On the Existence and Uniqueness of Global Solutions for the KdV Equation\n with Quasi-Periodic Initial Data
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Bibliographic record
Abstract
We consider the KdV equation $$ \\partial_t u +\\partial^3_x u +u\\partial_x u=0\n$$ with quasi-periodic initial data whose Fourier coefficients decay\nexponentially and prove existence and uniqueness, in the class of functions\nwhich have an expansion with exponentially decaying Fourier coefficients, of a\nsolution on a small interval of time, the length of which depends on the given\ndata and the frequency vector involved. For a Diophantine frequency vector and\nfor small quasi-periodic data (i.e., when the Fourier coefficients obey $|c(m)|\n\\le \\varepsilon \\exp(-\\kappa_0 |m|)$ with $\\varepsilon > 0$ sufficiently small,\ndepending on $\\kappa_0 > 0$ and the frequency vector), we prove global\nexistence and uniqueness of the solution. The latter result relies on our\nrecent work \\cite{DG} on the inverse spectral problem for the quasi-periodic\nSchr\\"{o}dinger equation.\n
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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