Discrete wave turbulence for the Benjamin-Bona-Mahony equation, Part I: oscillations for the correlations between the solutions and its initial datum

We investigate different problems regarding wave turbulence for the Benjamin-Bona-Mahony (BBM) equation in the context of discrete turbulence regime. In the part I, we investigate the behaviour of the correlations between the solution to the BBM equation at latter times with the initial datum.