Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

     

Nonlinear resonance in viscous films on inclined wavy planes

We study nonlinear resonance in viscous gravity-driven films flowing over undulated substrates. Numerical solution of the full, steady Navier–Stokes equations is used to follow the emergence of the first few free-surface harmonics with increasing wall amplitude, and to study their parametric dependence on film thickness, inertia and capillarity. Bistable resonance is computed for steep enough bottom undulations. As an analytic approach, we apply the integral boundary-layer method and derive an asymptotic equation valid for rather thin films. The analysis recovers the key numerical findings and provides qualitative understanding. It shows that higher harmonics are generated by a nonlinear coupling of the wall with lower-order harmonics of the free surface. It also accounts for bistable resonance, and produces a minimum model whose solution is similar to that of the Duffing oscillator.     

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Summary. We derive a set of asymptotically exact coupled amplitude-streaming flow ({CASF}) equations governing the evolution of weakly nonlinear nearly inviscid multimode Faraday waves and the associated streaming flow in finite geometries. The streaming flow is found to play a particularly important role near mode interactions. Such interactions come about either through a suitable choice of parameters or through breaking of degeneracy among modes related by symmetry. An example of the first case is provided by the interaction of two nonaxisymmetric modes in a circular container with different azimuthal wavenumbers. The second case arises when the shape of the container is changed from square to slightly rectangular, or from circular to slightly noncircular but with a plane of symmetry. The generation of streaming flow in each of these cases is discussed in detail and the properties of the resulting CASF equations are described. A preliminary analysis suggests that these equations can resolve discrepancies between existing theory and experimental results in the first two of the above cases.     

Finite element simulations of fully non‐linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation

In Ma, Wu, Eatock Taylor [Finite element simulation of fully non‐linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. International Journal for Numerical Methods in Fluids 2001], designated Part 1 hereafter, we have developed the methodology and solution procedure for simulating the three‐dimensional interaction between fixed bodies and steep waves based on a finite element method (FEM). This paper provides extensive numerical results and validation. The effectiveness of the radiation condition is investigated by comparing the results from short and long tanks; the accuracy of the computed data is confirmed through comparison with analytical solutions. The adopted mathematical model is also validated by comparing the obtained numerical results with experimental data. Various test cases, including non‐linear bichromatic and irregular waves and the interactions between waves and one or two cylinders, are analysed. Copyright © 2001 John Wiley & Sons, Ltd.     

On the plane problem of the flow around a submerged beam

We consider the problem of the steady flow of an ideal heavy fluid around a submerged beam. The problem is obtained from the free-boundary problem of the flow past a submerged obstacle in the limit of bodies of vanishing thickness. We introduce a special Sobolev space formulation of the problem in term of a perturbed stream function and prove its unique solvability for every value of the unperturbed flow velocity, with the possible exception of a discrete set depending on the geometry of the domain. The asymptotic properties of the solutions are discussed.     

Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

We present a general result of transverse nonlinear instability of 1d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1-d model and the transverse perturbation “have the same sign”. Our result applies to the generalized KP-I equation, the Nonlinear Schrödinger equation, the generalized Boussinesq system and the Zakharov–Kuznetsov equation and we hope that it may be useful in other contexts.     

Orbital Instability of Standing Waves for the Klein-Gordon-Zakharov System

O Introduction We consider the orbital instability of standing waves for the Klein-Gordon-Zakharov system with different propagation speeds in three space dimensions     

Invariant manifolds and the stability of traveling waves in scalar viscous conservation laws

The stability of traveling wave solutions of scalar viscous conservation laws is investigated by decomposing perturbations into three components: two far-field components and one near-field component. The linear operators associated to the far-field components are the constant coefficient operators determined by the asymptotic spatial limits of the original operator. Scaling variables can be applied to study the evolution of these components, allowing for the construction of invariant manifolds and the determination of their temporal decay rate. The large time evolution of the near-field component is shown to be governed by that of the far-field components, thus giving it the same temporal decay rate. We also give a discussion of the relationship between this geometric approach and previous results, which demonstrate that the decay rate of perturbations can be increased by requiring that initial data lie in appropriate algebraically weighted spaces.     

Detonation in gases

We review recent progress in gaseous detonation experiment, modeling, and simulation. We focus on the propagating detonation wave as a fundamental combustion process. The picture that is emerging is that although all propagating detonations are unstable, there is a wide range of behavior with one extreme being nearly laminar but unsteady periodic flow and the other chaotic instability with highly turbulent flow. We discuss the implications of this for detonation propagation and dynamic behavior such as diffraction, initiation, and quenching or failure.     

Flows formed after the passage of a discontinuous wave over a bottom drop

The solvability of the problem of the flow formed after a discontinuous wave has passed over a bottom drop is studied within the framework of the first approximation of shallow water theory. Solutions in which the total energy of the flow is either conserved or lost at the drop are considered. Stable self-similar solutions of five qualitatively different types are derived and their domains of existence are determined in the dimensionless parameter plane.