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Published January 1, 2003 | public
Journal Article Open

Reduction and reconstruction for self-similar dynamical systems


We present a general method for analysing and numerically solving partial differential equations with self-similar solutions. The method employs ideas from symmetry reduction in geometric mechanics, and involves separating the dynamics on the shape space (which determines the overall shape of the solution) from those on the group space (which determines the size and scale of the solution). The method is computationally tractable as well, allowing one to compute self-similar solutions by evolving a dynamical system to a steady state, in a scaled reference frame where the self-similarity has been factored out. More generally, bifurcation techniques can be used to find self-similar solutions, and determine their behaviour as parameters in the equations are varied.

Additional Information

© Institute of Physics and IOP Publishing Limited 2003. Received 16 August 2002, in final form 3 April 2003. Published 6 May 2003. Print publication: Issue 4 (July 2003). Recommended by F Otto. This work was partially supported by the AFOSR (Dynamics and Control) (CWR, IGK) and the National Science Foundation, ITR grants ACI-0204932 (JEM) and CTS-0205484 (IGK). KL is a postdoctoral fellow of the Fund for Scientific Research—Flanders (Belgium) (FWO—Vlaanderen). We would like to acknowledge helpful discussions with Professors D G Aronson and P G Kevrekidis.


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