Abstract
In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp(\({\cal F}{L^p}(\mathbb{T})\)), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp(\({\cal F}{L^p}(\mathbb{T})\)) for \(1 \leq p \leq {3 \over 2}\).
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Acknowledgments
T. Oh was supported by the ERC starting grant (no. 637995 “ProbDynDispEq”). The authors are grateful to the anonymous referee for a helpful comment that has improved the presentation of this paper.
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Oh, T., Wang, Y. Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces. JAMA 143, 723–762 (2021). https://doi.org/10.1007/s11854-021-0168-1
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DOI: https://doi.org/10.1007/s11854-021-0168-1