  The bi-Hamiltonian structure of the Benney hierarchy is revisited. We show
that the compatibility condition of the Poisson brackets provides the genus
zero free energy of a topological field theory coupled to 2d gravity. We
calculate the correlation functions via the Landau-Ginzburg formulation and
derive the string equation based on the twistor construction. Moreover, by
using the approach of Dubrovin and Zhang, we compute the genus one correction
of the Poisson brackets and compare them with the Oevel-Strampp's brackets of
the Kaup-Broer hierarchy.
