Related papers: Solving the unique continuation problem for Schr\"…
The cubic nonlinear Schr\"odinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of…
We introduce low regularity exponential-type integrators for nonlinear Schr\"odinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove…
We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schr\"odingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y.…
A stochastic Schr\"odinger equation is presented to describe simultaneous continuous measurement of the position and momentum of a non-relativistic particle. The equation is solved to yield a state localised in position and momentum…
We consider finite element approximations of ill-posed elliptic problems with conditional stability. The notion of {\emph{optimal error estimates}} is defined including both convergence with respect to mesh parameter and perturbations in…
This paper studies the problem of stability of a parameterized delay differential equations (DDE see equation (0.1)). After discretizing the DDE (0.1), we show that the problem can be equivalently casted into a semi-definite programming…
This article is concerned with the mathematical analysis of a class of a nonlinear fractional Schrodinger equations with a general Hartree-type integrand. We prove existence and uniqueness of global-in-time solutions to the associated…
In this work, we extend the equal-order stabilized scheme discussed in [Franca et al., Comput. Methods Appl. Mech. Engrg. 99 (1992) 209-233] to accommodate slip (i.e., Navier) boundary conditions for the stationary Navier-Stokes equations.…
In this paper we investigate the Cauchy problem for Schr\"odinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove $H^\infty$ well-posedness in the very weak sense under suitable assumptions of the…
We analyze the Schr\"odingerization method for quantum simulation of a general class of non-unitary dynamics with inhomogeneous source terms. The Schr\"odingerization technique, introduced in [31], transforms any linear ordinary and partial…
We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…
In this study a stabilized finite element method for solving advection-diffusion-reaction equation with spatially variable coefficients has been carried out. Here subgrid scale approach along with algebraic approximation to the sub-scales…
We consider the linear and non linear cubic Schr\"odinger equations with periodic boundary conditions, and their approximations by splitting methods. We prove that for a dense set of arbitrary small time steps, there exists numerical…
Orbital stability property for weakly coupled nonlinear Schr\"odinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated…
The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a…
High-frequency solutions of one or several Schr\"odinger-type equations are well known to differ very little from the plane wave solutions $\exp[\pm ik x]$. That is, the potential terms impact the envelope of a high-frequency plane wave by…
In this article, we study the increasing stability property for the determination of the potential in the Schr\"odinger equation from partial data. We shall assume that the inaccessible part of the boundary is flat and homogeneous boundary…
We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined…
In this paper we develop an abstract method to handle the problem of unique continuation for the Schr\"odinger equation $(i\partial_t+\Delta)u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique…
In this paper we consider the stabilization of non-fundamental unstable stationary solutions of the cubic nonlinear Schrodinger equation. Specifically we study the stabilization of radially symmetric solutions with nodes and asymmetric…