Solitary Wave Solution of Higher order Kadomstev-Petviashvili Equation for Complex Plasma
Apul Narayan Dev1* and Jnanjyoti Sarma2
1Department of Science and Humanities, College of Science and Technology, RUB, Bhutan.
2Department of Mathematics, R. G. Baruah College, Guwahati-781025, Assam.
*Corresponding Author E-mail: apul@cst.edu.bt
ABSTRACT:
The nonlinear wave phenomenon leading to the formation of coherent structures through solitary waves are studied in Kadomstev-Petviashvili (K-P) equation and modified K-P equations with the help of reductive perturbation technique. The K-P equation for complex plasma consisting of non-isothermal electrons and Maxwellian ions in cold dust with the effect of negative (positive) dust charge fluctuation is derived. Similarly different type mK-P equations for complex plasma consisting of trapped electrons and Maxwellian ions in cold dust with the effect of negative (positive) dust charge fluctuation are also derived by same reductive perturbation technique. The analytical solution of K-P equation and other higher order nonlinear wave equation such as mK-P equation are derived by well known tanh-method and dust acoustic solitary waves and shock waves propagation for different situation are reported, which could be relevant in case of different space and astrophysical plasmas including Saturn’s F-ring and also in laboratory observation for both positive and negative dust charge fluctuation.
KEYWORDS: Solitary wave, K-P equation, Reductive perturbation technique, tanh-method.
I. INTRODUCTION:
The integrability of nonlinear partial differential equation with higher order nonlinearity as well as variable coefficient of nonlinearity [1] has important effect in the field of differential equation. There are many analytical approaches to solve the nonlinear partial differential equation such as tanh method, sine-cosine method, Painleve analysis, Lie group analysis etc. The nonlinear partial differential equations like Korteweg-deVries (K-dV) equation, K-P equation, Zakharov-Kuznetsov(Z-K) equation, Burgers equation and K-dV-Burgers equation and many other are arise in fluid dynamics, magnetohydrodynamics (MHD) and plasma physics etc. Nonlinear localized structures like solitons and shock waves are formed in different type of plasma model due to the balance between nonlinearity, wave dispersion and dissipation. When the effect of dissipation is negligible compare to nonlinearity and dispersive, the system can be supportive for soliton. Meanwhile, when the effect of dissipation is dominant, the shock waves are formed. K-P [2] equation is the three dimensional form of K-dV equation which gives the solitary waves [3] solution. The modified K-P equation containing a square root nonlinearity for ion-acoustic waves in a multispecies plasma consisting of non-isothermal electron have been derive by Chakraborty and Das [4]. Duan [5] studied dust acoustic waves in a unmagnetized plasma in two dimensional geometry in the framework of the K-P equation. In this paper, we consider the cold dusty plasma or complex plasma with dust charge fluctuation. In section II, basic equation are written and using reductive perturbation method, we derive K-P equation in three dimension in presence of Maxwellian ions and non-thermal electron and solved with the help of well known tanh method, in section III, we derive different type of mK-P equation for non-isothermal electrons and solve, in section IV. Conclusions and figures are given in section V.
II. BASIC EQUATION:
We consider a collision-less unmagnetized plasma with non-thermal electrons and Maxwellian ions contaminated by massive dust particle. The basic equations, governing the dynamical system, are the equation of continuity and
momentum for the dust plasma are as follows [6].
CONCLUSION:
By considering dust charge fluctuation we derived K-P equation using the
RPT in presence of Maxwellian ions, non-thermal electrons. The soliton solution
was obtained using the tanh-method. It was shown that, the numbers of
non-thermal electrons (
) increases the small amplitude of
the solitary wave’s decreases. Using the similar technique we derived different
type of mK-P equation in presence of Maxwellian ions and non-isothermal electrons
and solve it. It was found that non-isothermal electrons are responsible for
dust acoustic solitary wave for large amplitude. It also clears that, for
, as 
increases
the amplitude of the solitary waves decreases. Similarly 
increases the amplitude of the solitary
waves decreases
ACKNOWLEDGEMENT:
One of the authors (A. N. D.) is gratefully acknowledged the College of Science and Technology, RUB, Bhutan, for financial support.
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