Longons from the nonlinear dispersion of Galerkin regularization
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915085874626560 |
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| author | Zhu, Jian-Zhou |
| author_facet | Zhu, Jian-Zhou |
| contents | Irregular compactons and peakons from some nonlinear dispersions can be regularized by another type of nonlinear dispersion, defined by a pseudo-differential operator in physical space for the Galerkin truncation preserving finite Fourier modes of wavenumbers no larger than $K$. This resembles yet differs from the Korteweg-de Vries (KdV) regularization of the Burgers-Hopf (BH) equation. The Galerkin-regularized compacton, peakon, KdV, and BH dynamics exhibit novel traveling waves and interacting solitonic structures amidst weaker, less-ordered components (`longons'). Quasi-periodic solutions are also constructed with on-torus invariants, towards a potential Kolmogorov-Arnold-Moser (KAM) argument with presumably whiskered tori. The latter are persistent against the approximation of the truncation by a type of linear dispersion models, resulting in similar longulent states. Time-dependent and stationary behaviors in the large-$K$ limit are addressed with numerical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Longons from the nonlinear dispersion of Galerkin regularization Zhu, Jian-Zhou Chaotic Dynamics Irregular compactons and peakons from some nonlinear dispersions can be regularized by another type of nonlinear dispersion, defined by a pseudo-differential operator in physical space for the Galerkin truncation preserving finite Fourier modes of wavenumbers no larger than $K$. This resembles yet differs from the Korteweg-de Vries (KdV) regularization of the Burgers-Hopf (BH) equation. The Galerkin-regularized compacton, peakon, KdV, and BH dynamics exhibit novel traveling waves and interacting solitonic structures amidst weaker, less-ordered components (`longons'). Quasi-periodic solutions are also constructed with on-torus invariants, towards a potential Kolmogorov-Arnold-Moser (KAM) argument with presumably whiskered tori. The latter are persistent against the approximation of the truncation by a type of linear dispersion models, resulting in similar longulent states. Time-dependent and stationary behaviors in the large-$K$ limit are addressed with numerical results. |
| title | Longons from the nonlinear dispersion of Galerkin regularization |
| topic | Chaotic Dynamics |
| url | https://arxiv.org/abs/2404.08583 |