Abstract
The results from the last chapter are extended to the three-dimensional situation. First, the three-dimensional quantum Zakharov system is derived from a two-time scales analysis. A Lagrangian formalism and the associated conservation laws are written down. Restricting to the adiabatic and semiclassical case, the system reduces to a quantum vector nonlinear Schrödinger equation (QVNLS) for the envelope electric field. A Lagrangian formalism for this QVNLS equation is used to investigate the behavior of Gaussian shaped solutions (Langmuir wave packets), by means of a time-dependent variational method. Quantum corrections are shown to prevent the collapse of Langmuir wave packets, in both two and three spatial dimensions. The conservation laws of the QVNLS equation are discussed. Finally, we discuss the oscillations of the width of the Langmuir wave packets, as a result from the interplay between classical refraction and quantum diffraction.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Akimoto, K., Rowland, H. L. and Papadopoulos, K.: Electromagnetic-radiation from strong Langmuir turbulence. Phys. Fluids 31, 2185 (1988)
Alinejad, H., Robinson, P. A., Cairns, I. H., Skjaeraasen, O. and Sobhanian, C.: Structure of Langmuir and electromagnetic collapsing wave packets in two-dimensional strong plasma turbulence. Phys. Plasmas 14, 082304–082314 (2007)
Cheung, P. Y. and Wong, A. Y.: Nonlinear evolution of electron-beam plasma interactions. Phys. Fluids 18, 1538–1548 (1985)
Dubois, D. F., Hanssen, A., Rose, H. A. and Russel, D.: Space and time distribution of HF excited Langmuir turbulence in the ionosphere – comparison of theory and experiment. J. Geophys. Res. 98, 17543–17567 (1993)
Garcia, L. G., Haas, F., Oliveira, L. P. L. and Goedert, J.: Modified Zakharov equations for plasmas with a quantum correction. Phys. Plasmas 12, 012302–012310 (2005)
Gibbons, J., Thornhill, S. G., Wardrop, M. J. and ter Haar, D. H.: On the theory of Langmuir solitons. J. Plasma Phys. 17, 153–170 (1977)
Goldman, M. V.: Strong turbulence of plasma waves. Rev. Mod. Phys. 56, 709–735 (1984)
Haas, F.: Variational approach for the quantum Zakharov system. Phys. Plasmas 14, 042309–042315 (2007)
Haas, F. and Shukla, P. K.: Quantum and classical dynamics of Langmuir wave packets. Phys. Rev. 79, 066402–066501 (2009)
Landman, M., Papanicolaou, G. C., Sulem, C., Sulem, P. L. and Wang, X. P.: Stability of isotropic self-similar dynamics for scalar-wave collapse. Phys. Rev. A 46, 7869–7876 (1992)
Li, L. H. and Li, X. Q.: Three-wave interactions in strong Langmuir turbulence. Phys. Fluids B 5, 3819–3822 (1993)
Malomed, B., Anderson, D., Lisak, M., Quiroga-Teixeiro, M. L. and Stenflo, L.: Dynamics of solitary waves in the Zakharov model equations. Phys. Rev. E 55, 962–968 (1997)
Marklund, M., Brodin, G. and Stenflo, L.: Electromagnetic wave collapse in a radiation background. Phys. Rev. Lett. 91, 163601–163605 (2003)
Papanicolaou, G. C., Sulem, C., Sulem, P. L. and Wang, X. P.: Singular solutions of the Zakharov equations for Langmuir turbulence. Phys. Fluids B 3, 969–980 (1991)
Pelletier, G., Sol, H. and Asseo, E.: Magnetized Langmuir wave-packets excited by a strong beam-plasma interaction. Phys. Rev. A 38, 2552–2563 (1988)
Robinson, P. A. and Newman, D. H.: Strong Langmuir turbulence generated by electron-beams – electric-field distributions and electron-scattering. Phys. Fluids B 2, 3120–3133 (1990)
Simpson, G., Sulem, C. and Sulem, P. L.: Arrest of Langmuir wave collapse by quantum effects. Phys. Rev. E 80, 056405–056414 (2009)
Stenflo, L.: Upper-hybrid wave collapse. Phys. Rev. Lett. 48, 1441–1441 (1982)
Thornhill, S. G. and ter Haar, D.: Langmuir turbulence and modulational instability. Phys. Rep. 43, 43–99 (1978)
Zakharov, V. E.: Collapse of Langmuir waves. Sov. Phys. J. Exp. Theor. Phys. 35, 908–914 (1972)
Zakharov, V. E., Mastryukov, A. F. and Sinakh, V. S.: 2-dimensional collapse of Langmuir waves. J. Exp. Theor. Phys. Letters 20, 3–4 (1974)
Zakharov, V. E.: Handbook of Plasma Physics, vol. 2, eds. M. N. Rosenbluth and R. Z. Sagdeev. Elsevier, New York (1984)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Haas, F. (2011). The Three-Dimensional Quantum Zakharov System. In: Quantum Plasmas. Springer Series on Atomic, Optical, and Plasma Physics, vol 65. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8201-8_8
Download citation
DOI: https://doi.org/10.1007/978-1-4419-8201-8_8
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-8200-1
Online ISBN: 978-1-4419-8201-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)