Related papers: Improved error estimates for low-regularity integr…
In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger equations $(i\partial_t+\mathcal{L}_{\mathbf{A}}^{\frac\alpha 2})u=0$ in the purely magnetic fields which includes the…
Consider the mass-critical nonlinear Schr\"odinger equations in both focusing and defocusing cases for initial data in $L^2$ in space dimension N. By Strichartz inequality, solutions to the corresponding linear problem belong to a global…
This paper presents a linear, decoupled, mass- and energy-conserving numerical scheme for the multi-dimensional coupled nonlinear Schr\"odinger (CNLS) system. The scheme combines the fourth-order compact difference approximation in space…
We generalize the Strichartz estimates for Schr\"odinger operators on compact manifolds of Burq, G\'erard and Tzvetkov [10] by allowing critically singular potentials $V$. Specifically, we show that their $1/p$--loss $L^p_tL^q_x(I\times…
We prove resolvent estimates for a Schr\"odinger operator with a short-range potential outside an obstacle with Dirichlet boundary conditions. As a consequence, we deduce integrability of the local energy for the wave equation, and…
The optimal $L^4$-Strichartz estimate for the Schr{\"o}dinger equation on the two-dimensional rational torus $\mathbb{T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach…
This paper develops and analyzes an optimal-order semi-discrete scheme and its fully discrete finite element approximation for nonlinear stochastic elastic wave equations with multiplicative noise. A non-standard time-stepping scheme is…
A nonlocal nonlinear Schr\"odinger (NLS) equation was recently found by the authors and shown to be an integrable infinite dimensional Hamiltonian equation. Unlike the classical (local) case, here the nonlinearly induced "potential" is $PT$…
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$…
We prove Strichartz estimates over large time scales for the Schrodinger equation set on irrational tori. They are optimal for Lebesgue exponents $p > 6$.
A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved,…
Numerical solving the Schr\"odinger equation with incommensurate potentials presents a great challenge since its solutions could be space-filling quasiperiodic structures without translational symmetry nor decay. In this paper, we propose…
We present an open-source Python implementation of an idealized high-order pseudo-spectral solver for the one-dimensional nonlinear Schr\"odinger equation (NLSE). The solver combines Fourier spectral spatial discretization with an adaptive…
We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be…
In this paper we focus on the validity of some fundamental estimates for time-degenerate Schr\"{o}dinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison…
We prove the local well-posedness of the initial boundary value problem for the nonlinear quadratic Schr\"odinger equation under low initial-boundary regularity assumption via the boundary integral operator method introduced by…
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…
The endpoint Strichartz estimate $\|e^{it\Delta} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $x=\rho\omega$, $\rho>0$,…
In this article we study the Schr\"odinger equation associated with Harmonic oscillator in the form of Strichartz type inequality. We give simple proofs for Strichartz type inequalities using purely the $L^2 \to L^p$ operator norm estimates…
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…