Related papers: Shock Reflection-Diffraction, von Neumann's Conjec…
Light injected into a spherical dielectric body may be confined very efficiently via the mechanism of total internal reflection. The frequencies that are most confined are called resonances. If the shape of the body deviates from the…
We consider two-dimensional Riemann boundary value problems of Euler equations for the Chaplygin gas with two piecewise constant initial data outside a convex cornered wedge. In self-similar coordinates, when the flow at the wedge corner is…
We are concerned with global solutions of multidimensional Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous…
As predicted by A. Einstein [Ann. Phys. (Leipzig) 17, 891 (1905)], the electromagnetic wave reflected at a moving mirror is frequency-upshifted and intensified as high as the mirror velocity is close to the speed of light in vacuum.…
For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is…
We are concerned with a two-dimensional ($2$-D) Riemann problem for compressible flows modeled by the pressure gradient system that is a $2$-D hyperbolic system of conservation laws. The Riemann initial data consist of four constant states…
This paper concerns the frequency domain problem of diffraction of a plane wave incident on an infinite right-angled wedge on which impedance (absorbing) boundary conditions are imposed. It is demonstrated that the exact…
We prove existence and uniqueness of a reaction-diffusion equation whose diffusivity is a non-linear functional of the boundary temperature. We do this by studying systems of one-dimensional reflecting diffusions whose noise is a function…
This paper is concerned with analysis of electromagnetic wave scattering by an obstacle which is embedded in a two-layered lossy medium separated by an unbounded rough surface. Given a dipole point source, the direct problem is to determine…
Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007),…
Despite the apparent simplicity, the problem of refraction of electromagnetic waves at the planar interface between two media has an incredibly rich spectrum of unusual phenomena. An example is the paradox that occurs when an…
Under the hypothesis of small deformations, the equations of 1D elastodynamics write as a 2 x 2 hyperbolic system of conservation laws. Here, we study the Riemann problem for convex and nonconvex constitutive laws. In the convex case, the…
The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of…
The propagation and refraction of waves in dispersive media are considered. A primary objective is to determine whether waves refract as monochromatic waves by Snell's law with phase velocity or as wave packets. The refraction of wave…
The purpose of this contribution is to show that some of the basic ideas of turbulence can be addressed in a deterministic setting instead of introducing random realizations of the fluid. Weak limits of oscillating sequences of solutions…
Even 100 years after its introduction by Louis de Broglie, the wave-nature of matter is often regarded as a mind-boggling phenomenon. To give an intuitive introduction to this field, we here discuss the diffraction of massive molecules…
The band gap, a key concept in solid-state physics, is traditionally explained by the Bragg diffraction of electron waves in the periodic potential of a crystal. Although widely accepted, this framework raises fundamental issues in…
This paper concerns the dynamic stability of the steady 3-D wave structure of a planar normal shock front intersecting perpendicularly to a planar solid wall for unsteady potential flows. The stability problem can be formulated as a free…
Let $\Omega\subset\mathbb{R}^n$ be a strictly convex domain with smooth boundary and diameter $D$. The fundamental gap conjecture claims that if $V:\bar\Omega\to\mathbb{R}$ is convex, then the spectral gap of the Schr\"odinger operator…
An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the…