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Cancellations of resonances and long time dynamics of cubic Schrödinger equation on \(\mathbb{T}\). (English) Zbl 1462.35355

Summary: 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 \(T = \frac{L^2}{\epsilon^4}\), if \(L\) is sufficiently large and size of initial data \(\epsilon\) 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q53 KdV equations (Korteweg-de Vries equations)
35A35 Theoretical approximation in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B34 Resonance in context of PDEs
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