Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

T. Buckmaster; P. Germain; Z. Hani; J. Shatah

doi:10.1007/s00222-021-01039-z

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

T. Buckmaster
, P. Germain
, Z. Hani
, J. Shatah

Mathematics

Research output: Contribution to journal › Article › peer-review

41 Scopus citations

Abstract

Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

Original language	English (US)
Pages (from-to)	787-855
Number of pages	69
Journal	Inventiones Mathematicae
Volume	225
Issue number	3
DOIs	https://doi.org/10.1007/s00222-021-01039-z
State	Published - Sep 2021

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00222-021-01039-z

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title = "Onset of the wave turbulence description of the longtime behavior of the nonlinear Schr{\"o}dinger equation",

abstract = "Consider the cubic nonlinear Schr{\"o}dinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.",

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T1 - Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

AU - Buckmaster, T.

AU - Germain, P.

AU - Hani, Z.

AU - Shatah, J.

PY - 2021/9

Y1 - 2021/9

N2 - Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

AB - Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

UR - https://www.scopus.com/pages/publications/85103045890

UR - https://www.scopus.com/pages/publications/85103045890#tab=citedBy

U2 - 10.1007/s00222-021-01039-z

DO - 10.1007/s00222-021-01039-z

M3 - Article

AN - SCOPUS:85103045890

SN - 0020-9910

VL - 225

SP - 787

EP - 855

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -

Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation

Abstract

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