Related papers: The initial-boundary value problem for some quadra…
We consider the Cauchy problem for a quadratic derivative nonlinear Schr\"odinger equation whose nonlinearity is a linear combination of $\partial_x (u^2)$ and $\partial_x (|u|^2)$. We prove the local well-posedness in the $L^2$-based…
Scattering for the mass-critical fractional Schr\"odinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is…
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6]…
We consider a number of linear and non-linear boundary value problems involving generalized Schr\"odinger equations. The model case is $-\Delta u=Vu$ for $u\in W_0^{1,2}(D)$ with $D$ a bounded domain in ${\bf R^n}$. We use the Sobolev…
In this paper, we study the partial data inverse boundary value problem for the Schrodinger operator at a high frequency k>=1 in a bounded domain with smooth boundary in Rn, n>=3. Assuming that the potential is known in a neighborhood of…
An efficient method is proposed for numerical solutions of nonlinear Schr\"{o}dinger equations in an unbounded domain. Through approximating the kinetic energy term by a one-way equation and uniting it with the potential energy equation,…
In this paper second order elliptic boundary value problems on bounded domains $\Omega\subset\dR^n$ with boundary conditions on $\partial\Omega$ depending nonlinearly on the spectral parameter are investigated in an operator theoretic…
We consider Calder\'{o}n's inverse boundary value problems for a class of nonlinear Helmholtz Schr\"{o}dinger equations and Maxwell's equations in a bounded domain in $\R^n$. The main method is the higher-order linearization of the…
We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing litterature are weaker regularity assumptions for the boundary data and…
It is shown that a large subset of initial data with finite energy ($L^2$ norm)evolves nearly linearly in nonlinear Schr\" odinger equation with periodic boundary conditions. These new solutions are not perturbations of the known ones such…
The purpose of the present work is to study the existence of solutions to initial value problems for nonlinear first order differential systems with nonlinear nonlocal boundary conditions of functional type. The existence results are…
We consider two nonlinear equations, the Localized Induction Equation and the cubic nonlinear Schr\"odinger Equation, and prove that the solvability of certain initial-boundary value problems for each equation is equivalent through the…
We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained.
In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in $\mathbb{R}^2$ with clamped boundary conditions. After establishing…
In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an…
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 prove that the Schroedinger map initial-value problem is locally well-posed for small data in the Sobolev spaces $H^\sigma$, $\sigma>(d+1)/2$.
We show some new local smoothing estimates of the fractional Schr\"odinger equations with initial data in $\alpha$-modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates…
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak…
In this paper we obtain improved local well-posedness results for the Schr\"odinger-KdV system on the half-line. We employ the Laplace-Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order…