Diophantine Conditions in Well-Posedness Theory of Coupled KdV-Type Systems: Local Theory
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Abstract
We consider the local well-posedness (LWP) problem of a one-parameter family of coupled Korteweg–de Vries-type systems in both the periodic and nonperiodic settings. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α ≠ 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish a sharp LWP of the system in , where is determined by the Diophantine characterization of certain constants derived from the coupling parameter α. We also present a sharp local (and global) result in . In the Appendix, we briefly discuss the LWP result in for α = 1 without the mean zero assumption, by introducing the vector-valued Xs,b spaces.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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