  We use algebraic Backlund transformations (BTs) to construct explicit
solutions of the modified 2+1 chiral model from $T^2\times R$ to SU(n), where
$T^2$ is a 2-torus. Algebraic BTs are parameterized by $z\in C$ (poles) and
holomorphic maps $\pi$ from $T^2$ to Gr$(k,C^n)$. We apply B\"acklund
transformations with carefully chosen poles and $\pi$'s to construct infinitely
many solutions of the 2+1 chiral model that are (i) doubly periodic in space
variables and periodic in time, i.e., triply periodic, (ii) homoclinic in the
sense that the solution $u$ has the same stationary limit $u_0$ as $t\to
\pm\infty$ and is tangent to a stable linear mode of $u_0$ as $t\to\infty$ and
is tangent to an unstable mode of $u_0$ as $t\to -\infty$.
