The Shallow-Water Models with Cubic Nonlinearity

In the present study several integrable equations with cubic nonlinearity are derived as asymptotic models from the classical shallow water theory. The starting point in our derivation is the Euler equation for an incompressible fluid with the simplest bottom and surface conditions. The approximate equations are obtained by working under suitable scalings that allow for the modeling of water waves of relatively large amplitude, truncating the asymptotic expansions of the unknowns to appropriate order, and introducing a special Kodama transformation. The so obtained equations exhibit cubic order nonlinearities and can be related to the following integrable systems: the Novikov equation, the modified Camassa-Holm equation, and a Camassa-Holm type equation with cubic nonlinearity. Analytically, the formation of singularities of the solution to some of these quasi-linear model equations is also investigated, with an emphasis on the understanding of the effect of the nonlocal higher order nonlinearities. In particular it is shown that one of the models accommodates the phenomenon of curvature blow-up.