Energy cascade for the Klein-Gordon lattice
Abstract
We study analytically the dynamics of a -dimensional Klein-Gordon lattice with periodic boundary conditions, for . We consider initial data supported on one low-frequency Fourier mode. We show that, in the continuous approximation, the resonant normal form of the system is given by a small-dispersion nonlinear Schr\"odinger (NLS) equation. By exploiting a result about the growth of Sobolev norms for solutions of small-dispersion NLS, we are able to describe an energy cascade phenomenon for the Klein-Gordon lattice, where part of the energy is transferred to modes associated to higher frequencies. Such phenomenon holds within the time-scale for which we can ensure the validity of the continuous approximation.
Cite
@article{arxiv.2304.07146,
title = {Energy cascade for the Klein-Gordon lattice},
author = {Stefano Pasquali},
journal= {arXiv preprint arXiv:2304.07146},
year = {2024}
}
Comments
Changes with respect to previous version: changes to the presentation of the main result in Section 2 and in Section 6, changes to Section 3 and Appendix A, some comments added, some typos corrected. arXiv admin note: text overlap with arXiv:1911.12648