Evolution of solitons and their reflection and tra

Authors: Renu Tomar Anjali Bhatnagar Hitendra K Malik Raj P Dahiya

Publish Date: 2014/07/05

Volume: 8, Issue: 3, Pages: 138-

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Abstract

Theoretical calculations are carried out for studying soliton’s reflection and transmission in an inhomogeneous plasma comprising ions two temperature electrons and negatively charged dust grains Using reductive perturbation technique relevant modified KdV equations are derived for the incident reflected and transmitted solitons Then a coupled equation is obtained based on these mKdV equations which is solved for the reflected soliton under the use of solutions of mKdV equations corresponding to the incident and transmitted solitons Based on the ratio of amplitudes of reflected and incident solitons reflection coefficient is examined under the effect of dust grain density the same is done for the transmission coefficient which is the ratio of amplitudes of transmitted and incident solitons The transmission of the solitons becomes weaker under the effect of stronger magnetic field and higher dust density However this leads to the stronger reflection of the solitonThe solitary structure is formed from an ion acoustic wave when the effects of nonlinearity and dispersion are balanced in the plasma Washimi and Taniuti 1 were the first to derive the wellknown KortewegdeVries KdV equation with the help of reductive perturbation technique RPT to describe the soliton behavior in the homogeneous plasma However plasma contributes an extra term in the usual KdV equation 2 3 4 5 when the density inhomogeneity is taken into account and then modified KdV mKdV equation is realized Accordingly the soliton behavior is modified in the inhomogeneous plasmas There are a large number of studies on the ion acoustic solitary waves in homogeneous plasmas 6 7 8 inhomogeneous plasmas 2 9 and magnetized plasmas 10 11 12 13 The ion acoustic waves and hence the solitons are found to reflect from a density gradient or the metal surface present in the plasma There have been a lot of experimental observations concerning solitons in different plasma models 14 15 16 17 18 19 20 21 22 23 The reflection of a planar ion acoustic soliton has been studied by Nishida 19 from a finite plane boundary The soliton propagation collision and reflection have been experimentally observed by Cooney et al 21 at a sheath in a multicomponent plasma they discussed a conservation law of soliton reflection and transmission Nagasawa and Nishida 20 studied the nonlinear reflection and refraction of the soliton from a metallic electrode in a doubleplasma DP deviceMost of the experiments on soliton reflection were conducted in plasmas by neglecting the dust grains which are charged by ions and electrons and are available in laboratory plasmas or spacerelated plasmas such as in planetary rings asteroid zones cometary tails and in lower parts of Earth’s ionosphere In addition lowtemperature technological plasmas are also contaminated by highly charged dust impurities Interestingly the dust grains may acquire either negative charge or positive charge 24 25 26 but the chances are higher that the grains acquire negative charge In most general situations the temperature of all the electrons does not remain the same and two groups of electrons with lower and higher temperature are found in the plasmas 27 28 29 In the two electron temperature plasmas the characteristics of ion acoustic waves and solitons are modified 27 28 29 30 31 due to different distributions of these electronsIn the present work we investigate the soliton propagation reflection and transmission in an inhomogeneous plasma which has two temperature electrons and negatively charged dust grains To study this we derive relevant mKdV equations for the incident reflected and transmitted waves and couple them at the point of reflection Finally the coupled equation is solved for finding the reflection and transmission coefficientsIn the above equations the motion of dust grains has been neglected in view of their verylowfrequency oscillations compared to the oscillations of ions and electrons The nonisothermality is taken through the expansion of nel where the parameter b l is given by bl=1Tel/Teff/π 27 28 29 30 ΩR=ε0min00B0 and Teff is the effective temperature of the plasma given by Teff=nelo+nehoTelTehneloTeh+nehoTel in view of two temperature electrons In Eqs 1–7 all the densities are normalized by the unperturbed plasma density n00 at some arbitrary reference point say x = z = 0 space coordinates x and z by the Debye length ε0Teff/n00e21/2 ion flow velocity by the ion acoustic speed Teff/mi1/2 and time t by the inverse of ion plasma frequency ωpi=n00e2/ε0mi1/2 where mi is the mass of the ion Finally the electric potential ϕ is normalized by Teff/eTo derive mKdV equations for the incident reflected and transmitted waves in the said inhomogeneous plasma we employ stretched coordinates based on the proposal of Asano and Taniuti 32 We use the subscripts I R and T for the cases of incident reflected and transmitted waves respectively In general the angles of incidence reflection and transmission should be different However for the sake of simplicity we assume that the direction of propagation of the reflected soliton is opposite to the directions of propagation of the incident and transmitted solitons ie the transmission of the soliton is considered in the same direction as that of the incident soliton Hence the stretched coordinates for the incident wave are taken as 5 11 27 33 ξ=ε1/4xsinθ+zcosθ/λ0tη=ε3/4xsinθ+zcosθ Accordingly the stretched coordinates for the transmitted wave are ξT=ε1/4xsinθ+zcosθ/λ0TtηT=ε3/4xsinθ+zcosθ and for the reflected wave ξR=ε1/4xsinθ+zcosθ/λ0Rt ηR=ε3/4xsinθ+zcosθ The power of ε determines the order or perturbation and higher lower power of ε means the slow fast variation of the physical quantities such as density velocity and potential This has been well established that these quantities should have fast variation in time and slow variation in space for the generation of solitary waves in inhomogeneous plasmas however opposite is true in homogeneous plasmas In plasmas without nonisothermal electrons the powers of ε are 12 and 32 for the timelike and spacelike stretched coordinates respectively On the other hand these powers are 14 and 34 in the inhomogeneous plasmas having nonisothermal electrons for the evolution of solitary waves as the physical quantities should have relatively faster variation with the time and space in the plasmas having nonisothermal electrons Under this situation only the effects of nonlinearity and dispersion are balanced which leads to the evolution of the solitons The fast variation is required due to the fact that the nonisothermal electrons have lower thermal velocity and can be easily trapped by the potential of the ion acoustic waveThe velocity components vx0B and vz0B correspond to the region where transmitted wave propagates Depending upon the magnitudes the phase velocity with positive sign in the above equations corresponds to the fast mode velocity λ0F and the one for negative sign corresponds to the slow mode velocity λ0S It means the present plasma supports two types of ion acoustic waves with different phase velocitiesThis equation represents the incident soliton having peak amplitude AI≡158αUL and the width WI=gI1=16βU3 It is clear that the soliton width would be real only when the coefficient β is positive for the positive velocity shift U Our calculations infer that the fast and slow waves evolve as density hill type structures only It means the plasma supports only the compressive solitary structures

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