Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schr{ö}dinger equations on $h\mathbb{Z}$
arXiv:1805.024685 citationsh-index: 11
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AI Analysis
Provides rigorous growth estimates for discrete Sobolev norms in a class of nonlinear lattice equations, relevant to numerical analysis and dispersive PDEs.
The paper proves polynomial bounds on the growth of high discrete Sobolev norms for cubic discrete nonlinear Schrödinger equations on a one-dimensional lattice, uniformly with respect to the grid stepsize, using higher modified energies.
We consider the discrete nonlinear Schr{ö}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.