Related papers: Diffractive Theorems for the Wave Equation with In…
Herein, we derive the fractional Laplacian operator as a means to represent the mean friction force arising in a turbulent flow: $ \rho \frac{D\bar{\bf u}}{Dt} = -\nabla p + \mu_\alpha \nabla^2\bar{\bf u} + \rho C_\alpha…
We consider the nonlinear Schr\"{o}dinger equation with a repulsive Dirac delta potential in one dimensional Euclidean space. We classify the global dynamics of even solutions with the same action as the high-frequency ground state standing…
For the wave equation associated to the Laplacian on a compact manifold with boundary with a conic metric (with respect to which the boundary is metrically a point) the propagation of singularities through the boundary is analyzed. Under…
The quantum mechanical description of phase remains a fundamental challenge, with theoretical efforts tracing from the early works of London and Dirac to discrete formalisms. In this work, we extend the action-angle formalism to the…
The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the…
We consider two formally determined inverse problems for the wave equation in more than one space dimension. Motivated by the fixed angle inverse scattering problem, we show that a compactly supported potential is uniquely determined by the…
We study the stability and dynamics of traveling-front solutions of a modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion…
This is the fourth article in a series where we succeed in enlarging the class of exactly solvable quantum systems. We do that by working in a complete set of square integrable basis that carries a tridiagonal matrix representation for the…
Spectral analysis for a class of Lagrangian-averaged Navier--Stokes (LANS) equations on the sphere is carried out. The equations arise from the Navier--Stokes equations by applying a Helmholtz filter of width $\alpha$ to the advecting…
Using the very basic physics principles, we have studied the implications of quantum corrections to classical electrodynamics and the propagation of electromagnetic waves and pulses. The initial nonlinear wave equation for the…
We discuss the propagation of electromagnetic waves on a rectangular lattice of polarizable point dipoles. For wavelengths long compared to the lattice spacing, we obtain the dispersion relation in terms of the lattice spacing and the…
We prove the sharp domain of dependence property for solutions to subelliptic wave equations for sums of squares of vector fields satisfying H\"ormander bracket condition. We deduce a unique continuation property for the square root of…
Streamer ionization fronts are pulled fronts propagating into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long time attractor out of a continuous family. A…
The present manuscript consists of inverse problems for a coupled system of wave equations with potential in $\mathbb{R}^3$. By establishing the fundamental solution to the aforementioned operator, we study the uniqueness aspects of the…
In this paper, we are concerned with deterministic wave generation in a hydrodynamic laboratory. A linear wavemaker theory is developed based on the fully dispersive water wave equations. The governing field equation is the Laplace equation…
This paper investigates solitary water waves propagating along the surface of a two-dimensional dielectric fluid with constant vorticity in the presence of an external electric field. We formulate the system as a nonlinear free boundary…
In this work we study the 3D Navier-Stokes equations, under the action of an external force and with the fractional Laplacian operator $(-\Delta)^{\alpha}$ in the diffusion term, from the point of view of variable Lebesgue spaces. Based on…
In this paper, we review the construction of periodic fundamental solutions and periodic layer potentials for various differential operators. Specifically, we focus on the Laplace equation, the Helmholtz equation, the Lam\'e system, and the…
Inverse problems for a diffusion equation containing a generalized fractional derivative are studied. The equation holds in a time interval $(0,T)$ and it is assumed that a state $u$ (solution of diffusion equation) and a source $f$ are…
For small perturbations of Minkowski space, we show that the square of the Lorentzian Dirac operator $P= -D^2$ has real spectrum apart from possible poles in a horizontal strip. Furthermore, for $\varepsilon>0$ we relate the poles of the…