Cancellations of Resonances and Long Time Dynamics of Cubic Schrödinger Equation on \mathbbT

2019/10/15 by Kexin Jin, Xiao Ma, Jin, Kexin +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #math-ph #math.AP #math.MP

paper · pdf · doi:10.48550/arxiv.1910.06875

arxiv created 2019/10/15 · openalex publication_date 2019/10/15 · arxiv updated 2019/10/16 · openalex created_date 2019/10/25 · openalex updated_date 2026/07/28

Abstract

We prove a vanishing property of the normal form transformation of the 1D cubic nonlinear Schrödinger (NLS) equation with periodic boundary conditions on [0,L]. We apply this property to quintic resonance interactions and obtain a description of dynamics for time up to T=(L²)/(ε⁴), if L is sufficiently large and size of initial data ε is small enough. Since T is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of 1D cubic NLS. Our approach can be adapted to other integrable systems without too many difficulties. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively.

Citations

• Nonlinear fractional Schrödinger equations in one dimension
• Normal forms and quadratic nonlinear Klein‐Gordon equations
• Long-term regularity of 3D gravity water waves

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