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arXiv · 2203.03500

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

Jean-Baptiste Casteras·Juraj Foldes·Gennady Uraltsev
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Abstract

In this paper, we study the local well-posedness of the cubic Schr\"odinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $\sigma \geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schr\"odinger equation (i.e. $\mathscr{L} = \Delta$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.

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Jean-Baptiste Casteras, Juraj Foldes, Gennady Uraltsev. 2022-03-07. Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators. https://arxiv.org/abs/2203.03500

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