Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies
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| Format: | Preprint |
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2025
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| author | Aratyn, H. Constantinidis, C. P. Gomes, J. F. Santiago, T. C. Zimerman, A. H. |
| author_facet | Aratyn, H. Constantinidis, C. P. Gomes, J. F. Santiago, T. C. Zimerman, A. H. |
| contents | We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions.
Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations.
Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models).
For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_10744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies Aratyn, H. Constantinidis, C. P. Gomes, J. F. Santiago, T. C. Zimerman, A. H. Exactly Solvable and Integrable Systems Mathematical Physics We propose a generalized Riemann-Hilbert-Birkhoff decomposition that expands the standard integrable hierarchy formalism in two fundamental ways: it allows for integer powers of Lax matrix components in the flow equations to be increased as compared to conventional models, and it incorporates constant non-zero vacuum (background) solutions. Two additional parameters control these features. The first one defines the grade of a semisimple element that underpins the algebraic construction of the hierarchy, where a grade-one semi-simple element recovers known hierarchies such as mKdV and AKNS. The second parameter distinguishes between zero and non-zero constant background (vacuum) configurations. Additionally, we introduce a third parameter associated with an ambiguity in the definition of the grade-zero component of the dressing matrices. While not affecting the decomposition itself, this parameter classifies different gauge realizations of the integrable equations (like for example, Kaup-Newell, Gerdjikov-Ivanov, Chen-Lee-Liu models). For various values of these parameters, we construct and analyze corresponding integrable models in a unified universal manner demonstrating the broad applicability and generative power of the extended formalism. |
| title | Generalized Riemann-Hilbert-Birkhoff Decomposition and a New Class of Higher Grading Integrable Hierarchies |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2507.10744 |