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

Faou, Erwan¹ ; Germain, Pierre² ; Hani, Zaher³

¹ INRIA & ENS Cachan Bretagne, Campus de Ker Lann, Avenue Robert Schumann, 35170 Bruz, France
² Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185
³ School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Journal of the American Mathematical Society, Tome 29 (2016) no. 4, pp. 915-982

Résumé

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.

Détail

DOI : 10.1090/jams/845

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