Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-23T11:47:29.976Z Has data issue: false hasContentIssue false

Collisions of acoustic solitons and their electric fields in plasmas at critical compositions

Published online by Cambridge University Press:  17 January 2019

Frank Verheest*
Affiliation:
Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281, B–9000 Gent, Belgium School of Chemistry and Physics, University of KwaZulu-Natal, Durban 4000, South Africa
Willy A. Hereman
Affiliation:
Department of Applied Mathematics and Statistics, Colorado School of Mines, Golden, CO 80401-1887, USA
*
Email address for correspondence: [email protected]

Abstract

Acoustic solitons obtained through a reductive perturbation scheme are normally governed by a Korteweg–de Vries (KdV) equation. In multispecies plasmas at critical compositions the coefficient of the quadratic nonlinearity vanishes. Extending the analytic treatment then leads to a modified KdV (mKdV) equation, which is characterized by a cubic nonlinearity and is even in the electrostatic potential. The mKdV equation admits solitons having opposite electrostatic polarities, in contrast to KdV solitons which can only be of one polarity at a time. A Hirota formalism has been used to derive the two-soliton solution. That solution covers not only the interaction of same-polarity solitons but also the collision of compressive and rarefactive solitons. For the visualization of the solutions, the focus is on the details of the interaction region. A novel and detailed discussion is included of typical electric field signatures that are often observed in ionospheric and magnetospheric plasmas. It is argued that these signatures can be attributed to solitons and their interactions. As such, they have received little attention.

Type
Research Article
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M. J. & Clarkson, P. A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press.Google Scholar
Anco, S. C., Ngatat, N. T. & Willoughby, M. 2011 Interaction properties of complex modified Korteweg–de Vries (mKdV) solitons. Physica D 240, 13781394.Google Scholar
Bernstein, I. B., Greene, J. M. & Kruskal, M. D. 1957 Exact nonlinear plasma oscillations. Phys. Rev. 108, 546550.Google Scholar
Cairns, R. A., Mamun, A. A., Bingham, R., Boström, R., Dendy, R. O., Nairn, C. M. C. & Shukla, P. K. 1995 Electrostatic solitary structures in non-thermal plasmas. Geophys. Res. Lett. 22, 27092712.Google Scholar
Das, G. C. 1975 Ion-acoustic solitary waves in multicomponent plasmas with negative-ions. IEEE Trans. Plasma Sci. 3, 168173.Google Scholar
Das, G. C. & Tagare, S. G. 1975 Propagation of ion-acoustic waves in a multi-component plasma. Plasma Phys. 17, 10251032.Google Scholar
Drazin, P. G. & Johnson, R. S. 1989 Solitons: An Introduction. Cambridge University Press.Google Scholar
Dubinov, A. E. & Kolotkov, D. Yu. 2012 Ion-acoustic super solitary waves in dusty multispecies plasmas. IEEE Trans. Plasma Sci. 40, 14291433.Google Scholar
Franz, J. R., Kintner, P. M. & Pickett, J. S. 1998 Polar observations of coherent electric field structures. Geophys. Res. Lett. 25, 12771280.Google Scholar
Franz, J. R., Kintner, P. M., Pickett, J. S. & Chen, L.-J. 2005 Properties of small-amplitude electron phase-space holes observed by Polar. J. Geophys. Res. 110, A09212.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett. 19, 10951097.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1974 Korteweg–de Vries equations and generalizations: methods for exact solutions. Commun. Pure Appl. Maths 27, 97133.Google Scholar
Harikrishnan, A., Kakad, A. & Kakad, B. 2018a Bernstein–Greene–Kruskal theory of electron holes in superthermal space plasma. Phys. Plasmas 25, 052901.Google Scholar
Harikrishnan, A., Kakad, A. & Kakad, B. 2018b Effects of wave potential on electron holes in thermal and superthermal space plasmas. Phys. Plasmas 25, 122901.Google Scholar
Harvey, P., Durniak, C., Samsonov, D. & Morfill, G. 2010 Soliton interaction in a complex plasma. Phys. Rev. E 81, 057401.Google Scholar
Hirota, R. 1971 Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 11921194.Google Scholar
Hirota, R. 1972 Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons. J. Phys. Soc. Japan 33, 14561458.Google Scholar
Hirota, R. 2004 The Direct Methods in Soliton Theory. Cambridge University Press.Google Scholar
Hutchinson, I. H. 2017 Electron holes in phase space: what they are and why they matter. Phys. Plasmas 24, 055601.Google Scholar
Kakad, A., Kakad, B. & Omura, Y. 2017 Formation and interaction of multiple coherent phase space structures in plasma. Phys. Plasmas 24, 060704.Google Scholar
Kakad, A., Lotekar, A. & Kakad, B. 2016 First-ever model simulation of the new subclass of solitons ‘Supersolitons’ in plasma. Phys. Plasmas 23, 110702.Google Scholar
Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39, 422443.Google Scholar
Kumar, S., Kumar Tiwari, S. & Das, A. 2017 Observation of the Korteweg–de Vries soliton in molecular dynamics simulations of a dusty plasma medium. Phys. Plasmas 24, 033711.Google Scholar
Lee, N. C. 2009 Small amplitude electron-acoustic double layers and solitons in fully relativistic plasmas of two-temperature electrons. Phys. Plasmas 16, 042316.Google Scholar
Lima, J. A. S., Silva, R. Jr. & Santos, J. 2000 Plasma oscillations and nonextensive statistics. Phys. Rev. E 61, 32603263.Google Scholar
Matsumoto, H., Kojima, H., Miyatake, T., Omura, Y., Okada, M., Nagano, I. & Tsutsui, M. 1994 Electrostatic solitary waves (ESW) in the magnetotail: BEN wave forms observed by GEOTAIL. Geophys. Res. Lett. 21, 29152918.Google Scholar
McFadden, J. P., Carlson, C. W., Ergun, R. E., Mozer, F. S., Muschietti, L., Roth, I. & Moebius, E. 2003 FAST observations of ion solitary waves. J. Geophys. Res. 108 (A4), 8018 doi:10.1029/2002JA009485.Google Scholar
Nakamura, Y. & Tsukabayashi, I. 2009 Modified Korteweg–de Vries ion-acoustic solitons in a plasma. J. Plasma Phys. 34, 401415.Google Scholar
Norgren, C., André, M., Vaivads, A. & Khotyaintsev, Y. V. 2015 Slow electron phase space holes: magnetotail observations. Geophys. Res. Lett. 42, 16541661.Google Scholar
Olivier, C. P., Verheest, F. & Hereman, W. A. 2018 Collision properties of overtaking supersolitons with small amplitudes. Phys. Plasmas 25, 032309.Google Scholar
Pickett, J. S., Chen, L.-J., Kahler, S. W., Santolík, O., Gurnett, D. A., Tsurutani, B. T. & Balogh, A. 2004 Isolated electrostatic structures observed throughout the Cluster orbit: relationship to magnetic field strength. Ann. Geophys. 22, 25152523.Google Scholar
Pickett, J. S., Chen, L.-J., Mutel, R. L., Christopher, I. W., Santolík, O., Lakhina, G. S., Singh, S. V., Reddy, R. V., Gurnett, D. A., Tsurutani, B. T. et al. 2008 Furthering our understanding of electrostatic solitary waves through Cluster multispacecraft observations and theory. Adv. Space Res. 41, 16661676.Google Scholar
Sagdeev, R. Z. 1966 Cooperative phenomena and shock waves in collisionless plasmas. In Reviews of Plasma Physics (ed. Leontovich, M. A.), vol. 4, pp. 2391. Consultants Bureau.Google Scholar
Saini, N. S. & Shalini 2013 Ion acoustic solitons in a nonextensive plasma with multi-temperature electrons. Astrophys. Space Sci. 346, 155163.Google Scholar
Summers, D. & Thorne, R. M. 1991 The modified plasma dispersion function. Phys. Fluids B 3, 18351847.Google Scholar
Tagare, S. G. 1986 Effect of ion-temperature on ion-acoustic solitons in a two-ion warm plasma with adiabatic positive and negative ions and isothermal electrons. J. Plasma Phys. 36, 301312.Google Scholar
Tsallis, C. 1988 Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52, 479487.Google Scholar
Vasyliunas, V. M. 1968 A survey of low-energy electrons in the evening sector of the magnetosphere with OGO 1 and OGO 3. J. Geophys. Res. 73, 28392884.Google Scholar
Verheest, F. 1988 Ion-acoustic solitons in multi-component plasmas including negative ions at critical densities. J. Plasma Phys. 39, 7179.Google Scholar
Verheest, F. 2000 Waves in Dusty Space Plasmas, pp. 109112. Kluwer Academic.Google Scholar
Verheest, F. 2010 Nonlinear acoustic waves in nonthermal dusty or pair plasmas. Phys. Plasmas 17, 062302.Google Scholar
Verheest, F. 2015 Critical densities for KdV-like acoustic solitons in multi-ion plasmas. J. Plasma Phys. 81, 905810605.Google Scholar
Verheest, F. & Hellberg, M. A. 2015 Electrostatic supersolitons and double layers at the acoustic speed. Phys. Plasmas 22, 012301.Google Scholar
Verheest, F., Hellberg, M. A. & Hereman, W. A. 2012a Head-on collisions of electrostatic solitons in nonthermal plasmas. Phys. Rev. E 86, 036402.Google Scholar
Verheest, F., Hellberg, M. A. & Hereman, W. A. 2012b Head-on collisions of electrostatic solitons in multi-ion plasmas. Phys. Plasmas 19, 092302.Google Scholar
Verheest, F., Hellberg, M. A. & Kourakis, I. 2013 Electrostatic supersolitons in three-species plasmas. Phys. Plasmas 20, 012302.Google Scholar
Watanabe, S. 1984 Ion acoustic soliton in plasma with negative ions. J. Phys. Soc. Japan 53, 950956.Google Scholar
Zabusky, N. J. & Kruskal, M. D. 1965 Interactions of solitons in a collsionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240243.Google Scholar