# Steady waves in flows over periodic bottoms

## TL;DR

This paper investigates the existence and persistence of steady waves in two-dimensional flows over periodic bottoms, proving multiple solutions near bifurcation points using a Dirichlet-Neumann operator framework.

## Contribution

It establishes the existence of at least two steady solutions near non-degenerate Stokes waves when a flat bottom is perturbed, extending bifurcation analysis.

## Key findings

- At least two steady solutions exist near bifurcation velocities.
- Persistence of steady waves close to non-degenerate Stokes waves.
- Application of Dirichlet-Neumann operator in wave formation analysis.

## Abstract

We study the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution when a flat bottom is perturbed, except for a sequence of velocities $c_{k}$. The main contribution is the proof that at least two steady solutions exist close to a non-degenerate $S^{1}$-orbit of non-constant steady waves when a flat bottom is perturbed. Consequently, we obtain persistence of at least two steady waves close to a non-degenerate $S^{1}$-orbit of Stokes waves bifurcating from the velocities $c_{k}$.