Computation of Travelling Combustion Waves in a Porous Medium

Numerical solutions for travelling combustion waves in a porous medium are sought. The algorithm of computation is based on a shooting method used in an existence proof. The numerical result suggests that there is a limit for the inlet gas velocity below which no travelling wave solution can be constructed.     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 1: Model equations

In this paper we formulate relatively simple models to describe the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. The models have good dispersive properties that are based on smooth quasi-homogeneous interpolation of the exact dispersion above flat bottom. This dispersive quality is then maintained in the second order nonlinear terms of uni-directional equations as known from the AB-equation. A linear coupling is employed to obtain bi-directional propagation which includes (interactions with) reflected waves.The derivation of the models is consistent with the basic variational formulation of surface waves without rotation. A subsequent spatial discretization that takes this variational structure into account leads to efficient and accurate codes, as will be shown in Part 2.     

Variational derivation of KdV-type models for surface water waves

Using the Hamiltonian formulation of surface waves, we approximate the kinetic energy and restrict the governing generalized action principle to a submanifold of uni-directional waves. Different from the usual method of using a series expansion in parameters related to wave height and wavelength, the variational methods retains the Hamiltonian structure (with consequent energy and momentum conservation) and makes it possible to derive equations for any dispersive approximation. Consequentially, the procedure is valid for waves above finite and above infinite depth, and for any approximation of dispersion, while quadratic terms in the wave height are modeled correctly. For finite depth this leads to higher-order KdV type of equations with terms of different spatial order. For waves above infinite depth, the pseudo-differential operators cannot be approximated by finite differential operators and all quadratic terms are of the same spatial order.     

Extreme wave phenomena in down-stream running modulated waves

Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive ‘extreme’ waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories.