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

TL;DR

This paper establishes polynomial bounds on the growth of discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on a one-dimensional lattice, uniformly across different grid sizes, using higher modified energies.

Contribution

It introduces a novel method to uniformly bound the growth of Sobolev norms in discrete NLS equations via higher modified energies.

Findings

Polynomial bounds on Sobolev norm growth are achieved.

Bounds are uniform with respect to grid stepsize h.

The approach applies to both focusing and defocusing nonlinearities.

Abstract

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.