  We find the general solution to Polchinski's classical scattering equations
for $1+1$ dimensional string theory. This allows efficient computation of
scattering amplitudes in the standard Liouville $\times$ $c=1$ background.
Moreover, the solution leads to a mapping from a large class of time-dependent
collective field theory backgrounds to corresponding nonlinear sigma models.
Finally, we derive recursion relations between tachyon amplitudes. These may be
summarized by an infinite set of nonlinear PDE's for the partition function in
an arbitrary time-dependent background.
