##
Hidden symmetries on partially unbounded domains

* Physica D * ** 143 ** (2000) 226-234

** Ian Melbourne **

To the memory of John David Crawford

** Abstract **

Systems of reaction-diffusion equations
posed on bounded rectangular domains with Neumann boundary conditions
often exhibit behavior that seems degenerate given
the physical symmetries of the problem. It is now well-understood
that Neumann boundary conditions lead to hidden symmetries that
are responsible for subtle changes in the generic bifurcations
of such systems.

In this article, we consider the analogous situation for
partially unbounded domains such as the strip **R**x[0,\pi].
We show that hidden symmetries due to assuming Neumann boundary conditions
have remarkable consequences for
the validity of Ginzburg-Landau equations which govern the local bifurcations.
A single Ginzburg-Landau equation (which is universal for general boundary
conditions on **R**x[0,\pi]) no longer suffices in general.
Instead, it is necessary to consider p coupled Ginzburg-Landau equations,
where p is an arbitrary positive integer.

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