Perturbation of Traveling Boussinesq Solitons by Periodic Bathymetry

TL;DR

This paper studies how periodic bathymetry affects traveling Boussinesq solitons using perturbation methods and numerical simulations, revealing the generation of dispersive waves and analyzing soliton stability.

Contribution

Introduces two perturbation approaches for analyzing Boussinesq solitons over periodic bathymetry, extending methods to non-periodic cases and comparing with numerical results.

Findings

Perturbations generate fourth-order dispersive waves behind solitons.

Analytic solutions match numerical simulations for various parameters.

Discussed stability of perturbed solitons over time.

Abstract

We investigate the perturbations induced by a periodic bathymetry on traveling Boussinesq solitons in a two-dimensional configuration. We present two perturbation approaches to solve the nonlinear, dispersive and non-autonomous differential equations of the model and compare the solutions with numerical simulations of the original system of equations. In the approximation for small periodic corrugations we built the solutions as modulated traveling waves using Fourier series. The coefficients of the series are solved using the Green function method and the pathordered exponential method. At the second order in the relative height of bed corrugations, we obtain the perturbation as the fourth-order linear dispersive waves generated by the modulated traveling soliton in its wake. In the second approach, we rewrite the Boussinesq system into a perturbed Korteweg-de Vries (KdV) nonlinear equation, and obtain the corresponding perturbed solitons. These analytic solutions are compared with the results of numerical simulations, for various parameters that characterize the effects of nonlinearity, dispersion, and bottom bathymetry. We also discuss the stability of the perturbed solitons in time. The perturbation approaches developed in this study are valid for any type of periodic bathymetries, and the method can readily be extended to non-periodic ones.