Time splitting method for nonlinear Schrödinger equation with rough initial data in L2

Abstract

We establish convergence results related to the operator splitting scheme on the Cauchy problem for the nonlinear Schrödinger equation with rough initial data in $L^2$,$\{\begin{array}{cc}i{\partial }_{t}u+\Delta u=\lambda |u{|}^{p}u,& (x,t)\in R^d\times R_{+},\\ u(x,0)=\varphi \left(x\right),& x\in R^d,\end{array}$ where $\lambda \in \{-1,1\}$ and $p>0$. While the Lie approximation $Z_L$ is known to converge to the solution u when the initial datum ϕ is sufficiently smooth, the convergence result for rough initial data is open to question. In this paper, for rough initial data $\varphi \in L^2\left(R^d\right)$, we prove the $L^2$ convergence of the filtered Lie approximation $Z_{flt}$ to the solution u in the mass-subcritical range, $0<p<\frac{4}{d}$. Furthermore, we provide a precise convergence result for radial initial data $\varphi \in L^2\left(R^d\right)$.