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We prove in this paper a long time existence result for a modifiedBoussinesq-Peregrine equation in one dimension, describing the motion of WaterWaves in shallow water, in the case of a non flat bottom. We first give a localexistence result for the original Boussinesq Peregrine equation as derived byBoussinesq and Peregrine in all dimensions. We then introduce a new model whichhas formally the same precision as the Boussinesq-Peregrine equation, and givea local existence result in all dimensions. We finally prove a long timeexistence result in dimension 1 for this new equation, without any assumptionon the smallness of the bathymetry, which is an improvement of the long timeexistence result for the Boussinesq systems in the case of flat bottom.

The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations