Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps
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arXiv
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2025
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| author | Caudrelier, Vincent Ossi, Nicholas J. Prinari, Barbara |
| author_facet | Caudrelier, Vincent Ossi, Nicholas J. Prinari, Barbara |
| contents | The goal of this work is to obtain a complete characterization of soliton and breather interactions in the integrable discrete Manakov (IDM) system, a vector generalization of the Ablowitz-Ladik model. The IDM system, which in the continuous limit reduces to the Manakov system (i.e., a 2-component vector nonlinear Schrodinger equation), was shown to admit a variety of discrete vector soliton solutions: fundamental solitons, fundamental breathers, and composite breathers. While the interaction of fundamental solitons was studied early on, no results are presently available for other types of soliton-breather and breather-breather interactions. Our study reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather. Conversely, the interaction of two fundamental breathers generically yields two fundamental breathers with polarization shifts, but may also result in a fundamental soliton and a fundamental breather. Composite breathers interact trivially both with each other and with a fundamental soliton or breather. Explicit formulas for the scattering coefficients that characterize fundamental and composite breathers are given. This allows us to interpret the interactions in terms of a refactorization problem and derive the associated Yang-Baxter maps describing the effect of interactions on the polarizations. These give the first examples of parametric Yang-Baxter maps of trigonometric type. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13228 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps Caudrelier, Vincent Ossi, Nicholas J. Prinari, Barbara Exactly Solvable and Integrable Systems Mathematical Physics Pattern Formation and Solitons The goal of this work is to obtain a complete characterization of soliton and breather interactions in the integrable discrete Manakov (IDM) system, a vector generalization of the Ablowitz-Ladik model. The IDM system, which in the continuous limit reduces to the Manakov system (i.e., a 2-component vector nonlinear Schrodinger equation), was shown to admit a variety of discrete vector soliton solutions: fundamental solitons, fundamental breathers, and composite breathers. While the interaction of fundamental solitons was studied early on, no results are presently available for other types of soliton-breather and breather-breather interactions. Our study reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather. Conversely, the interaction of two fundamental breathers generically yields two fundamental breathers with polarization shifts, but may also result in a fundamental soliton and a fundamental breather. Composite breathers interact trivially both with each other and with a fundamental soliton or breather. Explicit formulas for the scattering coefficients that characterize fundamental and composite breathers are given. This allows us to interpret the interactions in terms of a refactorization problem and derive the associated Yang-Baxter maps describing the effect of interactions on the polarizations. These give the first examples of parametric Yang-Baxter maps of trigonometric type. |
| title | Breather interactions in the integrable discrete Manakov system and trigonometric Yang-Baxter maps |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2503.13228 |