Evolution of Packets of Surface Gravity Waves over Strong Smooth Topography

Abstract

Wave packets in a smoothly inhomogeneous medium are governed by a nonlinear Schrödinger (NLS) equation with variable coefficients. There are two spatial scales in the problem: the spatial scale of the inhomogeneities and the distance over which nonlinearity and dispersion affect the packet. Accordingly, there are two limits where the problem can be approached asymptotically: when the former scale is much larger than the latter, and vice versa. In this paper, we examine the limit where the spatial scale of (periodic or random) inhomogeneities is much smaller than that of nonlinearity/dispersion (i.e., the latter effects are much weaker than the former). In this case, the packet undergoes rapid oscillations of the geometric-optical type, and also evolves slowly due to nonlinearity and dispersion. We demonstrate that the latter evolution is governed by an NLS equation with constant (averaged) coefficients. The general theory is illustrated by the example of surface gravity waves in a channel of variable depth. In particular, it is shown that topography increases the critical frequency, for which the nonlinearity coefficient of the NLS equation changes sign (in such cases, no steady solutions exist, i.e., waves with frequencies lower than the critical one disperse and cannot form packets).

Shallow-water soliton dynamics beyond the Korteweg–de Vries equation

“…A similar approach was later developed by Benilow and Howlin. This fission from the NLS has been found in other physical contexts [4,11].…”

“…Insertion of the trial function (11) into (9) and (10), use of the properties of the first order equation…”

Wave Packet Defocusing Due to a Highly Disordered Bathymetry

“…A similar behavior was observed in Benilov and Howlin in the case of a periodic topography. Another interesting result, regarding changes of the critical point q c is that of Grimshaw and Annenkov .…”

“…In Section we take a closer look into the influence of this new nonlinearity coefficient. It is found that as the topography amplitude increases the critical point, where the nonlinearity coefficient changes sign, moves to the right, in a similar fashion as in Benilov and Howlin . Note that, regarding the work of Grimshaw and Annenkov , the critical point moves to the left with an increasing wave amplitude, as opposed to the topography amplitude considered here.…”

“…Recently two articles on this problem were published in Studies of Applied Mathematics, both addressing critical point issues, which is the case in the present work. Benilov and Howlin [4] considered the evolution of surface wave packets over a strong (i.e., large amplitude), periodic topography. In particular, it was shown that the topography increases the critical frequency for which the nonlinearity coefficient of the NLS changes sign.…”