Derivation of a non-linear Pelnard-Considère equation

The Pelnard-Considère equation is a diffusion equation extensively used for the modelling of shoreline dynamics. However, it is a linear equation that cannot manage the growth of shoreline instabilities such as sand spits, cuspates,… We derive a new formalism from scratch, which introduces some kind of non-linearity and offer new striking opportunities to model shoreline features.