Journal of Nonlinear Mathematical Physics

Volume 19, Issue Supplement 1, November 2012

Research Article

1. Preface

Adrian Constantin
Pages: v - v
Research Article

2. Variational Derivation of the Green–Naghdi Shallow-Water Equations

Delia Ionescu-Kruse
Pages: 1 - 12
We consider the two-dimensional irrotational water-wave problem with a free surface and a flat bottom. In the shallow-water regime and without smallness assumption on the wave amplitude we derive, by a variational approach in the Lagrangian formalism, the Green–Naghdi equations (1.1). The second equation...
Research Article

3. The Two-Component Camassa–Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws

Marianna Euler, Norbert Euler, Thomas Wolf
Pages: 13 - 22
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa–Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor.43 (2010) 492001 (20pp)]....
Research Article

4. Experimental Study of the Velocity Field in Solitary Water Waves

Hung-Chu Hsu, Yang-Yih Chen, Chu-Yu Lin, Chia-Yan Cheng
Pages: 23 - 33
We describe experiments that have been conducted to investigate the velocity field in a solitary water wave. The horizontal and vertical velocity components were measured. The experimental results show that the horizontal velocity component is monotonically increasing with the distance to the wave crest...
Research Article

5. On Constant Vorticity Flows Beneath Two-Dimensional Surface Solitary Waves

Raphael Stuhlmeier
Pages: 34 - 42
We demonstrate that, for a two-dimensional, steady, solitary wave profile, a flow of constant vorticity beneath the wave must likewise be steady and two-dimensional, and the vorticity will point in the direction orthogonal to that of wave propagation. Constant vorticity is the hallmark of a harmonic...
Research Article

6. An Explicit Solution for Deep Water Waves With Coriolis Effects

Anca-Voichita Matioc
Pages: 43 - 50
We present an explicit solution for the geophysical equatorial deep water waves in the f-plane approximation.
Research Article

7. Regularity of Steady Periodic Capillary Water Waves with Constant Vorticity

Calin Iulian Martin
Pages: 51 - 57
We prove a regularity result for steady periodic travelling capillary waves of small amplitude at the free surface of water in a flow with constant vorticity over a flat bed.
Research Article

8. Dispersion Relations for Steady Periodic Water Waves of Fixed Mean-Depth with an Isolated Bottom Vorticity Layer

David Henry
Pages: 58 - 71
In this paper we obtain the dispersion relations for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified mean depth, and which exhibit discontinuous vorticity. We take as a model an isolated layer of constant nonzero vorticity adjacent to the flat bed, with irrotational...
Research Article

9. Integrable Models for Shallow Water with Energy Dependent Spectral Problems

Rossen Ivanov, Tony Lyons
Pages: 72 - 88
We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup–Boussinesq equation. We formulate the inverse...
Research Article

10. On Periodic Water Waves with Coriolis Effects and Isobaric Streamlines

Anca-Voichita Matioc, Bogdan-Vasile Matioc
Pages: 89 - 103
In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points, are Gerstner-type waves. Furthermore, for waves traveling over a flat bed,...
Research Article

11. Solitary Traveling Water Waves of Moderate Amplitude

Anna Geyer
Pages: 104 - 115
We prove the existence of solitary traveling wave solutions for an equation describing the evolution of the free surface for waves of moderate amplitude in the shallow water regime. This nonlinear third-order partial differential equation arises as an approximation of the Euler equations, modeling the...
Research Article

12. Solitary Waves in Open Channels with Abrupt Turns and Branching Points

André Nachbin, Vanessa da Silva Simões
Pages: 116 - 136
The dynamics of solitary waves is studied in intricate domains such as open channels with sharp-bends and branching points. Of particular interest, the wave characteristics at sharp-bends is rationalized by using the Jacobian of the Schwarz–Christoffel transformation. It is observed that it acts in a...
Research Article

13. A Problem in the Classical Theory of Water Waves: Weakly Nonlinear Waves in the Presence of Vorticity

Robin Stanley Johnson
Pages: 137 - 160
The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type (“soliton” equations), but with vorticity. The familiar...
Research Article

14. Geometrical Methods for Equations of Hydrodynamical Type

Joachim Escher, Boris Kolev
Pages: 161 - 178
We describe some recent results for a class of nonlinear hydrodynamical approximation models where the geometric approach gives insight into a variety of aspects. The main contribution concerns analytical results for Euler equations on the diffeomorphism group of the circle for which the inertia operator...
Research Article

15. Relating the Bottom Pressure and the Surface Elevation in the Water Wave Problem

B. Deconinck, K. L. Oliveras, V. Vasan
Pages: 179 - 189
An overview is presented of recent progress on the relation between the pressure at the bottom of the flat water bed and the elevation of the free water boundary within the context of the one-dimensional, irrotational water wave problem. We present five different approaches to this problem. All are compared...