Auto-Backlund transformations and New Exact Soliton Solutions of KdV Equation for Nonlinear Dust Acoustic Solitary Waves in Dust Plasma with Variable Dust Charge


  • Rafat Ibrahim Mathematics Department, Faculty of Science Beni-Suef University, Egypt





The nonlinear properties of dust acoustic solitary waves in unmagnetized dust plasma consistingof negative charged dust particles , Boltzmann distributed electrons and non-thermal distributedions with variable dust charge are investigated. By using the reductive perturbation theory, a Korteweg-de Veries (KdV) equation is derived. The Sagdeev’s potential are obtained in terms ofion acoustic velocity by using the auto-Backlund (BT) transformation, the modified G'/G expansion,the sine-cosine expansion method, the sinh-coshine expansion method, and the sech-tanh expansionmethod describing the nonlinear propagation of ion-acoustic solitary waves in unmagnetized dustplasma.

Keywords: dust plasma, KdV equation, Reductive perturbation method, solitary waves, BÄ acklund transformations, Traveling wave solutions.


S. K. El-Labany , and W. F. El-Taibany, "Dust acoustic solitary waves and double layers in a dusty plasma with an arbitrary streaming ion beam", Phys. Plasmas, vol.10, (2003), pp. 989-998.

N. N. Rao, P. K. Shukla, and M. Y. Yu, "Dust-acoust waves in dusty plasmas", Planet. Space Sci., Vol.38, (1990), pp.543-546.

A. Barkan, R. L. Merlino, and N. D'Angelo, "Laboratory observation of the dust-acoustic wave mode", Phys. Plasmas, Vol.2, (1995), pp. 3563-3565.

P.K. Shukla, and A. A. Mamun," Introduction to Dusty Plasma Physics", Institute of Physics, Bristol, UK, (2002).

R. L. Merlino, A. Barkan, C. Thompson, and N. D'Angeloet,"Experiments on waves and instabilities in dusty plasmas ", Plasma Phys. Control. Fusion, Vol.39, (1997), pp. A421-429.

N. N. Rao, "Dust-acoustic KdV Solitons in Weakly Non-ideal Dusty Plasmas", Physica Scripta, Vol.75, (1998), pp.1179-1181.

J. K. Xue, and L. P. Zhang, "Non-linear waves in complex plasma", Chaos, Solitons and Fractals, Vol.32, (2007), pp.592-597.

H. Alinejad, A. A. Mamun, "Dust-acoustic solitary waves in a strongly coupled inhomogeneous dusty plasma in the presence of polarization force and effective dust-temperature", Phys. Plasmas, Vol.18, (2011), pp.073706-073712.

P. K. Shukla, and A. A. Mamun, "Solitons, shocks and vortices in dusty plasmas", New J. Phys., Vol.5, (2003), pp.17-37.

S. Bagchi, A. P. Misra, K. R. Chowdhury,and A. R. Chowdhury,"Nonlinear oscillations in a magnetized dusty plasma with two-temperature trapped ions", Chaos, Solitons and Fractals, Vol.40, (2009), pp.758-765.

B. Das, and P. Chatterjee," Large amplitude double layers in dusty plasma with non-thermal electrons and two temperature isothermal ions" Phys. Lett. A, Vol.373, (2009),pp.1144-1147.

B. S. Xie, K. F. He, and Z. Q. Huang Z Q, "Dust-acoustic solitary waves and double layers in dusty plasma with variable dust charge and two-temperature ions ", Phys. Plasmas, Vol.6, (1999), pp.3808-3817.

R. Z. Sagdeev, "Cooperative Phenomena and Shock Waves in Collisionless Plasmas", Reviews of Plasma Physics, Vol.4, (1966), pp. 23-91.

M.M. Lin, W. S. Duan, "Dust acoustic solitary waves in a dusty plasmas with nonthermal ions", Chaos, Solitons and Fractals, Vol.33, (2007), pp.1189-1196.

M. H. M. Moussa and M. Rehab, "Auto-Bӓcklund transformation and similarity reductions to the variable coefficients variant Boussinesq system" Phys Lett A., Vol.372(9),(2008), pp. 1429-1434.

M.H.M. Moussa, and M. Rehab,"Two applications of the homogeneous balance method for solving the generalized Hirota-Satsume Coupled KdV system with variable coefficient", International journal of nonlinear science, Vol.7, (2009), pp. 29-38.

T. S. Gill,N. S. Saini and H. Kaur, "The Kadomstev-Petviashvili equation in dust plasma with variable dust charge and two temperature ions", Chaos, Soliton and Fractals, Vol. 28, (2006), pp. 1106-1111.

M. M. Lin and W.S.Duan, "The Kadomstev-Petviashvili (KP), MKP, and coupled KP equations for two-ion- temperature dusty plasmas", Chaos, Soliton and Fractals, Vol. 23,(2005), pp. 929-937.

Y. Q. Yang, Y.H. Wang, Lixin and X. P. Cheng, "Riccati-type Bӓcklund transformations of nonisospectral and generalized variable-coefficient KdV equations", Chin. Phys. B, Vol.23(3), (2014) pp. 030506-030508.

A. H. Khater, R.S. Ibrahim , A. B. Shamardan and D. K. Callebaut, "Bӓcklund transformations and Painlevé analysis : Exact solutions for a Grad-Shafranov type Magnetohydrodynamic Equilibrium', IMA J. Applied Math., Vol. 58,(1997), pp.51-69.

A. H. Khater, R.S. Ibrahim , O. H. El-Kalaawy and D. K. Callebaut, "Bӓcklund Transformations and Exact Soliton Solutions for Some Nonlinear Evolution Equations of ZS/AKNS system", Chaos Solitons and Fractals, Vol.9, (1998), pp.1847-1855.

A. Khater, D. Callebaut and R. S. Ibrahim, "Bӓcklund transformations and Painlevé analysis: exact solutions for the unstable nonlinear Schödinger equation modeling electron beam plasma", Phys. Plasmas, Vol.5(2), (1998), pp. 395-400.

O. H. EL-Kalaawy and R. B. Aldenari,"Painlevé analysis, Auto-Bӓcklund transformation and new exact solutions for improved modifed KdV equation", International Journal of Applied Mathematical Research, Vol.3(3), (2014), pp. 265-272.

A. R.Shehata , "The traveling wave solutions of the perturbed nonlinear Schrödinger equation and the cubic-quintic Ginzbury Landau equation using the modified (G'/G) expansion method", Applied Mathematics and Computation, Vol. 217, (2010), pp.1-10.

S. Khaleghizadeh and M. A. Mirzazadeh, "Exact complex solutions of the generalized Zakharov equation", World Applied Sciences Journal, Vol.21,(Special Issue of Applied Math) (2013), pp. 39-45.

Sh. Javadi, M. Miri Karbasaki and M. Jani, "Traveling wave solutions of a biological reaction-convection-diffusion equation model by using (G'/G) expansion method", Communications in Numerical Analysis, Vol.2013,(2013), pp.1-5.

A. Bekir and özkan Güner," Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method", Chin. Phys. B, Vol.22(11), (2013), pp.110202-110207.

E. M. E. Zayed and M.A.S. EL-Malky , "The generalized (G'/G) - expansion method for solving nonlinear partial differential equations in mathematical physics", Int. J. Contemp. Math. Sciences, Vol.6,(2011), pp. 263-278.

Harun-Or-Roshid, "N .Rahman and M.A.Akbar, "Traveling wave solutions of nonlinear Klein-Gordon equation by extended (G'/G) -expansion method", Annals of Pure and Applied Mathematics, Vol.3,(2013), pp. 10-16.

A.I.Singh , I.Tomba and K.S. Bhamra, "Solutions of Burger type nonlinear partial differential equations using (G'/G) method, International Journal of Advance Research, IJOAR. org., Vol.3, (2013),pp. 1-6.

Z. Ayati and J. Biazar, "Application of (G'/G)-expansion method to two concert problems", International Journal of Applied Mathematical Research, Vol.2(1), (2013), pp. 49-54.

Z. T. Fu, S. K. Liu and S. D. Liu, "New transformations and new approach to find exact solutions to nonlinear equations", Phys.Lett.A, Vol. 299, (2002), pp. 507-512.

Z.Yan , "A new sine-Gordan equation expansion algorithm to investigate some special nonlinear differential equations", MMRC, AMSS, Academia Sinica, Vol. 23,(2004), pp.300-310.

L.Z.Chang , Y.T.Pan and X.M.Ma,"New Exact traveling wave solutions of Davey-Stewartson equation", Journal of Computational Information Systems, Vol. 4,(2013), pp. 1687-1693.

L.Zhong and S.Tang, "Exact explicit traveling solutions for a (2+1)-dimensional generalized KP-BBME equation", British Journal of Mathematics and Computer Science, Vol. 3,(2013), pp. 655-663.

R. S. Ibrahim and O. H. El-Kalaawy, "Traveling Wave Solution for a Modified Kortewegde Vries Equation Modeling to Nonlinear Ion-Acoustic Waves in a Collisionless Plasma", Adv. Studies Theor. Phys., Vol. 2,(2008), pp.919 - 928.

A.J. Jawad, "Soliton solutions for the Boussinesq equations", J. Math. Comput. Sci.,Vol.3, (2013), pp.254-265.

A. R.Seadawy and A. Sayed, "Traveling wave solution of two dimensional nonlinear KdV-Burgers equation", Applied Mathematical Sciences, Vol.7,(2013), pp. 3367-3377.

R. S. Ibrahim "Tanh-travelling wave solutions and reduction of variable-coefficient unstable nonlinear Schrödinger equation to a quadrature in electron beam plasma", Applied Mathematical Sciences, Vol.2, (2008),pp.2157-216.

R. S. Ibrahim and O. H. El-Kalaawy "Extended tanh-function method and reduction of nonlinear Schrödinger-type equation to a quadrature", Chaos Solitons and Fractals, Vol.31, (2007), pp.1001-1008.

K. S. Al-Ghafri, "Analytic solutions of the Thomas equation by generalized tanh and travelling wave hypothesis methods", International Journal of Applied Mathematical Research, Vol.2(2),(2013), pp.274-278.

A. A. Mamun, S. M. Russell, C. A. Mendoza-Briceno, M. N. Alam MN, Datta TK, A. K. Das, "Multi-dimensional instability of electrostatic solitary structures in magnetized nonthermal dusty plasmas", Planet Space Sci, Vol. 48, (2000), pp. 163-173.

R. A. Cairns, A. A. Mamum, R. Bingham, R. Bostrom, R. O. Dendy, C. M. C. Nairn and P. K. Shukla, "Electrostatic solitary structures in nonthermal plasmas" Geophys Res Lett., Vol.22, (1995), pp. 2709-2712.

A. A. Mamun, "Effects of ion temperature on electrostatic solitary structures in nonthermal plasmas" Phys. Rev. E, Vol.55, (1997), pp. 1852-1857.

W. S. Daun, "The Kadomtsev-Petviashvili (KP) equation of dust acoustic waves for hot dust plasmas", Chaos, Solitons and Fractals, Vol.14, (2002), pp.503-506.

R. Hirota, "Exact solution of the Korteweg de Vries equation for multiple collisions of solitons", Phy. Rev. Lett., Vol. 27,(1971), pp.1192-1194.

P. G. Drazin, and R. S. Johnson, Solitons: an Introduction, Cambridge Univ. Press, Cambridge, (1989).

M. Remoissenet, Waves Called Solitons: Concepts and experiments, Springer-Verlag, Berlin, (1996).

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