  We consider the initial value problem for the generalized Allen-Cahn
equation, \[\partial_t \Phi = \Delta \Phi-\varepsilon^{-2} \Phi (\Phi^t \Phi -
I), \qquad x \in \Omega, \ t\geq 0,\] where $\Phi$ is an $n\times n$ real
matrix-valued field, $\Omega$ is a two-dimensional square with periodic
boundary conditions, and $\varepsilon > 0$. This equation is the gradient flow
for the energy, $E(\Phi) := \int \frac{1}{2} \|\nabla \Phi \|^2_F+\frac{1}{4
\varepsilon^2} \| \Phi^t \Phi - I \|^2_F$, where $\| \cdot \|_F$ denotes the
Frobenius norm. The primary contribution of this paper is to use asymptotic
methods to describe the solution of this initial value problem. If the initial
condition has single-signed determinant, at each point of the domain, at a fast
$O(\varepsilon^{-2} t)$ time scale, the solution evolves towards the closest
orthogonal matrix. Then, at the $O(t)$ time scale, the solution evolves
according to the $O_n$ diffusion equation. Stationary solutions to the $O_n$
diffusion equation are analyzed for $n=2$. If the initial condition has regions
where the determinant is positive and negative, a free interface develops. Away
from the interface, in each region, the matrix-valued field behaves as in the
single-signed determinant case. At the $O(t)$ time scale, the interface evolves
in the normal direction by curvature. At a slow $O(\varepsilon t)$ time scale,
the interface is driven by curvature and the surface diffusion of the
matrix-valued field. For $n=2$, the interface is driven by curvature and the
jump in the squared tangental derivative of the phase across the interface. In
particular, we emphasize that the interface when $n\geq 2$ is driven by surface
diffusion, while for $n=1$, the original Allen--Cahn equation, the interface is
only driven by mean curvature. A variety of numerical experiments are performed
to verify, support, and illustrate our analytical results.
