Unifying scheme for generating discrete integrable systems including inhomogeneous and hybrid models. (English) Zbl 1062.81084
Summary: A unifying scheme based on an ancestor model is proposed for generating a wide range of integrable discrete and continuum as well as inhomogeneous and hybrid models. They include in particular discrete versions of sine-Gordon, Landau-Lifshitz, nonlinear Schrödinger (NLS), derivative NLS equations, Liouville model, (non-)relativistic Toda chain, Ablowitz–Ladik model, etc. Our scheme introduces the possibility of building a novel class of integrable hybrid systems including multicomponent models like massive thirring, discrete self-trapping, two-mode derivative NLS by combining different descendant models. We also construct inhomogeneous systems like Gaudin model including new ones like variable mass sine-Gordon, variable coefficient NLS, Ablowitz-Ladik, Toda chains, etc. keeping their flows isospectral, as opposed to the standard approach. All our models are generated from the same ancestor Lax operator (or its \(q \to 1\) limit) and satisfy the classical Yang-Baxter equation sharing the same \(r\)-matrix. This reveals an inherent universality in these diverse systems, which become explicit at their action-angle level.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A12 | Discrete version of topics in analysis |
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