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The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system firstly appeared in [Dinvay, Dutykh, Kalisch 2018], where it was numerically shown to be stable and…
We study weak solutions for a class of free boundary problems which includes as a special case the classical problem of traveling waves on water of finite depth. We show that such problems are equivalent to problems in fixed domains and…
This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and…
This paper presents a pioneering investigation into the existence of traveling wave solutions for the two-dimensional Euler equations with constant vorticity in a curved annular domain, where gravity acts radially inward. This configuration…
Solitary electromagnetic waves propagating along the waveguides forming a rhombic one-dimensional lattice are considered. Two waveguides that are part of the unit cell are assumed to be made of an optical linear material, while the third…
Coherent states for equally spaced, homogeneous waveguide arrays are defined, in the infinite, semiinfinite and finite cases, and resolutions of the identity are constructed, using different methods. In the infinite case, which corresponds…
In this manuscript, we present a method to prove constructively the existence and spectral stability of solitary waves in both the Whitham and the capillary-gravity Whitham equations. By employing Fourier series analysis and computer-aided…
We report analytical solutions for spatial solitons supported by layers of a quadratically nonlinear material embedded into a linear planar waveguide. A full set of symmetric, asymmetric, and antisymmetric modes pinned to a symmetric pair…
The study of the possibility of existence of the non-propagating, trapped continental shelf waves (CSWs)along curved coasts reduces mathematically to a spectral problem for a self-adjoint operator pencil in a curved strip. Using the methods…
Diffraction of elastic waves is considered for a system consisting of two parallel arrays of thin (subwavelength) cylinders that are arranged periodically. The embedding media supports waves with all polarizations, one longitudinal and two…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
We prove that symmetric, doubly periodic, capillary-gravity water waves in finite depth bifurcating from non-uniform non-stagnant shear flows are necessarily two-dimensional to leading order. This is in stark contrast to the case of uniform…
This paper is devoted to the exponential stability for one-dimensional linear wave equations with in-domain localized damping and several types of Wentzell (or dynamic) boundary conditions. In a quite general boundary setting, we establish…
The paper presents new results on localisation and transmission of flexural waves in a structured plate containing a semi-infinite two-dimensional array of rigid pins. In particular, surface waves are identified and studied at the interface…
We prove existence results for travelling waves in discrete, damped, dc-driven sine-Gordon equations with periodic boundary conditions.
We study the spectrum of the Helmholtz equation in a two-dimensional infinite waveguide, containing a weak heterogeneity localized at an internal point, and obeying Dirichlet boundary conditions at its border. We prove that, when the…
In the dynamics generated by the suspension bridge equation, traveling waves are an essential feature. The existing literature focuses primarily on the idealized one-dimensional case, while traveling structures in two spatial dimensions…
This paper is devoted to study the wave propagation and its stability for a class of two-component discrete diffusive systems. We first establish the existence of positive monotone monostable traveling wave fronts. Then, applying the…
Recently, the Whitham and capillary-Whitham equations were shown to accurately model the evolution of surface waves on shallow water. In order to gain a deeper understanding of these equations, we compute periodic, traveling-wave solutions…
Recently, two different proofs for large and intermediate-size solitary waves of the nonlocally dispersive Whitham equation have been presented, using either global bifurcation theory or the limit of waves of large period. We give here a…