Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Published electronically: October 20, 2015
MathSciNet review: 3522607
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 35Q55, 37K05

Retrieve articles in all journals with MSC (2010): 35Q55, 37K05

Additional Information

Erwan Faou
Affiliation: INRIA & ENS Cachan Bretagne, Campus de Ker Lann, Avenue Robert Schumann, 35170 Bruz, France
MR Author ID: 656335

Pierre Germain
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185
MR Author ID: 758713

Zaher Hani
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
MR Author ID: 984928

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