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arXiv · 2409.10006

The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data

Fei Xu
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Abstract

In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schrödinger equation in higher dimensions will asymptotically approach the associated linear solution within a specific time scale. The proof is based on a combinatorial analysis method present through diagrams. Our results and methods apply to {\em arbitrary} space dimensions and general power-law nonlinearities of the form $\pm|u|^{2p}u$, where $1\leq p\in\mathbb N$.

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Fei Xu. 2024-09-25. The Weakly Nonlinear Schrödinger Equation in Higher Dimensions with Quasi-periodic Initial Data. https://arxiv.org/abs/2409.10006

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