Related papers: Making Waves
We consider propagating, spatially localised waves in a class of equations that contain variational and non-variational terms. The dynamics of the waves is analysed through a collective coordinate approach. Motivated by the variational…
Vector solitary waves are nonlinear waves of coupled polarizations that propagate with constant velocity and shape. In mechanics, they hold the potential to control locomotion, mitigate shocks and transfer information, among other…
Two methods are explained to exactly solve Maxwell's equations where permittivity, permeability and conductivity may vary in space. In the constitutive relations, retardation is regarded. If the material properties depend but on one…
A closed-form expression for the amplitudes of source waves in 2D discrete lattice with local and linear (waveguides) defects is derived. The numerical implementation of this analytic expression is demonstrated by several examples.
Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are…
This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a…
A nonlinear PDE featuring flux limitation effects together with those of the porous media equation (nonlinear Fokker-Planck) is presented in this paper. We analyze the balance of such diverse effects through the study of the existence and…
Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to…
Periodic waves in the fractional Korteweg-de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of…
This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolution-type operador from the…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
We study wave equations with energy dependent potentials. Simple analytical models are found useful to illustrate difficulties encountered with the calculation and interpretation of observables. A formal analysis shows under which…
We study waves in a viscoelastic rod whose constitutive equation is of generalized Zener type that contains fractional derivatives of complex order. The restrictions following from the Second Law of Thermodynamics are derived. The…
In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two…
We describe a map-based model which reproduces many of the behaviors seen in partial differential equations (PDE's). Like PDE's, we show that this model can support an infinite number of stationary solutions, traveling solutions, breathing…
We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in…
We describe the evolution of a paraxial electromagnetic wave characterizing by a non-uniform polarization distribution with singularities and propagating in a weakly anisotropic medium. Our approach is based on the Stokes vector evolution…
Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the…
We consider a system of two reaction-diffusion-advection equations describing the one dimensional directed motion of particles with superimposed diffusion and mutual alignment. For this system we show the existence of traveling wave…
A unified approach to the representation of solutions of linear PDE's with constant coefficients in high dimensions in terms of solutions of the same PDE's in lower dimensions is presented. It is based on the observation that if a function…