  We solve the large deviations of the Kardar-Parisi-Zhang (KPZ) equation in
one dimension at short time by introducing an approach which combines field
theoretical, probabilistic and integrable techniques. We expand the program of
the weak noise theory, which maps the large deviations onto a non-linear
hydrodynamic problem, and unveil its complete solvability through a connection
to the integrability of the Zakharov-Shabat system. Exact solutions, depending
on the initial condition of the KPZ equation, are obtained using the inverse
scattering method and a Fredholm determinant framework recently developed.
These results, explicit in the case of the droplet geometry, open the path to
obtain the complete large deviations for general initial conditions.
