The weakly nonlinear large-box limit of the 2D cubic nonlinear schrÖdinger equation

Erwan Faou, Pierre Germain, Zaher Hani

Research output: Contribution to journalArticle

Abstract

We consider the cubic nonlinear Schrödinger (NLS) equation set on a two-dimensional box of size L with periodic boundary conditions. By taking the large-box limit L→∞ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on R2 that approximates the dynamics of the frequency modes. The large-box limit and the weak nonlinearity limit are also performed in weak (or wave) turbulence theory, to which this work is related. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under the Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows one to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

Original languageEnglish (US)
Pages (from-to)915-982
Number of pages68
JournalJournal of the American Mathematical Society
Volume29
Issue number4
DOIs
StatePublished - Oct 1 2016

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Cubic equation
Nonlinear equations
Nonlinear Equations
Hamiltonians
Invariance
Fourier transforms
Turbulence
Periodic Boundary Conditions
Explicit Solution
Boundary conditions
Fourier transform
Nonlinearity
Symmetry
Approximation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The weakly nonlinear large-box limit of the 2D cubic nonlinear schrÖdinger equation. / Faou, Erwan; Germain, Pierre; Hani, Zaher.

In: Journal of the American Mathematical Society, Vol. 29, No. 4, 01.10.2016, p. 915-982.

Research output: Contribution to journalArticle

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