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


Authors: Erwan Faou, Pierre Germain and Zaher Hani
Journal: J. Amer. Math. Soc. 29 (2016), 915-982
MSC (2010): Primary 35Q55, 37K05
DOI: https://doi.org/10.1090/jams/845
Published electronically: October 20, 2015
MathSciNet review: 3522607
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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.


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Additional Information

Erwan Faou
Affiliation: INRIA & ENS Cachan Bretagne, Campus de Ker Lann, Avenue Robert Schumann, 35170 Bruz, France
Email: Erwan.Faou@inria.fr

Pierre Germain
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185
Email: pgermain@cims.nyu.edu

Zaher Hani
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: hani@math.gatech.edu

DOI: https://doi.org/10.1090/jams/845
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.
Article copyright: © Copyright 2015 American Mathematical Society