# The weakly nonlinear large box limit of the 2D cubic nonlinear Schrödinger equation

Erwan Faou, Pierre Germain, Zaher Hani

Research output: Contribution to journalArticlepeer-review

## Abstract

We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ 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. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under 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 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 Undefined 1308.6267 arXiv Published - Aug 28 2013

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