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On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $

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  • In this paper, we establish a convergence result for the operator splitting scheme $ Z_{\tau} $ introduced by Ignat [12], with initial data in $ H^1 $, for the nonlinear Schrödinger equation:

    $ \partial_t u = i \Delta u + i\lambda |u|^{p} u,\qquad u (x,0) = \phi (x), $

    where $ p >0 $, $ \lambda \in \{-1,1\} $ and $ (x,t) \in \mathbb{R}^d \times [0,\infty) $. We prove the $ L^2 $ convergence of order $ \mathcal{O}(\tau^{1/2}) $ for the scheme with initial data in the space $ H^1 (\mathbb{R}^d) $ for the energy-subcritical range of $ p $.

    Mathematics Subject Classification: Primary:35Q55, 65M15.


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