Large-amplitude solitary waves in dusty plasma are investigated, taking into account the dusty particle temperature. It is shown that finite dusty temperature restricts the region for the existence of solitary waves

Mukherjee, S.

Roychoudhury, R.

English

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Scholarworks@UTRGV Univ. of Texas RioGrande Valley

University of Texas Rio Grande Valley ScholarWorks @ UTRGV Physics and Astronomy Faculty Publications and Presentations College of Sciences 1997 Large-amplitude solitary waves in finite temperature dusty plasma R. Roychoudhury Soma Mukherjee The University of Texas Rio Grande Valley Follow this and additional works at: https://scholarworks.utrgv.edu/pa_fac  Part of the Astrophysics and Astronomy Commons, and the Physics Commons Recommended Citation Roychoudhury, R. and Mukherjee, Soma, "Large-amplitude solitary waves in finite temperature dusty plasma" (1997). Physics and Astronomy Faculty Publications and Presentations. 391. https://scholarworks.utrgv.edu/pa_fac/391 This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Physics and Astronomy Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact justin.white@utrgv.edu, william.flores01@utrgv.edu. ??????????????????????????????????????????????????????????????????? ????????????????????????????????????????????????????????????????????? ?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ?????????? ????????????????????????????????????? ????????????????????? ????????????????????????????????????????? ??????????????? ??????? ?????????????????????????? ???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????Large-amplitude solitary waves in finite temperature dusty plasmaR. Roychoudhury and Soma MukherjeePhysics & Applied Mathematics Unit, Indian Statistical Institute, Calcutta—700 035, IndiaReceived 27 January 1997; accepted 7 February 1997Large-amplitude solitary waves in dusty plasma are investigated, taking into account the dustyparticle temperature. It is shown that finite dusty temperature restricts the region for the existenceof solitary waves. © 1997 American Institute of Physics. S1070-664X9701806-5Recently a lot of interest1–8 for more references see Ref.8 is being shown in the study of dusty plasmas. Dusty plas-mas occur in nature in various forms. Some experimentalstudies9–12 of dusty plasmas have been made recently. Veryrecently Mamun et al.13 showed that a dusty plasma withinertial dust fluid and Boltzmann-distributed ions admitsonly negative solitary potentials associated with nonlineardust acoustic waves. In this Brief Communication we studylarge-amplitude solitary waves in a dusty plasma, taking intoaccount the temperature of dust particles that may not benegligible. We consider a two-fluid model of dusty plasmaswith Boltzmann-distributed ions and extremely massivenegatively charged inertial dusty grains. The one-dimensional equations governing the dynamics of dustyplasma arendt n ndud0, 1udt udndx ndpdx x , 22n2 nde, 3where nd is the dusty particle density normalized by the un-perturbed dust number density n0d , ud is the dusty particlevelocity normalized by the dust acoustic speed (cd)(Ti /md)1/2, and  is the electrostatic wave potential nor-malized by Ti /ei ,Ti being the ion temperature. The time andlength are normalized by the dust plasma period,wd1md/4	n0Zd2e21/2,Zd being the dust charge, and the Debye lengthDTi/4	Zdn0de21/2,respectively.Also, Td /Ti , Td being the dust particle temperature.We also take the equation of state aspnp0 . 4To find the Sagdeev potential we make all the dependentvariables depend on a single variable xVt , V being thesoliton velocity, normalized by cd . The equations 1–3written in terms of  yield the following set of ordinary dif-ferential equations:Vdndd dd ndud0, 5Vdndd uddndd nd2 dndd dd , 6d2d2 nde, 7where we have used the relation 4 and taken p01. From5 we obtainndV/Vud, 86 givesVudud2/21  VVu 11 , 9for 1 andVudud2/2lnV/Vud , 10for 1.In general, 9 is an implicit equation for ud in terms of. However for 3 the equation can be solved, and weobtainudV1/& V223v223212V21/2. 11The Sagdeev’s pseudopotential is defined byd2d2 . 12It is to be noted that to derive 8–11 we have used thefollowing boundary conditions: ud→0, nd→1, →0 when→ . Here  can be obtained from the set of Eqs. 5–7and is given by1VudeV/Vud, 13where we have used the boundary conditions→0, ()→0. For the existence of soliton solutions thefollowing conditions must be satisfied:d2d20; 14this gives1/V10 15andm0 16and2305Phys. Plasmas 4 (6), June 1997 1070-664X/97/4(6)/2305/2/$10.00 © 1997 American Institute of Physicsm00 , 17for a compressive rarefactive soliton, where m is the am-plitude of the soliton. In Fig. 1 the pseudopotential V() is plotted against  for various values of , viz., 0.01, 0.02, and 0.03 when V , the soliton velocity, is takento be 1.04.It can be seen that for 0.02 the soliton solution doesnot exist because though V() vanishes for a positive valueof  for 0.02 it remains positive between 0 andc where V(c)0, c0. The dependence of solitonamplitude m on  is shown in Fig. 2, where m is plottedagainst  for V1.1. It is seen that the amplitude decreaseswith the increase of .Finally, we give the small-amplitude expansion ofV() and the corresponding analytical form of the solitonsolution.If one neglects term of O(5), then V() is found to beVA12/2A23/6A34/24, 18whereA11/V21, 19A23V2/V231, 20A3V21 V24 3V22 V24  31V2V2  . 21The soliton solution of the differential equation 12 is givenby2a1a2a224a1a32 cosh2  1  , 22where a1A1/2, a2A2/6, a3A3/24, and 2/A1. Itcan be noted from the condition for existence of the solitonthat A1 is negative throughout the solution region. ThereforeA20 gives compressive and A20 gives rarefactive soli-tary waves. However this holds only for small-amplitudesolitary waves.To conclude, we have shown that finite dust temperaturerestricts the region of existence of solutions for solitary wavesolutions. For example, even for 0.0001, soliton solu-tions would not exist for V1.5, whereas for 0 solitonsolutions exist for V1.58. Also, we found that compressivesoliton solutions do not exist for any value of .ACKNOWLEDGMENTOne of us RR is grateful to DST, India for partial fi-nancial support.1E. C. Whipple, T. G. Northrop, and D. A. Mendis, J. Geophys. Res. 90,7405 1985.2C. B. Dwivedi, R. S. Tiwari, V. K. Sayal, and S. R. Sharma, J. PlasmaPhys. 41, 219 1989.3N. N. Rao, P. K. Shukla, and M. Y. Yu, Planet. Space Sci. 38, 543 1990.4V. N. Tsytovitch, G. E. Morfill, R. Bingham, and V. de Angelis, Com-ments Plasma Phys. Controlled Fusion 13, 153 1990.5B. P. Pandey, K. Avinash, and C. B. Dwivedi, Phys. Rev. E 49, 55991994.6P. K. Shukla and V. P. Silin, Phys. Sci. 45, 508 1990.7P. K. Shukla and H. U. Rahman, Phys. Plasmas 3, 430 1996.8C. B. Dwivedi and B. P. Pandey, Phys. Plasmas 2, 4134 1995.9C. Cui and J. Goree, IEEE Trans. Plasma Sci. PS-22, 151 1994.10R. C. Hazelton and E. J. Yadlowsky, IEEE Trans. Plasma Sci. PS-22, 911994.11I. Alexeff and M. Pace, IEEE Trans. Plasma Sci. PS-22, 136 1994.12A. Barkan, R. L. Merlino, and N. D. Angelo, Phys. Plasmas 2, 35631995.13A. A. Mamun, R. A. Cairns, and P. K. Shukla, Phys. Plasmas 3, 7021996.FIG. 1. Plot of the pseudopotential V() against  for various values of viz. 0.01, 0.02, and 0.03. The solid line is for 0.01, the middlebroken line is for 0.02, and the top broken line is for 0.03. HereV is 1.04 and 3 for all the cases.FIG. 2. Plot of m , the soliton amplitude against , for V1.1. Here also3.2306 Phys. Plasmas, Vol. 4, No. 6, June 1997 Brief Communications

Large-Amplitude Solitary Waves in Finite Temperature Dusty Plasma

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Large-Amplitude Solitary Waves in Finite Temperature Dusty Plasma

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