Abstract
We establish the functional relations between variations of the potential δu( x) in the scattering problem for the Lax operator of the Benjamin-Ono (BO) equation, on one hand, and the set of variations of the corresponding scattering data, on the other. The key operator in our calculations is the Green's function for the Lax operator of the BO. This Green's function is computed by using the analyticity properties of the Jost functions with respect to the spectral parameter. The results obtained allow us to establish the completeness of the “squared eigenfunctions” (SE) for the BO equation. We also prove the orthogonality relation of the SE. This enables us to determine the action-angle variables of the BO equation and their Poisson brackets. Thus, we can also prove the complete integrability of the BO equation as a Hamiltonian system.
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