For a given g ⊗ g-valued non-skew-symmetric non-dynamical classical r-matrices r(u, v) with spectral parameters, we construct the general form of ⊗g valued Lax matrices of finite-dimensional integrable systems satisfying linear r-matrix algebra. We show that the reduction in the corresponding finite-dimensional integrable systems is connected with "the special points" of the classical r-matrices in which they become degenerated. We also propose a systematic way of the construction of additional integrals of the Lax-integrable systems associated with the symmetries of the corresponding r-matrices. We consider examples of the Lax matrices and integrable systems that are obtained in the framework of the general scheme. Among them there are such physically important systems as generalized Gaudin systems in an external magnetic field, ultimate integrable generalization of Toda-type chains (including "modified" or "deformed" Toda chains), generalized integrable Jaynes-Cummings-Dicke models, integrable boson models generalizing Bose-Hubbard dimer models, etc.

Skrypnyk, T. (2016). Reductions in finite-dimensional integrable systems and special points of classical r-matrices. JOURNAL OF MATHEMATICAL PHYSICS, 57(12), 123504 [10.1063/1.4972021].

Reductions in finite-dimensional integrable systems and special points of classical r-matrices

Skrypnyk, T
2016

Abstract

For a given g ⊗ g-valued non-skew-symmetric non-dynamical classical r-matrices r(u, v) with spectral parameters, we construct the general form of ⊗g valued Lax matrices of finite-dimensional integrable systems satisfying linear r-matrix algebra. We show that the reduction in the corresponding finite-dimensional integrable systems is connected with "the special points" of the classical r-matrices in which they become degenerated. We also propose a systematic way of the construction of additional integrals of the Lax-integrable systems associated with the symmetries of the corresponding r-matrices. We consider examples of the Lax matrices and integrable systems that are obtained in the framework of the general scheme. Among them there are such physically important systems as generalized Gaudin systems in an external magnetic field, ultimate integrable generalization of Toda-type chains (including "modified" or "deformed" Toda chains), generalized integrable Jaynes-Cummings-Dicke models, integrable boson models generalizing Bose-Hubbard dimer models, etc.
Articolo in rivista - Articolo scientifico
Statistical and Nonlinear Physics; Mathematical Physics
English
2016
57
12
123504
123504
none
Skrypnyk, T. (2016). Reductions in finite-dimensional integrable systems and special points of classical r-matrices. JOURNAL OF MATHEMATICAL PHYSICS, 57(12), 123504 [10.1063/1.4972021].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/189884
Citazioni
  • Scopus 9
  • ???jsp.display-item.citation.isi??? 9
Social impact