Perturbation of Traveling Boussinesq Solitons by Periodic Bathymetry

TL;DR

This work addresses how a periodic bathymetry perturbs traveling Boussinesq solitons in a 2D setting by formulating a terrain-following Boussinesq model with non-autonomous coefficients. It develops two perturbation schemes for small bed corrugations: (i) a Fourier-based modulated traveling-wave approach using Green functions and path-ordered exponentials, and (ii) a mapping to a perturbed KdV equation to track soliton parameter evolution. The results show the perturbation produces a modulated soliton envelope and a wake of fourth-order linear dispersive waves, with good agreement against numerical simulations across diverse parameters and indications of soliton stability in many regimes; the methods extend to arbitrary periodic (and potentially non-periodic) bathymetries and can inform multi-soliton interactions. The analytic frameworks provide benchmarks for numerical methods and insight into Bragg-like scattering phenomena in nonlinear shallow-water waves.

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.

Figures (11)

• Figure 1: Evolution of a perturbed soliton $z_s(\xi,\tau)$ in the mathematical plane, for $\alpha=0.15$, $\beta=0.75$, and $\epsilon = 0.4$. Red: Perturbation solution. Black dotted: Numerical simulation. Blue: Profile shape of $M$ (not at the same vertical scale as $z_s$). Gray: Initial Boussinesq soliton of width $L_{sol}= 7 \pi$ and relative amplitude $\alpha=0.15$.
• Figure 2: Density map showing the soliton evolution and the wake in the soliton tail, from the COMSOL simulation for $\alpha=0.25, \beta=0.48$ and $\epsilon=0.263$. The initial soliton has a width $L_{sol}=4\pi$. Inset: Consecutive phases of the traveling soliton in the mathematical plane, showing the decrease and oscillations of the soliton amplitude from numerical calculations.
• Figure 3: Solution $z_{s}(x, t)$ at two moments of time. Parameters: $\alpha=0.15$, $\beta=0.6$, $\epsilon=0.1$, and $L_{sol}=14 \lambda_b$ for the initial soliton. Red: Analytical solution. Black dotted: Numerical simulation. Blue: bed corrugations (not at the same vertical scale as the waves). Inset: The solution shown at the faithful horizontal and vertical scales.
• Figure 4: Solution $z_s(x,t)$ at various times for $\alpha=0.75$, $\beta=0.35$, and $L_{sol}=3 \lambda_b /2$. Red: Analytical solution. Black: Numerical simulation. Blue: Corrugations. (a) Low corrugation $\epsilon=0.2$. (b) High corrugation $\epsilon=0.4$.
• Figure 5: Solution $z_s(x,t)$ for $\alpha=0.35$: (a) Unstable traveling soliton of $L_{sol}=3 \lambda_b$ and $\beta=0.35$ over high corrugations of $\epsilon=0.4$. (b) Stable traveling soliton of $L_{sol}=5 \lambda_b$ and $\beta=0.75$ over low corrugations of $\epsilon=0.2$. Red: Analytic solution. Black dotted: Numerical simulation. Blue: Bed corrugations.