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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2605.07669 |
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| _version_ | 1866913102933524480 |
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| author | Wan, Zhiqiang Wu, Wenji Zhang, Heng |
| author_facet | Wan, Zhiqiang Wu, Wenji Zhang, Heng |
| contents | We study the Cauchy problem for the improved Boussinesq equation \[
u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $ω\in\mathbb R^ν$, we prove local existence and uniqueness of classical spatially quasi-periodic solutions with the same frequency vector $ω$ in two Fourier-side classes. First, for exponentially decaying initial Fourier coefficients, we obtain a spatially quasi-periodic solution whose Fourier coefficients remain exponentially decaying on an explicit time interval. Second, for initial Fourier coefficients $c(n)$ and $d(n)$ satisfying the polynomial decay $
|c(n)|+|d(n)|\lesssim (1+|n|)^{-r},
\; r>ν+2, $
we prove that the corresponding spatially quasi-periodic solution preserves the same polynomial decay rate as the initial data. We also extend these results to the nonlinearity $u^p$ with integer $p \geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07669 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Cauchy problem for the improved Boussinesq equation with spatially quasi-periodic initial data Wan, Zhiqiang Wu, Wenji Zhang, Heng Analysis of PDEs We study the Cauchy problem for the improved Boussinesq equation \[ u_{tt}-u_{xx}-u_{xxtt}-(u^2)_{xx}=0 \] on the real line with spatially quasi-periodic initial data. For a non-resonant frequency vector $ω\in\mathbb R^ν$, we prove local existence and uniqueness of classical spatially quasi-periodic solutions with the same frequency vector $ω$ in two Fourier-side classes. First, for exponentially decaying initial Fourier coefficients, we obtain a spatially quasi-periodic solution whose Fourier coefficients remain exponentially decaying on an explicit time interval. Second, for initial Fourier coefficients $c(n)$ and $d(n)$ satisfying the polynomial decay $ |c(n)|+|d(n)|\lesssim (1+|n|)^{-r}, \; r>ν+2, $ we prove that the corresponding spatially quasi-periodic solution preserves the same polynomial decay rate as the initial data. We also extend these results to the nonlinearity $u^p$ with integer $p \geq 3$. |
| title | The Cauchy problem for the improved Boussinesq equation with spatially quasi-periodic initial data |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.07669 |