相关论文: Improved uniform error bounds of the time-splittin…
We establish optimal error bounds on time-splitting methods for the nonlinear Schr\"odinger equation with low regularity potential and typical power-type nonlinearity $ f(\rho) = \rho^\sigma $, where $ \rho:=|\psi|^2 $ is the density with $…
We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small electromagnetic potentials characterized by a dimensionless parameter $\varepsilon\in (0, 1]$ representing the…
We establish uniform error bounds of time-splitting Fourier pseudospectral (TSFP) methods for the nonlinear Klein--Gordon equation (NKGE) with weak power-type nonlinearity and $O(1)$ initial data, while the nonlinearity strength is…
Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where the nonlinearity strength is characterized by a dimensionless parameter…
We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schr\"odinger equation (NLSE) with semi-smooth nonlinearity $ f(\rho) = \rho^\sigma$, where $\rho=|\psi|^2$ is the density with $\psi$…
We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 <…
In this paper, we derive the improved uniform error bounds for the long-time dynamics of the $d$-dimensional $(d=2,3)$ nonlinear space fractional sine-Gordon equation (NSFSGE). The nonlinearity strength of the NSFSGE is characterized by…
This article is devoted to the construction of new numerical methods for the semiclassical Schr\"odinger equation. A phase-amplitude reformulation of the equation is described where the Planck constant epsilon is not a singular parameter.…
We establish the uniform error bound of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Schr\"odinger equation with wave operator (NLSW), in which the strength of the…
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…
In this paper, we conduct rigorous error analysis of the Lie-Totter time-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equation with a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodic boundary conditions…
An improved uniform error bound at $O\left(h^m+\varepsilon^2 \tau^2\right)$ is established in $H^{\alpha/2}$-norm for the long-time dynamics of the nonlinear space fractional Klein-Gordon equation (NSFKGE). A second-order exponential wave…
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 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…
We consider a version of the stationary phase method in one dimension of A. Erd\'elyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original…
We consider the nonlinear Schr{\"o}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding…
Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter…
We present different regularizations and numerical methods for the nonlinear Schr\"odinger equation with singular nonlinearity (sNLSE) including the regularized Lie-Trotter time-splitting (LTTS) methods and regularized Lawson-type…
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the…
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…