Optimal error bounds on the exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

Abstract

We establish optimal error bounds for the exponential wave integrator (EWI) applied to the nonlinear Schrödinger equation (NLSE) with $L^{\infty }$-potential and/or locally Lipschitz nonlinearity under the assumption of $H^2$-solution of the NLSE. For the semi-discretization in time by the first-order Gautschi-type EWI, we prove an optimal $L^2$-error bound at $O(\tau )$ with $\tau >0$ being the time step size, together with a uniform $H^2$-bound of the numerical solution. For the full-discretization scheme obtained by using the Fourier spectral method in space, we prove an optimal $L^2$-error bound at $O(\tau +h^2)$ without any coupling condition between $\tau$ and $h$, where $h>0$ is the mesh size. In addition, for $W^{1,4}$-potential and a little stronger regularity of the nonlinearity, under the assumption of $H^3$-solution, we obtain an optimal $H^1$-error bound. Furthermore, when the potential is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method which has the same error bound as the Fourier spectral method while its computational cost is similar to the standard Fourier pseudospectral method. Our new error bounds greatly improve the existing results for the NLSE with low regularity potential and/or nonlinearity. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.

arXiv comment26 pages, 8 figures

References in corpus (18)

• Quantum liquid droplets in a mixture of Bose-Einstein condensates
• Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation
• Observing the Rosensweig instability of a quantum ferrofluid
• Mathematical theory and numerical methods for Bose-Einstein condensation
• Ultradilute low-dimensional liquids
• Two-species mixture of quantum degenerate Bose and Fermi gases
• Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations
• Derivation of the Cubic Non-linear Schrödinger Equation from Quantum Dynamics of Many-Body Systems
• Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials
• Josephson effect between trapped Bose-Einstein condensates
• Dynamics of one-dimensional quantum droplets
• Adiabatically changing the phase-space density of a trapped Bose gas
• Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity
• Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation
• Resonance based schemes for dispersive equations via decorated trees
• Regularized numerical methods for the logarithmic Schrodinger equation
• Bose-Einstein condensation in a one-dimensional interacting system due to power-law trapping potentials
• Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

Cited by in corpus (5)

• Optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and nonlinearity
• An explicit and symmetric exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity
• Low regularity full error estimates for the cubic nonlinear Schrödinger equation
• An extended Fourier pseudospectral method for the Gross-Pitaevskii equation with low regularity potential
• A Lawson-time-splitting extended Fourier pseudospectral method for the Gross-Pitaevskii equation with time-dependent low regularity potential