  We present the first detailed numerical study of the Toda equations in $2+1$
dimensions in the limit of long wavelengths, both for the hyperbolic and
elliptic case. We first study the formal dispersionless limit of the Toda
equations and solve initial value problems for the resulting system up to the
point of gradient catastrophe.
  It is shown that the break-up of the solution in the hyperbolic case is
similar to the shock formation in the Hopf equation, a $1+1$ dimensional
singularity. In the elliptic case, it is found that the break-up is given by a
cusp as for the semiclassical system of the focusing nonlinear Schr\"odinger
equation in $1+1$ dimensions. The full Toda system is then studied for finite
small values of the dispersion parameter $\epsilon$ in the vicinity of the
shocks of the dispersionless Toda equations. We determine the scaling in
$\epsilon$ of the difference between the Toda solution for small $\epsilon$ and
the singular solution of the dispersionless Toda system. In the hyperbolic
case, the same scaling proportional to $\epsilon^{2/7}$ is found as in the
small dispersion limit of the Korteweg-de Vries and the defocusing nonlinear
Schr\"odinger equations. In the elliptic case, we obtain the same scaling
proportional to $\epsilon^{2/5}$ as in the semiclassical limit for the focusing
nonlinear Schr\"odinger equation. We also study the formation of dispersive
shocks for times much larger than the break-up time in the hyperbolic case. In
the elliptic case, an $L_{\infty}$ blow-up is observed instead of a dispersive
shock for finite times greater than the break-up time. The
$\epsilon$-dependence of the blow-up time is determined.
