On the splitting method for the nonlinear Schr\"odinger equation with initial data in $H^1$

TL;DR

This paper proves the convergence of a splitting numerical scheme for the nonlinear Schrödinger equation with initial data in H^1, establishing an order of convergence in L^2 norm for various dimensions and nonlinearities.

Contribution

It provides the first convergence proof of the splitting scheme for initial data in H^1 for the nonlinear Schrödinger equation.

Findings

L^2 convergence of order O(τ^{1/2}) established

Applicable for dimensions d=1,2,3 with specific p ranges

Extends previous results to initial data in H^1 space

Abstract

In this paper, we establish a convergence result for the operator splitting scheme $Z_{\tau}$ introduced by Ignat, with initial data in $H^1$, for the nonlinear Schr\"odinger equation : $$ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) =\phi (x), $$ where $(x,t) \in \mathbb{R}^d \times [0,\infty)$, with $0< p < 4/(d-2)$ for $d\geq3$ and $0< p<\infty$ for $d=1,2$. We prove the $L^2$ convergence of order $\mathcal{O}(\tau^{1/2})$ for this scheme with initial data in the space $H^1 (\mathbb{R}^d)$.