Journal of Nonlinear Mathematical Physics

Volume 15, Issue Supplement 2, August 2008

Water waves

Supplement reflecting the results of the water waves program organised by the Hamilton Mathematics Institute, Trinity College Dublin, Ireland.

Foreword

1. Algebraic Discretization of the Camassa-Holm and Hunter-Saxton Equations

Rossen I Ivanov
Pages: 1 - 12
The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving the H 1 and H 1 right-invariant metrics correspondingly. There is an analogy to the Euler equations in hydrodynamics, which describe geodesic flow for a...

2. Particle Tra jectories in Linearized Irrotational Shallow Water Flows

Delia Ionesco-Kruse
Pages: 13 - 27
We investigate the particle tra jectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending...

3. Blow-Up Phenomena and Decay for the Periodic Degasperis-Procesi Equation with Weak Dissipation

Shuyin Wu, Zhaoyang Yin
Pages: 28 - 49
In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato’s theorem and a precise blow-up scenario of the solutions is given. Then, several critera guaranteeing the blow-up...

4. On the Recovery of the Free Surface from the Pressure within Periodic Traveling Water Waves

Joachim Escher, Torsten Schlurmann
Pages: 50 - 57
We present a consistent derivation of the pressure transfer function for small amplitude waves within the framework of linear wave theory and discuss some nonlinear aspects.

5. On the Non-Dimensionalisation, Scaling and Resulting Interpretation of the Classical Governing Equations for Water Waves

Adrian Constantin, Robin Stanley Johnson
Pages: 58 - 73
In this note we describe the underlying principles — and pitfalls — of the process of non-dimensionalising and scaling the equations that model the classical problem in water waves. In particular, we introduce the two fundamental parameters (associated with amplitude and with wave length) and show how...

6. On the Fluid Motion in Standing Waves

Mats Ehrnstrom, Erik Wahlen
Pages: 74 - 86
This paper concerns linear standing gravity water waves on finite depth. We obtain qualitative and quantitative understanding of the particle paths within the wave.

7. On Gerstner's Water Wave

David Henry
Pages: 87 - 95
We present a simple approach showing that Gerstner’s flow is dynamically possible: each particle moves on a circle, but the particles never collide and fill out the entire region below the surface wave.

8. Global Dissipative Solutions of the Generalized Camassa-Holm Equation

Octavian G Mustafa
Pages: 96 - 115
A new approach to the analysis of wave-breaking solutions to the generalized Camassa-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of dissipative solutions is also built. The solutions have non-increasing H1 energy...

9. Geometric Differences between the Burgers and the Camassa-Holm Equations

Boris Kolev
Pages: 116 - 132
The Burgers equation and the Camassa-Holm equations can both be recast as the Euler equation for a right-invariant metric on the diffeomorphism group of the circle, the L 2-metric for Burgers and the H 1-metric for Camassa-Holm. Their geometric behaviors are however very different. We present a survey...

10. Water Waves near a Shoreline in a Flow with Vorticity: Two Classical Examples

Robin Stanley Johnson
Pages: 133 - 156
The equations that describe the classical problem of water waves-inviscid, no surface tension and constant pressure at the surface - are non-dimensionalised and scaled appropriately, and the two examples: traditional gravity waves and edge waves, are introduced. In addition each type of wave is allowed...

11. Fiber Bundle Description of Flow and Nonlinear Hydrodynamics on Circles

Andrei Ludu
Pages: 157 - 170
We introduce a differential geometry description of the path lines, stream lines and particles contours in hydrodynamics. We present a generalized form of a Korteweg-de Vries type of equation for the exterior of a circle. Nonlinearities from the boundary conditions, surface tension and the Euler equations...