Discrete second-order Ablowitz-Kaup-Newell-Segur equation and its modified form. (English. Russian original) Zbl 1515.37083
Theor. Math. Phys. 210, No. 3, 304-326 (2022); translation from Teor. Mat. Fiz. 210, No. 3, 350-374 (2022).
Summary: 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.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 39A14 | Partial difference equations |
Keywords:
second-order AKNS-type equations; discrete models; Cauchy matrix approach; continuum limit; solutionReferences:
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