  Extended shallow water wave equations are derived, using the method of
asymptotic expansions, from the Euler (or water wave) equations. These extended
models are valid one order beyond the usual weakly nonlinear, long wave
approximation, incorporating all appropriate dispersive and nonlinear terms.
Specifically, first we derive the extended Korteweg-de Vries (KdV) equation,
and then proceed with the extended Benjamin-Bona-Mahony and the extended
Camassa-Holm equations in (1+1)-dimensions, the extended cylindrical KdV
equation in the quasi-one dimensional setting, as well as the extended
Kadomtsev-Petviashvili and its cylindrical counterpart in (2+1)-dimensions. We
conclude with the case of the extended Green-Naghdi equations.
