In this paper the model 1D-GNσ is considered, which concerns the 1D Green–Naghdi equations with non-flat bottom and under the influence of surface tension, to be widely used in coastal oceanography to describe the propagation of large-wave amplitudes. The purpose of this paper is to show that the solution of 1D-GNσ can be made by the Picard iterative scheme, which proves that there is no loss of regularity of the solution relative to the initial condition.