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Stability analysis in order-preserving systems in the presence of symmetry

Published online by Cambridge University Press:  14 November 2011

Toshiko Ogiwara  and
Hiroshi Matano
Toshiko Ogiwara
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan
Hiroshi Matano
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo 153, Japan
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Abstract

Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 2 , 1999 , pp. 395 - 438
Copyright
Copyright © Royal Society of Edinburgh 1999

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