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

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- by Erwan Faou, Pierre Germain and Zaher Hani
- J. Amer. Math. Soc.
**29**(2016), 915-982 - DOI: https://doi.org/10.1090/jams/845
- Published electronically: October 20, 2015
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## 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 $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\mathbb {R}^2$ 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.## References

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## Bibliographic Information

**Erwan Faou**- Affiliation: INRIA & ENS Cachan Bretagne, Campus de Ker Lann, Avenue Robert Schumann, 35170 Bruz, France
- MR Author ID: 656335
- Email: Erwan.Faou@inria.fr
**Pierre Germain**- Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185
- MR Author ID: 758713
- Email: pgermain@cims.nyu.edu
**Zaher Hani**- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 984928
- Email: hani@math.gatech.edu
- Received by editor(s): March 10, 2014
- Received by editor(s) in revised form: July 21, 2015
- Published electronically: October 20, 2015
- Additional Notes: The first author was supported by the ERC Starting Grant project GEOPARDI

The second author was partially supported by NSF Grant DMS-1101269, a start-up grant from the Courant Institute, and a Sloan fellowship.

The third author was supported by a Simons Postdoctoral Fellowship and NSF Grant DMS-1301647. - © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 915-982 - MSC (2010): Primary 35Q55, 37K05
- DOI: https://doi.org/10.1090/jams/845
- MathSciNet review: 3522607