Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation

Abstract

Although tsunamis in the deep ocean are very long waves of quite small amplitudes, as they propagate shorewards into shallow water, nonlinearity becomes important and the structure of the leading waves depends on the polarity of the incident wave from the deep ocean. In this paper, we use a variable-coefficient Korteweg-de Vries equation to examine this issue, for an initial wave which is either elevation, or depression, or a combination of each. We show that the leading waves can be described by a reduction of the Whitham modulation theory to a solitary wave train. We find that for an initial elevation, the leading waves are elevation solitary waves with an amplitude which varies inversely with the depth, with a pre-factor which is twice the maximum amplitude in the initial wave. By contrast, for an initial depression, the leading wave is a depression rarefaction wave, followed by a solitary wave train whose maximum amplitude of the leading wave is determined by the square root of the mass in the initial wave.

Solitonic dispersive hydrodynamics: theory and observation

“…( 2). Equation ( 2) with c(a, u) = a/3 + u and f (a, u) = 2a/3 corresponds to the soliton limit of the KdV-Whitham system of modulation equations, shown in [26] to be equivalent to the soliton modulation equations determined by other means [24] with application to shallow water soliton propagation over topography in [24,[27][28][29]. The general case of Eq.…”

Solitonic Dispersive Hydrodynamics: Theory and Observation

“…(2). Equation 2with c(a, u) = a/3 + u and f (a, u) = 2a/3 corresponds to the soliton limit of the KdV-Whitham system of modulation equations, shown in [27] to be equivalent to the soliton modulation equations determined by other means [25] with application to shallow water soliton propagation over topography in [25,[28][29][30]. The general case of Eq.…”

Rogue waves on the double periodic background in Hirota equation

“…The nonlinear Schrödinger (NLS) equation describes the wave phenomena in ocean [1,2,3] and the propagation of optical pulse in optical fiber [4,5,6]. Considering the external factors such as depth of the sea, bottom friction and viscosity in the ocean and the femtosecond pulse propagation in fibers, several higher order NLS equations have been introduced in the literature [7,8].…”