相关论文: A second order low-regularity integrator for the n…

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schr\"odinger equation with low regularity. Our analysis shows that for any $s>\frac12$, the method converges with…

We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order,…

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…

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…

For the solution of the cubic nonlinear Schr\"odinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform…

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$…

In this paper, we establish the optimal convergence result of a second order exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV equation under rough initial data. The scheme is explicit and efficient to…

We develop resonance-based low-regularity numerical integrators for stochastic Schr"odinger equations with additive $Q$-Wiener noise, covering both the linear equation with rough potential and the cubic nonlinear case. For the linear…

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and…

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schr\"odinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows…

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schr\"odinger equation. The method can be implemented by using fast Fourier transform with $O(N\ln N)$ operations…

The numerical approximation of low-regularity solutions to the nonlinear Schr\"odinger equation is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing…

We propose and rigorously analyze a novel family of explicit low-regularity exponential integrators for the nonlinear Schr\"odinger (NLS) equation, based on a time-relaxation framework. The methods combine a resonance-based scheme for the…

We discuss normal forms of the completely resonant non-linear Schr\"odinger equation on a torus $\T^n$, with particular applications to quasi periodic solutions.

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we…

In this paper, we consider the numerical solution of the continuous disordered nonlinear Schr\"odinger equation, which contains a spatial random potential. We address the finite time accuracy order reduction issue of the usual numerical…

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schr\"odinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish…

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating…

In this paper, we propose and analyse a novel third-order low-regularity trigonometric integrator for the semilinear Klein-Gordon equation with non-smooth solution in the $d$-dimensional space, where $d=1,2,3$. The integrator is constructed…

We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in…