] Khater, A.H., Hassan, M.M., Temsah, R.S. 1998. Exact Solutions with Jacobi Elliptic Functions of Two Nonlinear Models for Ion-Acoustic Plasma Waves. J. Phys. Soc. Japan, 74, 1431-1435.
• [15] Khater, A.H., Hassan, M.M, Krishnan, E.V., Peng, Y.Z. 2008. Applications of elliptic functions to ionacoustic plasma waves. Eur. Phys. J. D, 50, 177-184.
• [16] Khater A.H., Hassan, M.M. 2011. Exact solutions expressible in hyperbolic and jacobi elliptic functions of some important equations of ion-acoustic waves. In Acoustic Waves-From Microdevices to Helioseismology.
• [17] Taha, W.M., Noorani, M.S.M., Hashim, I. 2013. New Exact Solutions of Ion-Acoustic Wave Equations by (G’/G)-Expansion Method. Journal of Applied Mathematics Volume, 810729.
• [18] Wang, M.L., Li, X., Zhang, J. 2008. The (G′/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolution Equations in Mathematical Phys. Phys. Lett. A., 372(4), 417-423.
• [19] Daghan, D., Yildiz, O., Toros, S. 2015. Comparison of (G′/G)−methods for finding exact solutions of the Drinfeld-Sokolov system. Mathematica Slovaca, 65(3), 607-632.
• [20] Daghan, D., Donmez, O. 2015. Investigating the effect of integration constants and various plasma parameters on the dynamics of the soliton in different physical plasmas. Phys. of Plas., 22, 072114.
• [21] Li, L-X., Wang, M.L. 2009. The (G′/G)-Expansion Method and Traveling Wave Solutions for a Higher-Order Nonlinear Schrödinger Equation. Appl. Math. and Comput., 208(2), 440-445.
• [22] Li, L-X., Li, E.Q., Wang, M.L. 2010. The (G′/G,1/G)-expansion method and its application to traveling wave solutions of the Zakharov equations. Appl. Math. A J. Chin. Univ., 25, 454-462.
• [23] Zayed, E.M.E., Abdelaziz, M.A.M. 2013. The two variables (G′/G/G)-expansion method for solving the nonlinear KdV mKdV equation. Math. Probl. Eng., 2012, 725061.
• [24] Zayed, E.M.E., Hoda Ibrahim, S.A., Abdelaziz, M.A.M. 2012. Traveling wave solutions of the nonlinear 3 + 1-dimensional Kadomtsev Petviashvili equation using the two variables (G′/G,1/G)-expansion method. J. Appl. Math., 560531.
• [25] Demiray, S., Unsal, O., Bekir, A. 2015. Exact solutions of nonlinear wave equations using (G′/G,1/G)-expansion method. Journal of the Egyptian Mathematical Society, 23(1), 78-84.
• [26] Daghan, D., Donmez, O. 2016. Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas. Brazilian Journal of Physics, 46, 321-333.
• [27] Gill, T.S., Bedi, C., Saini, N.S. 2011. Higher order nonlinear effects on wave structures in a fourcomponent dusty plasma with nonisothermal electrons. Phys. of Plas., 18, 043701.
• [28] Abdel, H.I., Tantawy, M. 2014. Exact Solutions of the Schamel-Korteweg-de Vries Equation with Time Dependent Coefficients. Information Sciences Letters, 3(3), 103-109.
• [29] Kalaawy, O.H., Ibrahim, R.S. 2008. Exact Solutions for nonlinear Propagation of slow ion acoustic monotonic double layers and a Solitary Hole in Semirelativistic Plasma. Phys. of Plas., 15, 072303.
• [30] Moslem, W.M., Sabry, R., Shukla, P.K. 2010. Three dimensional cylindrical Kadomtsev-Petviashvili equation in a very dense electronpositron-ion plasma. Phys. of Plas., 17, 032305.