Dynamics of the Infinite Discrete Nonlinear Schrödinger Equation

The discrete nonlinear Schrödinger equation on \({\mathbb Z}^d\) ${\mathbb{Z}}^d$ , \(d \ge 1\) $d\ge 1$ is an example of a dispersive nonlinear wave system. Being a Hamiltonian system that conserves also the \(\ell ^2({\mathbb Z}^d)\) ${\ell }^2({\mathbb{Z}}^d)$ -norm, the well-posedness of the corresponding Cauchy problem follows for square-summable initial data. In this paper, we prove that the well-posedness continues to hold for initial data that can grow towards infinity, namely anything that has at most a certain power law growth far away from the origin. The growth condition is loose enough to guarantee that, at least in dimension \(d=1\) $d=1$ , initial data sampled from any reasonable equilibrium distribution of the defocusing DNLS satisfies it almost surely.