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Stability of obliquely propagating plane solitons of the Zakharov–Kuznetsov equation

Published online by Cambridge University Press:  13 March 2009

M. A. Allen  and
G. Rowlands
M. A. Allen
Affiliation:
Department of Physics, University of Warwick, Coventry CV4 7AL, U.K.
G. Rowlands
Affiliation:
Department of Physics, University of Warwick, Coventry CV4 7AL, U.K.
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Abstract

By determining to first order the growth rate of a small, long-wavelength, perturbation to a Zakharov–Kuznetsov plane soliton moving at an angle α to the magnetic field, it has been found that such solitons are unstable for α α0( ∼ 38 °). To determine the stability for angles greater than α0, one needs the growth rate to higher order. The conventional approach generates a second-order growth rate that is singular at α = α0. We rigorously obtain an expression that is bounded at this point, by developing a method in which exponentially secular terms that arise are regrouped before their subsequent elimination. We then show that these solitons are unstable for all α, although the growth rate is small for α>α and goes to zero as α→½π. The relevant linearized equation is solved numerically, and excellent agreement between analytical and numerical results is obtained.

Type
Research Article
Information
Journal of Plasma Physics , Volume 53 , Issue 1 , February 1995 , pp. 63 - 73
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Allen, M. A. & Rowlands, G. 1993 J. Plasma Phys. 50, 413.CrossRefGoogle Scholar
Das, K. P. & Verheest, F. 1989 J. Plasma Phys. 41, 139.CrossRefGoogle Scholar
Frycz, P., Infeld, E. & Samson, J. C. 1992 Phys. Rev. Lett. 69, 1057.CrossRefGoogle Scholar
Kuznetsov, E. A., Spector, M. D. & Fal'kovich, G. E. 1984 Physica D 10, 379.Google Scholar
Laedke, E. W. & Spatschek, K. H. 1982 J. Plasma Phys. 28, 469.CrossRefGoogle Scholar
Press, W. H., Flannery, B. P., Teukolsky, S. A. & Vetterling, W. T. 1992 Numerical Recipes in C, 2nd edn., p. 757. Cambridge University Press.Google Scholar
Shivamoggi, B. K., Rollins, D. K. & Fanjul, R., 1993 Physica Scripta 47, 15.CrossRefGoogle Scholar
Wolfram, S. 1991 Mathematica. Addison-Wesley.Google Scholar
Zakharov, V. E. & Kuznetsov, E. A. 1974 Soviet Phys. JETP 39, 285.Google Scholar