Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map
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| Format: | Preprint |
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2015
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| _version_ | 1866911788801458176 |
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| author | Tsuchida, Takayuki |
| author_facet | Tsuchida, Takayuki |
| contents | The action of a Bäcklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Bäcklund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrödinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Bäcklund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1505_07924 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map Tsuchida, Takayuki Exactly Solvable and Integrable Systems Mathematical Physics Pattern Formation and Solitons The action of a Bäcklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Bäcklund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrödinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Bäcklund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices. |
| title | Integrable discretization of the vector/matrix nonlinear Schrödinger equation and the associated Yang-Baxter map |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Pattern Formation and Solitons |
| url | https://arxiv.org/abs/1505.07924 |