Hidden symmetries on partially unbounded domains

Physica D 143 (2000) 226-234

Ian Melbourne

To the memory of John David Crawford


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 Rx[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 Rx[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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