In the paper "[Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$](https://arxiv.org/pdf/math/0301260.pdf)" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning that the NLS
$$i\partial\_t\phi(x,t)+\Delta\phi(x,t)=\vert\phi(x,t)\vert^2\phi(x,t)$$
with initial datum $\phi\_0\in H^s(\mathbb{R}^3)$ is globally-in-time well-posed for $s\geq 1$. 


And the precise argument has been discussed in [this thread.](https://mathoverflow.net/questions/207750/hs-norm-of-a-solution-of-a-nonlinear-schr%C3%B6dinger-equation)


I am wondering whether the discrete NLS is also globally well-posed
$$i\partial\_t u(t,n) = -\Delta\_{\text{disc}} u(t,n) + \lvert{u(t,n)}\rvert^2 u(t,n), \qquad t\in \mathbb{R},\ n\in\mathbb{Z},$$
where $\Delta\_{\text{disc}}u(u) = u(n+1) - 2u(n) + u(n-1)$?



Nonlinear Schrödinger equation with discrete Laplacian
My impression is, that in the discrete case the interesting questions and tools are sometimes different from the continuous case.


Studying the discrete Laplacion also means that one is interested in discrete solution. For example, in one dimension people are often investigating the discrete nonlinear Schrödinger equation
$$i\partial\_t u(t,n) = -\Delta\_{\text{disc}} u(t,n) + \lvert{u(t,n)}\rvert^2 u(t,n), \qquad t\in \mathbb{R},\ n\in\mathbb{Z},$$
where $\Delta\_{\text{disc}}u(u) = u(n+1) - 2u(n) + u(n-1)$. As you have suspected, the global-in-time well-posedness theory is much easier than in the continuous case. For example, in the space $\ell^2(\mathbb{Z})$, the existence and uniqueness of local solutions to the above equations follows from the classical Picard-Lindelöf theorem (regard the equation as an ODE with $C^1$-right-hand side in the Hilbert space $\ell^2(\mathbb{Z})$). The conservation of the $\ell^2$-norm (check $\frac{d}{dt} \lVert u(t)\rVert\_2^2 = 0$) shows that there is no blow-up and hence the solution exists globally. Similar arguments work in weighted $\ell^2$-spaces, see e.g. Lemma 2 in the paper [Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation](https://doi.org/10.1137/080737654 "Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation.") by Kevrekidis, Pelinovsky and Stefanov.


You say that tools such as classical Strichartz estimates are gone in the discrete case. Such questions are also studied in the literature, see for example the paper [Dispersion estimates for one-dimensional discrete Schrödinger and wave equations](https://arxiv.org/abs/1403.7803 "Dispersion Estimates for One-dimensional Discrete Schrödinger and Wave Equations") by Egorova, Kopylova, and Teschl. They prove dispersive estimates (and deduce Strichartz estimates) in the discrete setting.


