Optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

Abstract

We establish optimal error bounds on time-splitting methods for the nonlinear Schrödinger equation with low regularity potential and typical power-type nonlinearity [Formula: see text], where [Formula: see text] is the density with [Formula: see text] the wave function and [Formula: see text] the exponent of the nonlinearity. For the first-order Lie–Trotter time-splitting method, optimal [Formula: see text]-norm error bound is proved for [Formula: see text]-potential and [Formula: see text], and optimal [Formula: see text]-norm error bound is obtained for [Formula: see text]-potential and [Formula: see text]. For the second-order Strang time-splitting method, optimal [Formula: see text]-norm error bound is established for [Formula: see text]-potential and [Formula: see text], and optimal [Formula: see text]-norm error bound is proved for [Formula: see text]-potential and [Formula: see text] (or [Formula: see text]). Compared to those error estimates of time-splitting methods in the literature, our optimal error bounds either improve the convergence rates under the same regularity assumptions or significantly relax the regularity requirements on potential and nonlinearity for optimal convergence orders. A key ingredient in our proof is to adopt a new technique called regularity compensation oscillation (RCO), where low frequency modes are analyzed by phase cancellation, and high frequency modes are estimated by regularity of the solution. Extensive numerical results are reported to confirm our error estimates and to demonstrate that they are sharp.