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In this article we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous…
This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of…
We present a wide class of novel solitary and periodic waves in a non-centrosymmetric waveguide exhibiting second- and third-order nonlinearities. We show the existence of bright, gray, and W-shaped solitary waves as well as periodic waves…
In this work we present the results of a study of the possibility of using a homogeneous basis and a new generalization of coupled modes theory to describe non-periodic structured waveguides. It was shown that for the studied…
We analyze the existence and stability of nonlinear localized waves in a periodic medium described by the Kronig-Penney model with a nonlinear defect. We demonstrate the existence of a novel type of stable nonlinear band-gap localized…
We prove a principle of linearized stability for traveling wave solutions to neural field equations posed on the real line. Additionally, we provide the existence of a finite dimensional invariant center manifold close to a traveling wave,…
We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is…
We present a large-amplitude existence theory for two-dimensional solitary waves propagating through a two layer body of water. The domain of the fluid is bounded below by an impermeable flat ocean floor and above by a free boundary at…
In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan, et al. (J. Opt. Soc. Am. B…
Pinning and depinning of wavefronts are ubiquitous features of spatially discrete systems describing a host of phenomena in physics, biology, etc. A large class of discrete systems is described by overdamped chains of nonlinear oscillators…
The distance among two counter-rotating vortex filaments satisfies a beam-type of equation according to the model derived in [15]. This equation has an explicit solution where two straight filaments travel with constant speed at a constant…
We consider the stabilization problem on a manifold with boundary for a wave equation with measure-valued linear damping. For a wide class of measures, containing Dirac masses on hypersurfaces as well as measures with fractal support, we…
In this paper, we examine in detail the principal branches of solutions that arise in vector discrete models with nonlinear inter-component coupling and four wave mixing. The relevant four branches of solutions consist of two single mode…
This paper is concerned with the existence and the regularity of global solutions to the linear wave equation associated with two-point type boundary conditions. We also investigate the decay properties of the global solutions to this…
For steady-state and some other types of mechanical waves of an arbitrary form, intensity and nature, propagating in a free uniform waveguide, we present the following. Relations for the axial momentum as it directly follows from the…
We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schr\"odinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the…
We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field,…
Bound states in the continuum (BICs) exist in a variety of physical systems where they appear as lossless propagating states surrounded by radiating modes. However, in the case of open systems, they coexist with continuous families of…
We study existence and non-existence of constant scalar curvature metrics conformal and arbitrarily close to homogeneous metrics on spheres, using variational techniques. This describes all critical points of the Hilbert-Einstein functional…
Spiral waves are striking self-organized coherent structures that organize spatio-temporal dynamics in dissipative, spatially extended systems. In this paper, we provide a conceptual approach to various properties of spiral waves. Rather…