Conservation laws and non-decaying solutions for the Benney–Luke equation

Abstract

A long wave multi-dimensional approximation of shallow-water waves is the bi-directional BenneyLuke (BL) equation. It yields the well-known Kadomtsev-Petviashvili (KP) equation in a quasi one-directional limit. A direct perturbation method is developed; it uses underlying conservation laws to determine the slow evolution of parameters of two space-dimensional, non-decaying solutions to the BL equation. These non-decaying solutions are perturbations of recently studied web solutions of the KP equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the KP and the BL equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the BL equation are also studied.

Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

“…These values are approximate. and regular reflections, which confirms the conclusions obtained by Ablowitz and Curtis (2013) regarding the ability of the Benney-Luke model to predict reflection of obliquely incident solitary waves. Our simulations do not allow determination of the exact value of the interaction parameter at the transition from Mach to regular reflection, but currently the maximal amplification is reached at κ = 0.9733, which is very close to the predicted maximal amplification at κ = 1.0.…”

“…The scaling from BenneyLuke to KP is not exact but asymptotic, with a truncation at second order, which leads to a slight difference in the final wave amplification. This observation agrees with the conclusions of Ablowitz and Curtis (2013) on the asymptotic amplification of the stem wave in the case of the Benney-Luke model. The shift is probably also increased by the mesh resolution, which could be optimized to get a better estimate of the incident-wave amplitude in order to limit the error caused by its approximation.…”

“…Our simulations do not allow determination of the exact value of the interaction parameter at the transition from Mach to regular reflection, but currently the maximal amplification is reached at κ = 0.9733, which is very close to the predicted maximal amplification at κ = 1.0. The maximal amplification obtained herein is α w = 3.6, which is higher than the amplifications obtained with most previous models and experiments (Kodama et al, 2009;Li et al, 2011;Tanaka, 1993;Funakoshi, 1980), but still slightly lower than the expected 3.9 amplification from Ablowitz and Curtis (2013). This agrees with the conclusion of Ablowitz and Curtis (2013) concerning the impact of on the amplification near κ = 1.…”

“…Figure 8 shows a comparison between predictions Eqs. (3) and (39) Variation of is an alternative choice to the one made in the work of Ablowitz and Curtis (2013), where, for a specific , they compute simulations with varying amplitude and angle of incidence; this choice enabled them to show that the smallamplitude parameter has only a weak impact on the amplification of the stem wave for κ < 1 but limits the amplification, with a decrease of O( ) close to the resonant case κ = 1, leading, for example, to a maximal wave amplification of 3.9 when epsilon = 0.1. Despite this asymptotic limitation in the wave amplification, the purpose of the present simulations is to model wave amplification in various sea states, with various depths of water and characteristic wave heights, and we do so by using different values of , recalling that the smallamplitude parameter is the quotient between the characteristic wave height and the water depth.…”

“…The numerical scheme could not simulate the highest amplitudes that Miles predicts for κ ≈ 1. Recently, Ablowitz and Curtis (2013) The purpose of the present work is to derive and apply a stable numerical scheme able to estimate the solution over a long distance of propagation, in order to model highamplitude waves and to confirm the transition from regular to Mach reflection happening for κ ≈ 1. We develop a model similar to the one of Benney and Luke (1964), which is an asymptotic approximation of the potential-flow equations for small-amplitude and long waves.…”