Inverse problem for the third order operator with small periodic coefficients.

We consider the non-self-adjoint third order operator with real periodic coefficients. We construct the mapping from the set of coefficients of the operator into the set of spectral data. This mapping is similar to the corresponding mapping for the Hill operator constructed by E.Korotyaev. In order to construct our mapping we use the transformation of H.McKean reducing the spectral problem for the third order operator to the spectral problem for the Hill operator with an energy dependent potential. We prove that in the neighborhood of the zero our mapping is analytic and one-to-one. We apply the result for solving the good Boussinesq equation. This is a joint work with E.Korotyaev.