arXiv subjects

Evgueni Dinvay

Publications and source records attributed to Evgueni Dinvay.

21 records · Page 2Linked to original sources

arXiv2019

A comparative study of bi-directional Whitham systems

In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties. In the current work, we review some of the existing fully dispersive systems. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

physics.comp-ph↗

arXiv2018

On well-posedness of a dispersive system of the Whitham--Boussinesq type

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in [4]. It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument.

math.AP↗

arXiv2018

Babenko's equation for periodic gravity waves on water of finite depth: derivation and numerical solution

The nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko's equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $2 π$-periodic Hilbert transform $\mathcal{C}$ used in the equation for deep water, the equation obtained here contains a certain operator $\mathcal{B}_r$, which is the sum of $\mathcal{C}$ and a compact operator whose dependence on the parameter involves on the depth of water. Numerical computations are based on an equivalent form of Babenko's equation derived by virtue of the spectral decomposition of the operator $\mathcal{B}_r \D / \D t$. Bifurcation curves and wave profiles of the extreme form are obtained numerically.

math.AP↗