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We perform numerical experiments on the Serre-Green-Naghdi (SGN) equationsand a fully dispersive "Whitham-Green-Naghdi" (WGN) counterpart in dimension 1.In particular, solitary wave solutions of the WGN equations are constructed andtheir stability, along with the explicit ones of the SGN equations, is studied.Additionally, the emergence of modulated oscillations and the possibility of ablow-up of solutions in various situations is investigated.  We argue that a simple numerical scheme based on a Fourier spectral methodcombined with the Krylov subspace iterative technique GMRES to address theelliptic problem and a fourth order explicit Runge-Kutta scheme in time allowsto address efficiently even computationally challenging problems.

Numerical study of the Serre-Green-Naghdi equations and a fully dispersive counterpart