  The dynamics of the poles of the two--soliton solutions of the modified
Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are determined. A
consequence of this study is the existence of classes of smooth,
complex--valued solutions of this equation, defined for $-\infty < x < \infty$,
exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite
time.
