Journal of Nonlinear Mathematical Physics

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 in (1.1) is a transport equation, the free surface is advected by the fluid flow. We show that the first equation of the system (1.1) yields the critical points of an action functional in the space of paths with fixed endpoints, within the Lagrangian formalism.

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)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy–Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).

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 line but the vertical velocity is not monotonic.

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 velocity field, and the simplified vorticity equation is used along with maximum principles to derive the results.

We present an explicit solution for the geophysical equatorial deep water waves in the f-plane approximation.

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.

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 flow above the layer.

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 problem as a Riemann–Hilbert problem with a ℤ₂ reduction group. The soliton solutions are explicitly obtained.

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, we prove that there are only laminar flow solutions with these properties.

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 unidirectional propagation of surface water waves. We give a description of the solitary wave profiles by performing a phase plane analysis and study some qualitative features of the solutions.

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 similar fashion as a topography in other wave models previously studied. Previous numerical studies are revisited. A new very efficient algorithm is described, which computationally solves the problem in much more general channel configurations than presented in the literature. Also the conformal mapping naturally leads to a new strategy regarding the geometrical singularity at the sharp-bends. Finally preliminary results illustrate the use of the mapping's Jacobian at a channel's branching point. A future goal of this study regards deducing accurate reduced (one-dimensional) models for the reflection and transmission of solitary waves on graphs/networks.

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 problem of propagation governed by the Korteweg–de Vries (KdV) equation is introduced, but allowing for an arbitrary distribution of vorticity. The effect of the constant vorticity on the solitary wave is described. The corresponding problem for the Nonlinear Schrödinger (NLS) equation is briefly mentioned but not explored here. The problem of two-way propagation (admitting head-on collisions), as described by the Boussinesq equation, is examined next. This leads to a new equation: the Boussinesq-type equation valid for constant vorticity. However, this cannot be transformed into an integrable Boussinesq equation (as is possible for the corresponding KdV and NLS equations). The solitary-wave solution for this new equation is presented. A description of the Camassa–Holm equation for water waves, with constant vorticity, with its solitary-wave solution, is described. Finally, we outline the problem of propagation of small-amplitude, large-radius ring waves over a flow with vorticity (representing a background flow in one direction). Some properties of this flow, for constant vorticity, are described.

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 is a nonlocal operator.

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 to (1) numerical data for Stokes waves, one-dimensional traveling wave solutions of the full irrotational water wave problem, and (2) experimental data for high-amplitude waves in a long, narrow (i.e. one-dimensional) wave tank.