Abstract
By introducing shift relations satisfied by a matrix \(\boldsymbol{r}\), we propose a generalized Cauchy matrix scheme and construct a discrete second-order Ablowitz���Kaup–Newell–Segur equation. A modified form of this equation is given. By applying an appropriate skew continuum limit, we obtain the semi-discrete counterparts of these two discrete equations; in the full continuum limit, we derive continuous nonlinear equations. Solutions, including soliton solutions, Jordan-block solutions, and mixed solutions, of the resulting discrete, semi-discrete, and continuous Ablowitz–Kaup–Newell–Segur-type equations are presented. The reductions to discrete, semi-discrete, and continuous nonlinear Schrödinger equations and modified nonlinear Schrödinger equation are also discussed.
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Notes
The dAKNS equation (3.14) is different from the one introduced in [10]. That equation can be deduced from the shift relations of the variable \( \boldsymbol{S} (a,b)= \,^t {\boldsymbol{c}} (b \boldsymbol{I} + \boldsymbol{K} )^{-1}( \boldsymbol{I} + \boldsymbol{M} )^{-1}(a \boldsymbol{I} + \boldsymbol{K} )^{-1} \boldsymbol{r} \) with \(a,b \in \mathbb{C}\).
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We are very grateful to the referees for the invaluable and expert comments.
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This project is supported by the National Natural Science Foundation of China (No. 12071432).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 210, pp. 350-374 https://doi.org/10.4213/tmf10159.
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Zhang, S., Zhao, SL. & Shi, Y. Discrete second-order Ablowitz–Kaup–Newell–Segur equation and its modified form. Theor Math Phys 210, 304–326 (2022). https://doi.org/10.1134/S0040577922030023
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DOI: https://doi.org/10.1134/S0040577922030023