  We investigate theoretically and numerically the dynamics of long-living
oscillating coherent structures - bi-solitons - in the exact and approximate
models for waves on the free surface of deep water. We generate numerically the
bi-solitons of the approximate Dyachenko-Zakharov equation and fully nonlinear
equations propagating without significant loss of energy for hundreds of the
structure oscillation periods, which is hundreds of thousands of characteristic
periods of the surface waves. To elucidate the long-living bi-soliton complex
nature we apply an analytical-numerical approach based on the perturbation
theory and the inverse scattering transform (IST) for the one-dimensional
focusing nonlinear Schr\"odinger equation model. We observe a periodic energy
and momentum exchange between solitons and continuous spectrum radiation
resulting in repetitive oscillations of the coherent structure. We find that
soliton eigenvalues oscillate on stable trajectories experiencing a slight
drift on a scale of hundreds of the structure oscillation periods so that the
eigenvalue dynamic is in good agreement with predictions of the IST
perturbation theory. Based on the obtained results, we conclude that the IST
perturbation theory justifies the existence of the long-living bi-solitons on
the surface of deep water which emerge as a result of a balance between their
dominant solitonic part and a portion of continuous spectrum radiation.
