Related papers: Explicit Dirichlet-neumann Operator For Water Wave…
We consider a Dirichlet to Neumann operator $\mathcal{L}_a$ arising in a model for water waves, with a nonlocal parameter $a\in(-1,1)$. We deduce the expression of the operator in terms of the Fourier transform, highlighting a local…
We provide the first proof of local well-posedness for the two-dimensional gravity water wave equations with spatially quasi-periodic initial conditions. We represent the solution using holomorphic coordinates, which are equivalent to a…
In this article we shall go over recent work in proving dispersive and Strichartz estimates for the Dirichlet-wave equation. We shall discuss applications to existence questions outside of obstacles and discuss open problems.
We put forward a conjecture about an universal asymptotical behaviour of the symbol of the Dirichlet-to-Neumann operator (considered as a pseudodifferential operator) in the 2D exterior problem for the Hemholtz equation. The conjecture is…
In this article we deal with a class of geometric inverse problem for bottom detection by one single measurement on the free surface in water--waves. We found upper and lower bounds for the size of the region enclosed between two different…
Dirichlet-Neumann Operators (DNOs) are important to the formulation, analysis, and simulation of many crucial models found in engineering and the sciences. For instance, these operators permit moving-boundary problems, such as the classical…
In this short note, we derive a system of two nonlocal equations for the water-wave problem following the work of [AFM06]. Specifically, we consider a fluid with a one-dimensional free surface for an irrotational fluid both with, and…
We compare the effectiveness of solving Dirichlet-Neumann problems via the Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit formulation, the dual AFM formulation (AFM*), a boundary integral collocation method (BIM),…
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…
The classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free…
The computation of the Dirichlet-Neumann operator for the Laplace equation is the primary challenge for the numerical simulation of the ideal fluid equations. The techniques used commonly for 2D fluids, such as conformal mapping and…
We consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a…
We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace…
We show that the fractional wave operator, which is usually studied in the context of hypersingular integrals but had not yet appeared in mathematical physics, can be constructed as the Dirichlet-to-Neumann map associated with the…
This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded $d$-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the…
A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet-Neumann operators. The equations for the surface…
The system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet -- Neumann operator of the unperturbed free surface. This…
The Dirichlet problem for the wave equation is a classical example of a problem which is not well-posed. Nevertheless, it has been used to model internal waves oscillating sinusoidally in time, in various situations, standing internal waves…
We establish a shape-derivative formula for the Dirichlet-to-Neumann operator on a compact manifold. Then, we apply this formula to obtain the well-posedness in H 1 under a specific Rayleigh-Taylor condition to an equation describing cell…
In this work, we prove the existence of wave operator for the following generalized derivative nonlinear Schr\"odinger equation \begin{align*} i\partial_t u+\partial_x^2 u +i |u|^{2\sigma}\partial_x u=0, \end{align*} with…