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Snakes, ladders, and isolas of localized patterns

Beck, M, Knobloch, J, Lloyd, DJB, Sandstede, B and Wagenknecht, T (2009) Snakes, ladders, and isolas of localized patterns SIAM Journal on Mathematical Analysis, 41 (3). 936 - 972. ISSN 0036-1410

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Abstract

Stable localized roll structures have been observed in many physical problems and model equations, notably in the 1D Swift–Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two “snaking” solution branches, so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many “ladder” branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction.

Item Type: Article
Additional Information: Copyright 2009 Society for Industrial and Applied Mathematics.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Symplectic Elements
Date Deposited: 16 Mar 2012 10:46
Last Modified: 23 Sep 2013 19:18
URI: http://epubs.surrey.ac.uk/id/eprint/294642

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