p}^{(f)}$ at late time. In these asymptotic regions the blow-up moduli $z_{(\t ,n,a)}$ are constant while they move around the time $x\simeq 1$ to facilitate the transition between the two free solutions.

One can now compute the relative volume $\e_{(\t ,a)}$ of the blow-ups by inserting the solution  into Eq. . As before, we find that the required condition $\e_{(\t ,a)}\ll 1$ is satisfied only in two finite time windows at $x\ll 1$ and $x\gg 1$. At the endpoints of these windows the volume of the blow-ups becomes sizeable and control over the approximation is lost.

More complicated cases
======================

An obvious generalisation of the previous example is to consider a situation that corresponds to an $SU(2)^n$ Toda model, for some integer $n$. This amounts to having a subset of blow-up moduli evolve non-trivially for which the associated characteristic vectors ${\bf P}_i$ are orthogonal, that is, $${\bf P}_i\cdot {\bf P}_j = 2\d_{ij}\; . \label{ortho}$$ Inspection of Table 1 shows such cases can indeed be realised. For example, allowing the moduli of a certain type $\t$ to evolve non-trivially, while moduli of the other two types are being kept constant by setting $\z_{\t ',n,a}=0$ in Eq.  for $\t
'\neq\t$, leads to 