Related papers: Limits of Gaudin Systems: Classical and Quantum Ca…
The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the…
We consider the Vlasov-Poisson-Landau system, a classical model for a dilute collisional plasma interacting through Coulombic collisions and with its self-consistent electrostatic field. We establish global stability and well-posedness near…
We consider non-ultra local linear Poisson algebras on a continuous line . Suitable combinations of representations of these algebras yield representations of novel generalized linear Poisson algebras or "boundary" extensions. They are…
We study the class of nonlinear Klein-Gordon-Maxwell systems describing a standing wave (charged matter field) in equilibrium with a purely electrostatic field. We improve some previous existence results in the case of an homogeneous…
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum $R$-matrices. Here we study the simplest case -- the 11-vertex $R$-matrix and related ${\rm gl}_2$ rational…
We show that a recently introduced generalized scheme of quantum mechanics has connections to Li\'{e}nard and Levinson-Smith classes of nonlinear systems. For the Li\'{e}nard type, which has coefficients of odd and odd symmetry, we…
We review and give detailed description for ${\rm gl}_{NM}$ Gaudin models related to holomorphic vector bundles of rank $NM$ and degree $N$ over elliptic curve with $n$ punctures. Then we introduce their generalizations constructed by means…
In complete analogy with the classical situation (which is briefly reviewed) it is possible to define bi-Hamiltonian descriptions for Quantum systems. We also analyze compatible Hermitian structures in full analogy with compatible Poisson…
Coupling the fields may lead to the emergence of new phenomena. In the realm of classical fields and nonlinear systems, extensive research has been conducted on their solitary and soliton solutions. In the conducted studies, typically two…
The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to be…
Using the point fusion procedure we obtain the new integrable systems from the Elliptic Schlesinger system (ESS). These new systems have the pole orders higher than one in the matrix of the Lax operator. Quadratic Poisson algebras on the…
A new class of 1D discrete nonlinear Schr${\ddot{\rm{o}}}$dinger Hamiltonians with tunable nonlinerities is introduced, which includes the integrable Ablowitz-Ladik system as a limit. A new subset of equations, which are derived from these…
Mixed quantum-classical models have been proposed in several contexts to overcome the computational challenges of fully quantum approaches. However, current models typically suffer from long-standing consistency issues, and, in some cases,…
Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with…
We review one dimensional matrix theory and its variations, collective field theory and quantum phase space description. In the planar limit, these theories become classical and can be easily analyzed. With these descriptions, one…
The finite-size behaviours of the homogeneous sine-Gordon models are analysed in detail, using the thermodynamic Bethe ansatz. Crossovers are observed which allow scales associated with both stable and unstable quantum particles to be…
Without wasting time and effort on philosophical justifications and implications, we write down the conditions for the Hamiltonian of a quantum system for rendering it mathematically equivalent to a deterministic system. These are the…
The use of exactly-solvable Richardson-Gaudin (R-G) models to describe the physics of systems with strong pair correlations is reviewed. We begin with a brief discussion of Richardson's early work, which demonstrated the exact solvability…
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction…
We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture,…