Related papers: Numerical inverse scattering transform for the der…
As a classical time-stepping method, it is well-known that the Strang splitting method reaches the first-order accuracy by losing two spatial derivatives. In this paper, we propose a modified splitting method for the 1D cubic nonlinear…
This paper is concerned with direct and inverse scattering by a locally perturbed infinite plane (called a locally rough surface in this paper) on which a Neumann boundary condition is imposed. A novel integral equation formulation is…
We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schr\"odinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume…
Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we…
In this work, we extend the Riemann-Hilbert (RH) method in order to study the coupled modified Korteweg-de Vries equation (cmKdV) under nonzero boundary conditions (NZBCs), and successfully find its solutions with their various dynamic…
We analyze the inverse problem to reconstruct the shape of a three dimensional homogeneous dielectric obstacle from the knowledge of noisy far field data. The forward problem is solved by a system of second kind boundary integral equations.…
A typical procedure to integrate numerically the time dependent Schr\"o\-din\-ger equation involves two stages. In the first one carries out a space discretization of the continuous problem. This results in the linear system of differential…
Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent…
The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work,…
This paper presents a general expression for a number-theoretic Hilbert transform (NHT). The transformations preserve the circulant nature of the discrete Hilbert transform (DHT) matrix together with alternating values in each row being…
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schr\"odinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} <…
We study statistical inverse learning in the context of nonlinear inverse problems under random design. Specifically, we address a class of nonlinear problems by employing gradient descent (GD) and stochastic gradient descent (SGD) with…
In many Direct and Inverse Scattering problems one has to use a parameter-fitting procedure, because analytical inversion procedures are often not available. In this paper a variety of such methods is presented with a discussion of…
A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse…
We develop a Riemann-Hilbert approach to the inverse scattering transform method for the short pulse (SP) equation $u_{xt}=u+\frac{1}{6}(u^3)_{xx}$ with zero boundary conditions (as $|x|\to\infty$). This approach is directly applied to the…
Using the two-dimensional nonlinear Schr\"odinger equation (NLS) as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation from its small-data scattering behavior. We prove that under…
The inverse scattering transform for the defocusing-defocusing coupled Hirota equations is strictly discussed with non-zero boundary conditions at infinity including non-parallel boundary conditions, specifically referring to the asymptotic…
Stochastic gradient descent (SGD) and its variants are widely used and highly effective optimization methods in machine learning, especially for neural network training. By using a single datum or a small subset of the data, selected…
We introduce a numerical approach to computing the Schr\"odinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schr\"odinger (NLS) equation. In exploiting this nonlinear transform we are able to…
We present a novel numerical method to the time-harmonic inverse medium scattering problem of recovering the refractive index from near-field scattered data. The approach consists of two stages, one pruning step of detecting the scatterer…