The fourth-order Schr\"{o}dinger equation on lattices

Jiawei Cheng

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arXiv · 2403.07445

The fourth-order Schr\"{o}dinger equation on lattices

Jiawei Cheng

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Abstract

In this paper, we study the fourth-order Schr\"{o}dinger equation \begin{equation*} i \partial_t u + {\Delta}^2 u - \gamma \Delta u = \pm |u|^{s-1}u \end{equation*} on the lattice $\mathbb{Z}^d$ with dimensions $d=1,2$ and parameter $\gamma \in \mathbb{R}$. In order to establish sharp dispersive estimates, we consider the fundamental solution as an oscillatory integral and analyze the Newton polyhedron of its phase function. Furthermore, we prove Strichartz estimates which yield the existence of global solutions to nonlinear equations with small data.

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Jiawei Cheng. 2024-03-12. The fourth-order Schr\"{o}dinger equation on lattices. https://arxiv.org/abs/2403.07445

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