### 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 R^{2} 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 language | English (US) |
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Pages (from-to) | 915-982 |

Number of pages | 68 |

Journal | Journal of the American Mathematical Society |

Volume | 29 |

Issue number | 4 |

DOIs | |

State | Published - Oct 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the American Mathematical Society*,

*29*(4), 915-982. https://doi.org/10.1090/jams/845