Journal of Nonlinear Mathematical Physics
Volume 19, Issue Supplement 1, November 2012
Pages: v - v
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...
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)]....
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...
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...
Pages: 43 - 50
We present an explicit solution for the geophysical equatorial deep water waves in the f-plane approximation.
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.
8. Dispersion Relations for Steady Periodic Water Waves of Fixed Mean-Depth with an Isolated Bottom Vorticity Layer
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...
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...
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,...
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...
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...
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...
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...
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...