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We consider the discrete nonlinear Schr{\"o}dinger equations on a one dimensional lattice of mesh h, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.

discrete and continuous dynamical systems

Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ h\mathbb{Z} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;