Related papers: Increasing stability of the first order linearized…
We consider the inverse coefficient problem of simultaneously determining the space dependent electromagnetic potential, the zero-th order coupling term and the first order coupling vector of a two-state Schr\"odinger equation in a bounded…
In the paper, we study the problem of recovering the Sturm--Liouville operator with frozen argument from its spectrum and additional data. For this inverse problem, we establish a substantial property of the uniform stability, which…
We consider a nonlinear Schr\"odinger equation with double power nonlinearity \begin{align*} i\partial_t u+\Delta u-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the…
We are concerned with the existence of solutions to the following nonlinear Schr\"odinger system in $\mathbb{R}^3$: \begin{equation*} \left\{ \begin{aligned} -\Delta u_1 + (x_1^2+x_2^2)u_1&= \lambda_1 u_1 + \mu_1 |u_1|^{p_1 -2}u_1 + \beta…
The initial value problem is considered for a higher order nonlinear Schr\"odinger equation with quadratic nonlinearity. Results on existence and uniqueness of weak solutions are obtained. In the case of an effective at infinity additional…
In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lam\'e parameters with certain regularity assumptions. In…
We consider two inverse problems for the multi-channel two-dimensional Schr\"odinger equation at fixed positive energy, i.e. the equation $-\Delta \psi + V(x)\psi = E \psi$ at fixed positive $E$, where $V$ is a matrix-valued potential. The…
Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where…
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schr\"odinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when…
Inverse nodal problem on diffusion operator is the problem of finding the potential functions and parameters in the boundary conditions by using nodal data. In particular, we solve the reconstruction and stability problems using nodal set…
We consider inverse boundary value problems for the Schrodinger equations in two dimensions. Within less regular classes of potentials, we establish a conditional stability estimate of logarithmic order. Moreover we prove the uniqueness…
We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local…
We consider the stability of Robust Optimization problems with respect to perturbations in their uncertainty sets. We focus on Linear Optimization problems, including those with a possibly infinite number of constraints, also known as…
In this paper, we for the first time prove local solvability and stability of an inverse spectral problem for higher-order ($n > 3$) differential operators with distribution coefficients. The inverse problem consists in the recovery of…
We prove stability estimates for the problem of recovering the nonlinearity from scattering data. We focus our attention on nonlinear Schr\"odinger equations of the form \[ (i\partial_t+\Delta)u = a(x)|u|^p u \] in three space dimensions,…
We consider the cubic nonlinear Schr\"odinger equation with an exceptional potential. We obtain a sharp time decay for the global in time solution and we get the large time asymptotic profile of small solutions. We prove the existence of…
In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schr\"odinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we…
Amplitude expansions are used to determine steady states of a semi-infinite solid subject to the Grinfeld instability in systems with a fixed (wave)length. We present two methods to obtain high-order weakly nonlinear results. Using the…
We investigate the properties of standing waves to a nonlinear Schr\"odinger equation with inverse-square potential on the half-line. We first establish the existence of standing waves. Then, by a variational characterization of the ground…
Control and state estimation of nonlinear systems satisfying a Lipschitz continuity condition have been important topics in nonlinear system theory for over three decades, resulting in a substantial amount of literature. The main criticism…