Related papers: Explicit complex-valued solutions of the 2D eikona…
We use a one-scale similarity analysis which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations, whose exact solutions are generally difficult to obtain. We also…
We consider methods for constructing explicit solutions of the non-stationary Lam\'e equation, which is a generalization of the classical Lam\'e equation, that has appeared in works on integrable models, conformal field theory, high energy…
We propose a novel numerical algorithm for computing the electronic structure related eigenvalue problem of incommensurate systems. Unlike the conventional practice that approximates the system by a large commensurate supercell, our…
We study the inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map at selected frequency as the data. We develop an explicit reconstruction of the wavespeed using a multi-level nonlinear projected…
This work proposes an extension of the 1-D Hilbert Huang transform for the analysis of images. The proposed method consists in (i) adaptively decomposing an image into oscillating parts called intrinsic mode functions (IMFs) using a mode…
Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the…
This paper is concerned with the asymptotic expansions of the amplitude of the solution of the Helmholtz equation. The original expansions were obtained using a pseudo-differential decomposition of the Dirichlet to Neumann operator. This…
Deterministic neural operators perform well on many PDEs but can struggle with the approximation of high-frequency wave phenomena, where strong input-to-output sensitivity makes operator learning challenging, and spectral bias blurs…
We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential…
We develop a method for constructing exact cosmological solutions of the Einstein equations based on representing them as a second-order linear differential equation. In particular, the method allows using an arbitrary known solution to…
Wave fields obeying the 2D Helmholtz equation on branched surfaces (Sommerfeld surfaces) are studied. Such surfaces appear naturally as a result of applying the reflection method to diffraction problems with straight scatterers bearing…
In this paper we study nonlinear Helmholtz equations with sign-changing diffusion coefficients on bounded domains. The existence of an orthonormal basis of eigenfunctions is established making use of weak T-coercivity theory. All…
A system of commutative hyperbolic complex numbers in 2 dimensions is studied in this paper. Exponential and trigonometric forms are obtained for these hyperbolic twocomplex numbers. Expressions are given for the elementary functions of…
Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the…
This paper presents a new approach for the computation of eigenvalues of the generalized spheroidal wave equations. The novelty of the present method is in the use of the analytical derivatives of the eigenvalues to minimize losses in…
In this paper, we present a new compositing approach to obtain stylized reflections and refractions with a simple control. Our approach does not require any mask or separate 3D rendering. Moreover, only one additional image is sufficient to…
In this paper, we investigate whether Variational Principles can be associated with the Helmholtz equation subject to impedance (absorbing) boundary conditions. This model has been extensively studied in the literature from both…
We find an explicit formula for the elliptic stable envelope in the case of the Hilbert scheme of points on a complex plane. The formula has a structure of a sum over trees in Young diagrams. In the limit we obtain the formulas for the…
After a short outline of the factorization and Grassmann picture of the one-dimensional (1D) Fokker-Planck (FP) equation, we consider a class of spatially-inhomogeneous solutions of the 2D FP equation with symmetric 2D (super)potentials. We…
We consider the inverse problem of reconstructing inhomogeneities by performing a finite number of scattering measurements of acoustic type in the time-harmonic setting. We set up the reconstruction as a fully discrete variational problem…