Time splitting method for nonlinear Schrödinger equation with rough initial data in $L^2$

TL;DR

It is proved that the convergence of the filtered Lie approximation $Z_{flt}$ to the solution $u$ in the mass-subcritical range, $\max\left\{1,\frac{2}{d}\right\} \leq p< \frac{4}{d}$.

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 $L^2$, $$ \left\{ \begin{array}{ll} i\partial_t u +Δu = λ|u|^{p} u, & (x,t) \in \mathbb{R}^d \times \mathbb{R}_+, u (x,0) =φ(x), & x\in\mathbb{R}^d, \end{array} \right. $$ where $λ\in \{-1,1\}$ 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 $φ\in L^2 (\mathbb{R}^d)$, 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 $φ\in L^2 (\mathbb{R}^d)$.

Theorems & Definitions (15)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Remark 1.5
• Theorem 1: Local well-posedness of $u$ and $Z_{flt}$
• Theorem 2: Time-continuous and time-discrete Strichartz estimates
• Lemma 2.1
• Lemma 2.2
• Proposition 3.1