Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS

2011/03/28 by Zihua Guo, Guo, Zihua, Soonsik Kwon +3 · 6 citations
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #math.AP #math.DS #msc:35Q55

paper · pdf · doi:10.48550/arxiv.1103.5271

28 pages. In Section 3, we now use (ordered) trees for indexing multilinear terms appearing in the process (instead of assuming that the time derivative falls on the first factor as in the previous version.)

openalex publication_date 2011/03/28 · arxiv created 2011/09/06 · arxiv updated 2011/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L²(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L²(T) with initial data in L²(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s ≥ 1/6.

Citations

• Normal forms and quadratic nonlinear Klein‐Gordon equations

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• Nonsqueezing property of the coupled KdV type system without Miura transform
• Birkhoff normal form via decorated trees
• Nonlinear PDEs with modulated dispersion IV: normal form approach and unconditional uniqueness
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