Energy cascade for the Klein-Gordon lattice
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866912103681490944 |
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| author | Pasquali, Stefano |
| author_facet | Pasquali, Stefano |
| contents | We study analytically the dynamics of a $d$-dimensional Klein-Gordon lattice with periodic boundary conditions, for $d \leq 3$. 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 Schrödinger (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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07146 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Energy cascade for the Klein-Gordon lattice Pasquali, Stefano Dynamical Systems Mathematical Physics Analysis of PDEs We study analytically the dynamics of a $d$-dimensional Klein-Gordon lattice with periodic boundary conditions, for $d \leq 3$. 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 Schrödinger (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. |
| title | Energy cascade for the Klein-Gordon lattice |
| topic | Dynamical Systems Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2304.07146 |