  A hydrodynamic approach is used to calculate an asymptotics of the Emptiness
Formation Probability - the probability of a formation of an empty space in the
ground state of a quantum one-dimensional many body system. Quantum
hydrodynamics of a system is represented as a Euclidian path integral over
configurations of hydrodynamic variables. In the limit of a large size of the
empty space, the probability is dominated by an instanton configuration, and
the problem is reduced to the finding of an instanton solution of classical
hydrodynamic equations. After establishing a general formalism, we carry out
this calculation for several simple systems -- free fermions with an arbitrary
dispersion and Calogero-Sutherland model. For these systems we confirm the
obtained results by comparison with exact results known in Random Matrix
theory. We argue that the nonlinear hydrodynamic approach might be useful even
in cases where the linearized hydrodynamics fails.
