  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.
