Related papers: Monotonicity and local uniqueness for the Helmholt…
We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice $\mathbb{Z}^d$, $d \ge 1$. We consider this problem for the cases with and without phase information. We prove uniqueness results…
The paper introduces and characterizes new notions of Lipschitzian and H\"olderian full stability of solutions to general parametric variational systems described via partial subdifferential and normal cone mappings acting in Hilbert…
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We…
Generalized impedance boundary conditions are effective, approximate boundary conditions that describe scattering of waves in situations where the wave interaction with the material involves multiple scales. In particular, this includes…
Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier…
In this paper, we consider a class of variational problems with integral functionals involving nonlocal gradients. These models have been recently proposed as refinements of classical hyperelasticity, aiming for an effective framework to…
This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the…
In this paper, we consider the following singularly perturbed fractional Kirchhoff problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N}…
We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of…
We prove some existence (and sometimes also uniqueness) of weak solutions to some stationary equations associated to the complex Schr\''{o}dinger operator under the presence of a singular nonlinear term. Among other new facts, with respect…
In this paper, a new model is proposed for the inverse random source scattering problem of the Helmholtz equation with attenuation. The source is assumed to be driven by a fractional Gaussian field whose covariance is represented by a…
We show that the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains strongly differs from the one of solutions to elliptic problems modelled upon the Laplace-Poisson equation with zero boundary data.…
Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not…
The $q$-Whittaker function $W_\lambda(\mathbf{x};q)$ associated to a partition $\lambda$ is a $q$-analogue of the Schur function $s_\lambda(\mathbf{x})$, and is defined as the $t=0$ specialization of the Macdonald polynomial…
We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs…
The inverse scattering problem of the three-dimensional Schroedinger equation is considered at fixed scattering energy with spherically symmetric potentials. The phase shifts determine the potential therefore a constructive scheme for…
In this paper, we focus on a new wave equation described wave propagation in the attenuation medium. In the first part of this paper, based on the time-domain space fractional wave equation, we formulate the frequency-domain equation named…
In this work, we mainly study the general $N$-soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation by utilizing the Riemann-Hilbert (RH) method. For the initial value belonging to Schwarz space, we firstly obtain the…
A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…
We study the stationary Stokes system with variable coefficients in the whole space, a half space, and on bounded Lipschitz domains. In the whole and half spaces, we obtain a priori $\dot W^1_q$-estimates for any $q\in [2,\infty)$ when the…