John Norbury et al 2002 Nonlinearity 15 2077 doi:10.1088/0951-7715/15/6/315
John Norbury1, Juncheng Wei2 and Matthias Winter3
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We study singular patterns in a particular system of parabolic partial differential equations which consist of a Ginzburg–Landau equation and a mean field equation. We prove the existence of the three simplest concentrated periodic stationary patterns (single spikes, double spikes, double transition layers) by composing them of more elementary patterns and solving the corresponding consistency conditions. In the case of spike patterns we prove stability for sufficiently large spatial periods by first showing that the eigenvalues do not tend to zero as the period goes to infinity and then passing in the limit to a nonlocal eigenvalue problem which can be studied explicitly. For the two other patterns we show instability by using the variational characterization of eigenvalues.
02.30.Jr Partial differential equations
35Kxx Parabolic equations and systems (See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35)
Issue 6 (November 2002)
Received 13 November 2001, in final form 5 July 2002
Published 14 October 2002
John Norbury et al 2002 Nonlinearity 15 2077
C.D. Orth et al 1996 Nucl. Fusion 36 75
J H In et al 2009 Plasma Sources Sci. Technol. 18 045029
E A O Saettone et al 2003 J. Phys. D: Appl. Phys. 36 842
Carel W E van Eijk 2002 Phys. Med. Biol. 47 R85
Jan Hannig et al 2003 Metrologia 40 177
Julie Bernauer et al 2005 Phys. Biol. 2 S17
H Kobeissi and M Korek 1993 J. Phys. B: At. Mol. Opt. Phys. 26 L35
Yu Huang-Zhong and Peng Jun-Biao 2008 Chinese Phys. Lett. 25 1411
S A Bogacz 2003 J. Phys. G: Nucl. Part. Phys. 29 1723