k_a\t +\t_a \\
 \r_i &=& \sqrt{2}\ln\cosh (y_i)
          -\frac{1}{\sqrt{2}}\ln\left(\frac{16\z_i^2}{k_i^2}\right)\; ,\end{aligned}$$ where $\t_A$ and $k_A$ are constants and $$y_i = \frac{k_i}{\sqrt{2}}(\t -\t_i)$$ are rescaled time coordinates. The Hamiltonian constraint in  amounts to the condition $$\sum_Ak_A^2 = 4E\; .$$ As expected the modes $\r_a$ evolve freely. Each of the other modes $\r_i$ evolves similarly to what has been found for the single $SU(2)$ Toda model in the previous section. This can be seen explicitly by converting the above solution to proper time using  and comparing with Eq. . This means that each mode interpolates between two regions of simple power-law evolution at early and late time. The blow-up moduli $z_i$ are constant in these asymptotic regions and their evolution at intermediate times facilitates the transition. This can be seen explicitly from their solution $$z_i=\frac{d_i}{2}\left(\tanh (y_i)+1\right)+z_{0i}\; ,$$ with $z_{0i}$ and $d_i$ constants representing the value of the modulus at early time and its total change respectively. This solution is obtained by inserting the above solution for $\bph$ into Eq. .

From Eq.  one finds the relative volume $$\e_i = \frac{\sqrt{2}}{3}\left(\frac{z_{1i}}{d_i}e^{y_i}+\frac{z_{0i}}{d_i}
        e^{-y_i}\right)^2$$ of the blow-ups. As in all previous cases, $\e_i$ can only be kept small, $\e_i\ll 1$, for a finite proper time at either 