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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.24222 |
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| _version_ | 1866917972543537152 |
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| author | de Suzzoni, Anne-Sophie Stingo, Annalaura Touati, Arthur |
| author_facet | de Suzzoni, Anne-Sophie Stingo, Annalaura Touati, Arthur |
| contents | In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We rigorously derive the effective dynamics for the correlations associated to the solution, in the limit where $L\to\infty$ and $\varepsilon\to 0$ according to some power law. The main novelty of our work is that, due to the absence of invariances, trivial resonances always take precedence over quasi-resonances. The derivation of the nonlinear effective dynamics is justified up time to $δT$ , where $T =\varepsilon^{-2}$ is the appropriate timescale and $δ$ is independent of $L$ and $\varepsilon$. We use Feynmann interaction diagrams, here adapted to a normal form reduction and to the coupled nature of our real-valued system. We also introduce a frequency decomposition at the level of the diagrammatic and develop a new combinatorial tool which allows us to work with the Klein-Gordon dispersion relation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_24222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wave turbulence for a semilinear Klein-Gordon system de Suzzoni, Anne-Sophie Stingo, Annalaura Touati, Arthur Analysis of PDEs In this article we consider a system of two Klein-Gordon equations, set on the $d$-dimensional box of size $L$, coupled through quadratic semilinear terms of strength $\varepsilon$ and evolving from well-prepared random initial data. We rigorously derive the effective dynamics for the correlations associated to the solution, in the limit where $L\to\infty$ and $\varepsilon\to 0$ according to some power law. The main novelty of our work is that, due to the absence of invariances, trivial resonances always take precedence over quasi-resonances. The derivation of the nonlinear effective dynamics is justified up time to $δT$ , where $T =\varepsilon^{-2}$ is the appropriate timescale and $δ$ is independent of $L$ and $\varepsilon$. We use Feynmann interaction diagrams, here adapted to a normal form reduction and to the coupled nature of our real-valued system. We also introduce a frequency decomposition at the level of the diagrammatic and develop a new combinatorial tool which allows us to work with the Klein-Gordon dispersion relation. |
| title | Wave turbulence for a semilinear Klein-Gordon system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.24222 |