Water wave packets over variable depth

In this paper we develop a higher-order nonlinear Schrodinger equation with variable coefficients to describe how a water wave packet will deform and eventually be destroyed as it propagates shoreward from deep to shallow water. It is well-known that in the framework of the usual nonlinear Schrodinger equation, a wave packet can only exist in deep water, more precisely when kh > 1.363 where k is the wavenumber and h is the depth. Using a combination of asymptotic analysis and numerical simulations we find that in the framework of the higher-order nonlinear Schrodinger equation, the wave packet can penetrate into shallow water kh < 1.363 or not even reach kh > 1.363, depending on the sign of the initial value in deep water of certain parameter of the wave packet which measures its speed.