455 research outputs found
Separation of variables and the XXZ Gaudin magnet
No full textKalnins, E.G.; Kuznetsov, V.B.; Miller, Jr., W.. (1994). Separation of variables and the XXZ Gaudin magnet. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2719
Backlund transformations for many-body systems related to KdV
No full textWe present Backlund transformations (BTs) with parameter for certain classical integrable n-body systems, namely the many-body generalised Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the fact that all these systems may be obtained as particular reductions (stationary or restricted flows) of the KdV hierarchy; alternatively they may be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide exact time-discretizations of the original (continuous) systems, preserving the Lax matrix and hence all integrals of motion, and satisfy the spectrality property with respect to the Backlund parameter
Quadrics on complex Riemannian spaces of constant curvature, separation of variables and the Gaudin magnet
No full textKalnins, E.G.; Kuznetsov, V.B.; Miller, Jr., Willard. (1993). Quadrics on complex Riemannian spaces of constant curvature, separation of variables and the Gaudin magnet. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/2447
Laurent Polynomials and Superintegrable Maps
No full textThis article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations
Bethe ansatz solutions of the Bose-Hubbard dimer
No full textThe Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates. One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods. Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field. Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The work was funded by the Australian Research Council under Discovery Project DP0557949
Restricted Flows and the Soliton Equation with Self-Consistent Sources
No full textThe KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Bäcklund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions
On Classical r-Matrix for the Kowalevski Gyrostat on so(4)
No full textWe present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).The authors thank E. Sklyanin and N. Reshetikhin for very useful conversations on the subject of this paper. I.V.K. wishes to thank the London Mathematical Society for support his visit to England and V.B. Kuznetsov for hospitality at the University of Leeds
On Classical r-Matrix for the Kowalevski Gyrostat on so(4)
No full textWe present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).The authors thank E. Sklyanin and N. Reshetikhin for very useful conversations on the subject of this paper. I.V.K. wishes to thank the London Mathematical Society for support his visit to England and V.B. Kuznetsov for hospitality at the University of Leeds
Restricted flows and the soliton equation with self-consistent sources
No full textThe KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Bäcklund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The authors are grateful to the referees for the valuable comments. This work is supported by the Chinese Basic Research Project “Nonlinear Science”. R.L. Lin is supported in part by “Scientific Foundation for Returned Overseas Chinese Scholars, Ministry of Education”
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