  This paper presents an extended version of the celebrated Serre-Green-Naghdi
(SGN) system. This extension is based on the well-known Bona-Smith-Nwogu trick
which aims to improve the linear dispersion properties. We show that in the
fully nonlinear setting it results in modifying the vertical acceleration. Even
if this technique is well-known, the effect of this modification on the
nonlinear properties of the model is not clear. The first goal of this study is
to shed some light on the properties of solitary waves, as the most important
class of nonlinear permanent solutions. Then, we propose a simple adaptive
strategy to choose the optimal value of the free parameter at every instance of
time. This strategy is validated by comparing the model prediction with the
reference solutions of the full Euler equations and its classical counterpart.
Numerical simulations show that the new adaptive model provides a much better
accuracy for the same computational complexity.
