Numerical integrators for continuous disordered nonlinear Schrödinger equation

Abstract

In this paper, we consider the numerical solution of the continuous disordered nonlinear Schrödinger equation, which contains a spatial random potential. We address the finite time accuracy order reduction issue of the usual numerical integrators on this problem, which is due to the presence of the random/rough potential. By using the recently proposed low-regularity integrator (LRI) from (33, SIAM J. Numer. Anal., 2019), we show how to integrate the potential term by losing two spatial derivatives. Convergence analysis is done to show that LRI has the second order accuracy in $L^2$-norm for potentials in $H^2$. Numerical experiments are done to verify this theoretical result. More numerical results are presented to investigate the accuracy of LRI compared with classical methods under rougher random potentials from applications.

arXiv comment23 pages, 11 figures

References in corpus (16)

• Mathematical theory and numerical methods for Bose-Einstein condensation
• Destruction of Anderson localization by a weak nonlinearity
• Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations
• Universal spreading of wavepackets in disordered nonlinear systems
• High order symplectic integrators for perturbed Hamiltonian systems
• Anderson Localization of Expanding Bose-Einstein Condensates in Random Potentials
• Delocalization of wave packets in disordered nonlinear chains
• A comprehensive proof of localization for continuous Anderson models with singular random potentials
• Strong Convergence Rate of Splitting Schemes for Stochastic Nonlinear Schrödinger Equations
• Shock waves in disordered media
• The continuous Anderson hamiltonian in dimension two
• Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity
• Localization of a matter wave packet in a disordered potential
• Resonance based schemes for dispersive equations via decorated trees
• Solitonization of the Anderson Localization
• Effective noise theory for the Nonlinear Schrödinger Equation with disorder