## 3 Citations

### An Existence Theory for Small-Amplitude Doubly Periodic Water Waves with Vorticity

- Mathematics, PhysicsArchive for Rational Mechanics and Analysis
- 2020

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the…

### Traveling water waves — the ebb and flow of two centuries

- PhysicsQuarterly of Applied Mathematics
- 2022

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of…

### A variational formulation for steady surface water waves on a Beltrami flow

- MathematicsProceedings of the Royal Society A
- 2020

This paper considers steady surface waves ‘riding’ a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be…

## References

SHOWING 1-10 OF 42 REFERENCES

### Variational formulations for steady water waves with vorticity

- PhysicsJournal of Fluid Mechanics
- 2006

For free-surface water flows with a vorticity that is monotone with depth, we show that any critical point of a functional representing the total energy of the flow adjusted with a measure of the…

### A Hamiltonian Formulation of Water Waves with Constant Vorticity

- Physics, Mathematics
- 2007

We show that the governing equations for two-dimensional water waves with constant vorticity can be formulated as a canonical Hamiltonian system, in which one of the canonical variables is the…

### A Bifurcation Theory for Three-Dimensional Oblique Travelling Gravity-Capillary Water Waves

- MathematicsJ. Nonlinear Sci.
- 2003

This article presents a rigorous existence theory for small-amplitude threedimensional travelling water waves using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom.

### Non-existence of three-dimensional travelling water waves with constant non-zero vorticity

- MathematicsJournal of Fluid Mechanics
- 2014

Abstract We prove that there are no three-dimensional bounded travelling gravity waves with constant non-zero vorticity on water of finite depth. The result also holds for gravity–capillary waves…

### Traveling Two and Three Dimensional Capillary Gravity Water Waves

- MathematicsSIAM J. Math. Anal.
- 2000

The main results of this paper are existence theorems for traveling gravity and cap- illary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any…

### A variational principle for a fluid with a free surface

- PhysicsJournal of Fluid Mechanics
- 1967

The full set of equations of motion for the classical water wave problem in Eulerian co-ordinates is obtained from a Lagrangian function which equals the pressure. This Lagrangian is compared with…

### An exact solution for equatorially trapped waves

- Mathematics, Physics
- 2012

[1] We present an exact solution of the nonlinear governing equations for geophysical water waves in the β-plane approximation near the Equator. The solution describes in the Lagrangian framework…

### On three-dimensional Gerstner-like equatorial water waves

- GeologyPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2017

A number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave–current interactions, in the equatorial region are surveyed.

### Existence of solutions for three dimensional stationary incompressible Euler equations with nonvanishing vorticity

- Mathematics
- 2009

In this paper, solutions with nonvanishing vorticity are established for the three dimensional stationary incompressible Euler equations on simply connected bounded three dimensional domains with…