  We present the first exact solution for the time dependent equations of the
macroscopic fluctuation theory (MFT) for the symmetric simple exclusion process
by combining a generalization of the canonical Cole-Hopf transformation with
the inverse scattering method. For the step initial condition with two
densities, the associated Riemann-Hilbert problem is solved to determine
exactly the optimal density profile and the response field which produce a
required fluctuation, both at initial and final times. The large deviation
function of the current is derived and coincides with the formula obtained
previously by microscopic calculations. This provides the first analytic
confirmation of the validity of the MFT for an interacting model in the time
dependent regime.
