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We consider the discrete nonlinear Schr{\"o}dinger equations on a onedimensional lattice of mesh h, with a cubic focusing or defocusingnonlinearity. We prove a polynomial bound on the growth of the discrete Sobolevnorms, uniformly with respect to the stepsize of the grid. This bound is basedon a construction of higher modified energies.

Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schr{\"o}dinger equations on $h\mathbb{Z}$