Monotone increasing solutions of the Painlevé 1 equation y″ = y2 + x and their role in the stability of the plasma-sheath transition

European Journal of Applied Mathematics

Volumn 13, Issue 6, 2002, Pages 663-680

  Slemrod, M ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036906052     PISSN: 09567925     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0956792502004977     Document Type: Article

References (16)

1. Bender, C.M. & Orszag, S.A. (1978) Advanced Mathematical Methods for Scientists and Engineers McGraw Hill.
2. Dafermos, C.M. (1978) Asymptotic behavior of solutions of evolution equations. In: Crandall, M.G. (ed.), Nonlinear Evolution EqUations, pp. 103-124. Academic Press.
3. Franklin, R.N. & Ockendon, J.R. (1970) Asymptotic matching of plasma and sheath in an active low pressure discharge. J. Plasma Physics, 4, 371.
4. Godyak, V. & Sternberg, N. (1990) Smooth plasma-sheath transition in a hydrodynamic model. IEEE Trans. Plasma Sci. 18(1), 159-168.
5. Hille, E. (1968) Lectures on Ordinary Differential Equations Addison-Wesley.
6. Holmes, P. & Spence, D. (1984) On a Painlevé-type boundary value problem. Quart. J. Mechanics & Appl. Math. 37, 525-538.
7. Ince, E.L. (1944) Ordinary Differential Equations. Dover.
8. Joshi, N. & Kitaev, A.V. (2001) On Boutroux's tritronquée solutions of the first Painlevé equation. 2001 Studies in Applied Mathematics, 107, 253-291.
9. Kolmogorov, A.N. & Fomin, S.V. (1970) Introductory Real Analysis. Dover.
10. Lieberman, M.A. & Lichtenberg, A.J. (1994) Principles of Plasma Discharges and Materials Processing. Wiley.
11. Riemann, K.-U. (1991) The Bohm criterion and sheath formation. J. Physics D: Appl. Phys. 24, 493-518.
12. Riemann, K.-U. (1997) The influence of collisions on the plasma sheath transition. Physics of Plasmas, 4, 4158-4166.
13. Royden, H.L. (1968) Real Analysis. Macmillan.
14. Slemrod, M. & Sternberg, N. Quasi-neutral limit for the Euler-Poisson system. Preprint.
15. Sternberg, N. & Godyak, V. (1996) Approximation of the bounded plasma problem by the plasma and the sheath models. Physica D, 97, 498-508.
16. Su, C.H. (1967) Kinetic theory of a highly negative Langmuir probe. Proceedings VIIIth International Conference on Phenomena in Ionized Gases, Vienna, Vol. I, 569-580.