# Solitary water waves created by variations in bathymetry

## TL;DR

This paper investigates how periodic bathymetry affects water wave dispersion, leading to the formation of solitary waves through a balance of effective dispersion and nonlinearity, with potential experimental observation.

## Contribution

It introduces a homogenized model showing bathymetry-induced dispersion, derives a KdV-type equation for weakly nonlinear waves, and demonstrates the formation of solitary waves due to this effect.

## Key findings

- Bathymetry creates significant effective dispersion.
- Balance of dispersion and nonlinearity produces solitary waves.
- Bathymetry-induced dispersion can surpass traditional water wave dispersion.

## Abstract

We study the flow of water waves over bathymetry that varies periodically along one direction. We derive a linearized, homogenized model and show that the periodic bathymetry induces an effective dispersion, distinct from the dispersion inherently present in water waves. We relate this dispersion to the well-known effective dispersion introduced by changes in the bathymetry in non-rectangular channels. Numerical simulations using the (non-dispersive) shallow water equations reveal that a balance between this effective dispersion and nonlinearity can create solitary waves. We derive a KdV-type equation that approximates the behavior of these waves in the weakly-nonlinear regime. We show that, depending on geometry, dispersion due to bathymetry can be much stronger than traditional water wave dispersion and can prevent wave breaking in strongly nonlinear regimes. Computational experiments using depth-averaged water wave models %, as well as confirm the analysis and suggest that experimental observation of these solitary waves is possible.