Geodesic
Discrete second-order Ablowitz--Kaup--Newell--Segur equation and its modified form
Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 3, pp. 350-374

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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.
Keywords: second-order AKNS-type equations, discrete models, Cauchy matrix approach, continuum limit
Mots-clés : solution. 
@article{TMF_2022_210_3_a1,
     author = {Shuai Zhang and Song-Lin Zhao and Ying Shi},
     title = {Discrete second-order {Ablowitz--Kaup--Newell--Segur} equation and its modified form},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {350--374},
     publisher = {mathdoc},
     volume = {210},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a1/}
}
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Shuai Zhang; Song-Lin Zhao; Ying Shi. Discrete second-order Ablowitz--Kaup--Newell--Segur equation and its modified form. Teoretičeskaâ i matematičeskaâ fizika, Tome 210 (2022) no. 3, pp. 350-374. http://geodesic.mathdoc.fr/item/TMF_2022_210_3_a1/