  We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+
u(1-u^2)\quad (x,t)\in \mathbb{R}\times (-\infty, 0].$$ The steady state $w(x)
=\tanh (x/\sqrt{2})$, connects, as a "transition layer" the stable phases $-1$
and $+1$. We construct a solution $u$ with any given number $k$ of transition
layers between $-1$ and $+1$. At main order they consist of $k$ time-traveling
copies of $w$ with interfaces diverging one to each other as $t\to -\infty$.
More precisely, we find $$ u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(x-\xi_j(t))
+ \frac 12 ((-1)^{k-1}- 1)\quad \hbox{as} t\to -\infty, $$ where the functions
$\xi_j(t)$ satisfy a first order Toda-type system. They are given by
$$\xi_j(t)=\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log(-t)+\gamma_{jk},\quad
j=1,...,k,$$ for certain explicit constants $\gamma_{jk}.$
