A generalization of the beam problem: Connection to multi-component Camassa-Holm dynamics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918143425773568 |
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| author | Beals, Richard Szmigielski, Jacek |
| author_facet | Beals, Richard Szmigielski, Jacek |
| contents | We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete densities, we prove that the associated matrix Weyl function admits a Stieltjes-type continued fraction expansion with Clifford-valued coefficients. By mapping the problem from the finite interval to the real line, we uncover a direct link to a multi-component generalization of the Camassa-Holm equation. This yields a vectorized form of the Camassa-Holm equation invariant under arbitrary orthogonal group actions. As an illustration, we examine the dynamics of a two-atom (two-peakon) matrix measure in the special case of a Clifford algebra with two generators and Minkowski signature. Our analysis shows that, even when peakon waves remain spatially separated, they can engage in long-range, synchronized energy exchange. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14639 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalization of the beam problem: Connection to multi-component Camassa-Holm dynamics Beals, Richard Szmigielski, Jacek Exactly Solvable and Integrable Systems Mathematical Physics We extend the Euler-Bernoulli beam problem, formulated as a matrix string equation with a matrix-valued density, to a setting where the density takes values in a Clifford algebra, and we analyze its isospectral deformations. For discrete densities, we prove that the associated matrix Weyl function admits a Stieltjes-type continued fraction expansion with Clifford-valued coefficients. By mapping the problem from the finite interval to the real line, we uncover a direct link to a multi-component generalization of the Camassa-Holm equation. This yields a vectorized form of the Camassa-Holm equation invariant under arbitrary orthogonal group actions. As an illustration, we examine the dynamics of a two-atom (two-peakon) matrix measure in the special case of a Clifford algebra with two generators and Minkowski signature. Our analysis shows that, even when peakon waves remain spatially separated, they can engage in long-range, synchronized energy exchange. |
| title | A generalization of the beam problem: Connection to multi-component Camassa-Holm dynamics |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2509.14639 |